TIDES
I. On the Nature of Tides.
§ 1. Definition of Tide.
WHEN, as occasionally happens, a ship in the open sea meets a short succession of waves of very unusua! magnitude, we hear of tidal waves; and the large wave caused by an earthquake is commonly so described. The use of the term “tide” in this connexion is certainly incorrect, but it has perhaps been fostered by the fact that such waves impress their records on automatic tide-gauges, as, for example, when the wave due to the volcanic outbreak at Krakatoa was thus distinctly traceable in South Africa, and perhaps even faintly at Brest. We can only adequately define a tide by reference to the cause which produces it. A tide then is a rise and fall of the water of the sea produced by the attraction of the sun and moon. A rise and fall of the sea produced by a regular alternation of day and night breezes, by regular rainfall and evaporation, or by any influence which the moon may have on the weather cannot strictly be called a tide. Such alternations may, it is true, be inextricably involved with the rise and fall of the true astronomical tide, but we shall here distinguish them as meteorological· tides. These movements are the result of the action of the sun, as a radiating body, on the earth. Tides in the atmosphere would be shown by a regular rise and fall in the barometer, but such tides are undoubtedly very minute, and we shall not discuss them in this article, merely referring the reader to the Mécanique Celeste of Laplace, bks. i. and xiii. There are, however, very strongly marked diurnal and semi-diurnal inequalities of the barometer due to atmospheric meteorological tides. Sir William Thomson in an interesting speculation shows that the interaction of these quasi-tides with the sun is that of a thermodynamic engine, whereby there is caused a minute secular acceleration of the earth’s rotation. This matter is, however, beyond the scope of the present article. We shall here extend the term “tide” to denote an elastic or viscous periodic deformation of a solid or viscous globe under the action of tide-generating forces. In the technical part of the article by the term “a simple tide” we shall denote a spherical harmonic deformation of the water on the surface of the globe, or of the solid globe itself, multiplied by a simple harmonic function of the time.
§ 2. General Description of Tidal Phenomena.
If we live by the sea or on an estuary, we see that the water rises and falls nearly twice a day; speaking more exactly, the average interval from high water to high water is about 12 h 25 m , so that the average retardation from day to day is about 50 m . The times of high water are then found to bear an intimate relation with the moon’s position. Thus at Ipswich high water occurs when the moon is nearly south, at London Bridge when it is southwest, and at Bristol when it is east-south-east. For a very rough determination of the time of high water it is sufficient to add the solar time of high water on the days of new and full moon (called the “establishment of the port”) to the time of the moon’s passage over the meridian, either visibly above or invisibly below the horizon. The interval between the moon’s passage over the meridian and high water varies sensibly with the moon’s age. From new moon to first quarter, and from full moon to third quarter (or rather from and to a day later than each of these phases), the interval diminishes from its average to a mini mum, and then increases again to the average; and in the other two quarters it increases from the average to a maximum, and then diminishes again to the average.
The range of the rise and fall of water is also subject to great variability. On the days after new and full moon the range of tide is at its maximum, and on the day after the first and third quarter at its minimum. The maximum is called “spring tide” and the minimum “neap tide,” and the range of spring tide is usually between two and three times as great as that of neap tide. At many ports, however, especially non-European ones, two successive high waters are of unequal heights, and the law of variability of the difference is somewhat complex; a statement of that law will be easier when we come to consider tidal theories. In considering any tide we find, especially in estuaries, that the interval from high to low water is longer than that from low to high water, and the difference between the intervals is greater at spring than at neap.
In a river the current continues to run up stream for some considerable time after high water is attained and to run down similarly after low water. Much confusion has been occasioned by the indiscriminate use of the term “tide” to denote a tidal current and a rise of water, and it has often been incorrectly inferred that high water must have been attained at the moment of cessation of the upward current. The distinction between “rising and falling” and “flowing and ebbing” must be carefully maintained in rivers, whilst it vanishes at the seaboard. If we examine the progress of the tide-wave up a river, we find that high water occurs at the sea earlier than higher up. If, for instance, on a certain day it is high water at Margate at noon, it is high water at Gravesend at a quarter past two, and at London Bridge a few minutes before three. The interval from low to high water diminishes also as we go up the river; and at some distance up certain rivers—as, for example, the Severn—the rising water spreads over the flat sands in a roaring surf and travels up the rivpr almost like a wall of water. This kind of sudden rise is called a “bore.” In other cases where the difference between the periods of rising and falling is considerable, there are, in each high water, two or three rises and falls. A double high water exists at Southampton.
When an estuary contracts considerably, the range of tide becomes largely magnified as it narrows; for example, at the entrance of the Bristol Channel the range of spring tides is about 18 feet, and at Chepstow about 50 feet. This augmentation of the height of the tide-wave is due to the concentration of the energy of motion of a large mass of water into a narrow space. At oceanic ports the tidal phenomena are much less marked, the range of tide being usually only 2 or 3 feet, and the interval from high to low water sensibly equal to that from low to high water. The changes from spring to neap tide and the relation of the time of high water to the moon’s transit remain, however, the same as in the case of the river tides.
In long and narrow seas, such as the English Channel, the tide in mid-channel follows the same law as at a station near the mouth of a river, rising and falling in equal times; the current runs in the direction analogous to up stream for three hours before and after high water, and down stream for the same period before and after low water. But near the sides of channels and near the mouths of bays the changes of the currents are very complex; and near the headlands separating two bays there is usually at certain times a very swift current, termed a “race.”
In inland seas, such as the Mediterranean, the tides are nearly insensible except at the ends of long bays. Thus at Malta the tides are not noticed by the ordinary observer, whilst at Venice they are conspicuous.
The effect of a strong wind on the height of tide is generally supposed to be very marked, especially in estuaries. In the case of an exceptional gale, when the wind veered round appropriately, Airy states that the water has been known to depart from its predicted height at London by as much as 5 feet. The effect of wind will certainly be different at each port. The discrepancy of opinion on this subject appears to be great,—so much so that we hear of some observers concluding that the effect of the wind is insensible. Variations in barometric pressure also cause departures from the predicted height of water, high barometer corresponding to decrease of height of water. Roughly speaking, an inch of the mercury column will correspond to something less than a foot of water, but the effect seems to vary much at different ports.
§ 3. General Explanation of the Cause of Tides.
The moon attracts every particle of the earth and ocean, and by the law of gravitation the force acting on any particle is directed towards the moon’s centre, and is jointly proportional to the masses of the particle and of the moon, and inversely proportional to the square of the distance between the particle and the moon’s centre. If we imagine the earth and ocean subdivided into a number of small portions or particles of equal mass, then the average, both as to direction and intensity, of the forces acting on these particles is equal to the force acting on that particle which is at the earth’s centre. For there is symmetry about the line joining the centres of the two bodies, and, if we divide the earth into two portions by an ideal spherical surface passing through the earth’s centre and having its centre at the moon, the portion remote from the moon is a little larger than the portion towards the moon, but the nearer portion is under the action of forces which are a little stronger than those acting on the further portion, and the resultant of the weaker forces on the larger portion is exactly equal to the resultant of the stronger forces on the smaller. If every particle of the earth and ocean were being urged by equal and parallel forces, there would be no cause for relative motion between the ocean and the earth. Hence it is the departure of the force acting on any particle from the average which constitutes the tidegenerating force. Now it is obvious that on the side of the earth towards the moon the departure from the average is a small force directed towards the moon; and on the side of the earth away from the m∞n the departure is a small force directed away from the moon. Also these two departures are very nearly equal to one another, that on the near side being so little greater than that on the other that we may neglect the excess. All round the sides of the earth along a great circle perpendicular to the line joining the moon and earth, the departure is a force directed inwards towards the earth’s centre. Thus we see that the tidal forces tend to pull the water towards and away from the moon, and to depress the water at right angles to that direction. If we could neglect the rotations of the bodies, and could consider the system as at rest, we should find that the water was in equilibrium when elongated into a prolate ellipsoid with its long axis directed towards and away from the moon.
But it must not be assumed that this would be the case when there is motion. For, suppose that the ocean consisted of a canal round the equator, and that an earthquake or any other cause were to generate a great wave in the canal, this wave would travel along it with a velocity de pendent on the depth. If the canal were about 13 miles deep, the velocity of the wave would be about 1000 miles an hour, and with depth about equal to the depth of our seas the velocity of the wave would be about half as great. We may conceive the moon’s tide-generating force as making a wave in the canal and continually outstripping the wave it generates, for the moon travels along the equator at the rate of about 1000 miles an hour, and the sea is less than 13 miles deep. The resultant oscillation of the ocean must therefore be the summation of a series of partial waves generated at each instant by the moon and always falling behind her, and the aggregate wave, being the same at each instant, must travel 1000 miles an hour so as to keep up with the moon.
Now it is a general law of frictionless oscillation that, if a slowly varying periodic force acts on a system which would oscillate quickly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction of the excursion; but, if a quickly varying periodic force acts on a system which would oscillate slowly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction opposite to that of the excursion. An example of the first is a ball hanging by a short string, which we push slowly to and fro; the ball will never quit contact with the hand, and will agree with its excursions. If, however, the ball is hanging by a long string we can play at battledore and shuttlecock with it, and it always meets our blows. The latter is the analogue of the tides, for a free wave in our shallow canal goes slowly, whilst the moon’s tide-generating action goes quickly. Hence, when the system is left to settle into steady oscillation, it is low water under and opposite to the moon, whilst the forces are such as to make it high water at those times.
If we consider the moon as revolving round the earth, the water assumes nearly the shape of an oblate spheroid with the minor axis pointed to the moon. The rotation of the earth in the actual case introduces a complexity which it is not easy to unravel by general reasoning. We can see, however, that if water moves from a lower to a higher latitude it arrives at the higher latitude with more velocity from west to east than is appropriate to its latitude, and it will move accordingly on the earth’s surface. Following out this conception, we see that an oscillation of the water to and fro between south and north must be accompanied by an eddy. Laplace’s solution of the difficult problem involved in working out this idea will be given below.
The conclusion at which we have arrived about the tides of an equatorial canal is probably more nearly true of the tides of a globe partially covered with land than if we were to suppose the ocean at each moment to assume the prolate figure of equilibrium. In fact, observation shows that it is more nearly low water than high water when the moon is on the meridian. If we consider how the oscillation of the water would appear to an observer carried round with the earth, we see that he will have low water twice in the lunar day, somewhere about the time when the moon is on the meridian, either above or below the horizon, and high water half way between the low waters.
If the sun be now introduced, we have another similar tide of about half the height, and this depends on solar time, giving low water somewhere about noon and midnight. The superposition of the two, modified by friction and by the interference of land, gives the actually observed aggregate tide, and it is clear that about new and full moon we must have spring tides and at quarter moons neap tides, and that (the sum of the lunar and solar tidegenerating forces being about three times their difference) the range of spring tide will be about three times that of neap tide.
So far we have supposed the luminaries to move on the equator; now let us consider the case where the moon is not on the equator. It is clear in this case that at any place the moon’s zenith distance at the upper transit is different from her nadir distance at the lower transit. But the tide-generating force is greater the smaller the zenith or nadir distance, and therefore the forces are different at successive transits. This was not the case when the moon was deemed to move on the equator. Thus there is a tendency for two successive lunar tides to be of unequal heights, and the resulting inequality of height is called a “diurnal tide.” This tendency vanishes when the moon is on the equator; and, as this occurs each fortnight, the lunar diurnal tide is evanescent once a fortnight. Similarly in summer and winter the successive solar tides are generally of unequal height, whilst in spring and autumn this difference is inconspicuous.
One of the most remarkable conclusions of Laplace’s theory of the tides, on a globe covered with ocean to a uniform depth, is that the diurnal tide is everywhere nonexistent. But this hypothesis differs much from the reality, and in fact at some ports the diurnal tide is so large that during two portions of each lunation there is only one great high water and one great low water in each twenty-four hours, whilst in other parts of the lunation the usual semi-diurnal tide is observed.
§ 4. Historical Sketch.
In 1687 Newton laid the foundation for all that has since been added to the theory of the tides when he brought his grand generalization of universal gravitation to bear on the subject. Kepler had indeed at an earlier date recognized the tendency of the water of the ocean to move towards the centres of the sun and moon, but he was unable to submit his theory to calculation. Galileo expresses his regret that so acute a man as Kepler should have produced a theory -which appeared to him to reintroduce the occult qualities of the ancient philosophers. His own explanation referred the phenomenon to the rotation and orbital motion of the earth, and he considered that it afforded a principal proof of the Copernican system.
In the 19th corollary of the 66th proposition of book i. of the Principia, Newton introduces the conception of a canal circling the earth, and he considers the influence of a satellite on the water in the canal. He remarks that the movement of each molecule of fluid must be accelerated in the conjunction and opposition of the satellite with the molecule, and retarded in the quadratures, so that the fluid must undergo a tidal oscillation. It is, however, in propositions 26 and 27 of book iii. that he first determines the tidal force due to the sun and moon. The sea is here supposed to cover the whole earth, and to assume at each instant a figure of equilibrium, and the tide-generating bodies are supposed to move in the equator. Considering only the action of the sun, he assumes that the figure is an ellipsoid of revolution with its major axis directed towards the sun, and he determines the ellipticity of such an ellipsoid. High solar tide then occurs at noon and midnight, and low tide at sunrise and sunset. The action of the moon produces a similar ellipsoid, but of greater ellipticity. The superposition of these ellipsoids gives the principal variations of tide. He then proceeds to consider the influence of latitude on the height of tide, and to discuss other peculiarities of the phenomenon. Observation shows, however, that spring tides occur a day and a half after syzygies, and Newton falsely attributed this to the fact that the oscillations would last for some time if the attractions of the two bodies were to cease.
The Newtonian hypothesis, although it fails in the form which he gave to it, may still be made to represent the tides, if the lunar and solar ellipsoids have their major axes always directed towards a fictitious moon and sun, which are respectively at constant distances from the true bodies; these distances are such that the syzygies of the fictitious planets occur about a day or a day and a half later than the true syzygies. In fact, the actual tides may be supposed to be generated directly by the action of the real sun and m∞n, and the wave may be imagined to take a day and a half to arrive at the port of observation. This period has accordingly been called “the age of the tide.” In what precedes the planets have been supposed to move in the equator; but the theory of the two ellipsoids cannot be reconciled with the truth when they move in orbits inclined to the equator. At equatorial ports the theory of the ellipsoids would at spring tides give morning and evening high waters of nearly equal height, whatever the declinations of the bodies. But at a port in any other latitude these high waters would be of very different heights, and at Brest, for example, when the declinations of the bodies are equal to the obliquity of the elliptic, the evening tide would be eight times as great as the morning tide. Now observation shows that at this port the two tides are nearly equal to one another, and that their greatest difference is not a thirtieth of their sum.
Newton here also offered an erroneous explanation of the phenomenon. In fact, we shall see that by Laplace’s dynamical theory the diurnal tide is evanescent when the ocean is of uniform depth over the earth. At many nonEuropean ports, however, the diurnal tide is very important, and thus as an actual means of prediction the dynamical theory, where the ocean is treated as of uniform depth, may be hardly better than the equilibrium theory.
In 1738 the Academy of Sciences of Paris offered, as a subject for a prize, the theory of the tides. The authors of four essays received prizes, viz., Daniel Bernoulli, Euler, Maclaurin, and Cavalieri. The first three adopted not only the theory of gravitation but also Newton’s method of the superposition of the two ellipsoids. Bernoulli’s essay contained an extended development of the conception of the two ellipsoids, and, under the name of the equilibrium theory, it is commonly associated with his name. Laplace gives an account and critique of the essays of Bernoulli and Euler in the Mécanique Celeste. The essay of Maclaurin presented little that was new in tidal theory, but is notable as containing those theorems concerning the attraction of ellipsoids which we now know by his name. In 1746 D’Alembert wrote a paper in which he treated the tides of the atmosphere; but this work, like Maclaurin’s, is chiefly remarkable for the importance of collateral points.
The theory of the tidal movements of an ocean was therefore, as Laplace remarks, almost untouched when in 1774 he first undertook the subject. In the Mécanique Céleste he gives an interesting account of ⅛e manner in which he was led to attack the problem. We shall give below the investigation of the tides of an ocean covering the whole earth; the theory is substantially Laplace’s, although presented in a somewhat different form. This theory, although very wide, is far from representing the tides of our ports. Observation shows, in fact, that the irregular distribution of land and water and the variable depth of the ocean produce an irregularity in the oscillations of the sea of such complexity that the rigorous solution of the problem is altogether beyond the power of analysis. Laplace, however, rested his discussion of tidal observation on this principle— The state of oscillation of a system of bodies in which the primitive conditions of moνe- nient have disappeared through friction is coperiodic νηth the forces acting on the system. Hence, if the sea is solicited by a periodic force expressed as a coefficient multiplied by the cosine of an angle which increases proportionately with the time, there results a partial tide, also expressed by the cosine of an angle which increases at the same rate; but the phase of the angle and the coefficient of the cosine in the expression for the height may be very different from those occurring in the corresponding term of the equilibrium theory. The coefficients and the constants or epochs of the angles in the expressions for the tide are only derivable from observation. The action of the sun and moon is expressible in a converging series of similar cosines; whence there arise as many partial tides, which by the principle of superposition may be added together to give the total tide at any port. In order to unite the several constants of the partial tides Laplace considers each tide as being produced by a fictitious satellite moving uniformly on the equator. Sir W. Thomson and others have followed Laplace in this conception; but in the present article we shall not do so. The difference of treatment is in reality only a matter of phraseology, and the proper motion of each one of Laplace’s astres fictifs is at once derivable from the argument (or angle under the sign of cosine), which we shall here associate with the partial tides.
Subsequently to Laplace the most important workers in this field were Sir John Lubbock (senior), Whewell, and Airy. The work of Lubbock and Whewell (see § 34 below) is chiefly remarkable for the coordination and analysis of enormous masses of data at various ports, and the construction of trustworthy tide-tables and of cotidal maps. Airy contributed an important review of the whole tidal theory. He also studied profoundly the theory of waves in canals, and explained the effects of frictional resistances on the progress of tidal and other waves. Of other authors whose work is of great importance we shall speak below.
Amongst all the grand work which has been bestowed on this difficult subject, Newton, notwithstanding his errors, stands out first, and next to him we must rank Laplace. However original any future contribution to the science of the tides may be, it would seem as though it must perforce be based on the work of these two.
A complete list of works bearing on the theory of the tides, from the time of Newton down to 1881, is contained in vol. ii. of the Bibliographie de l' Astronomie by Houzeau and Lancaster (Brussels, 1882). This list does not contain papers on the tides of particular ports, and we are not aware of the existence of any catalogue of works on practical observation, reduction of observations, prediction, and tidal instruments. References are, however, given below to several works on these points.
II. Tide-geneeating Forces.
§ 5. Investigation of Tide-generating Potential and Forces.
We have already given a general explanation of the nature of tide-generating forces; we now proceed to a rigorous investigation.
If a planet is attended by a single satellite, the motion of any body relatively to the planet’s surface is found by the process described as reducing the planet’s centre to rest. The planet’s centre will be at rest if every body in the system has impressed on it a velocity equal and opposite to that of the planet’s centre; and this is accomplished by impressing on every body an acceleration equal and opposite to that of the planet’s centre.
Let M, m be the masses of the planet and the satellite; r the radius vector of the satellite, measured from the planet’s centre; ρ the radius vector, measured fronr the same point, of the particle whose motion we wish to determine; and z the angle between r and p. The satellite moves in an elliptic orbit about the planet, and the acceleration relatively to the planet’s centre of the satellite is {M+m) ∣ r 3, towards the planet along the radius vector r. Now the centre of inertia of the planet and satellite remains fixed in space, and the centre of the planet describes an orbit round that centre of inertia similar to that described by the satellite round the planet, but with linear dimensions reduced in the proportion of m to M+m.
Hence the acceleration of the planet’s centre is nιft 3 ∙ towards the centre of inertia of the two bodies. Thus, in order to reduce the planet’s centre to rest, we apply to every particle of the system an acceleration τn∕r 2 parallel to r, and directed from satellite to planet.
Now take a set of rectangular axes fixed in the planet, and let M 1 r, M 2 r, M 3 γ be the coordinates of the satellite referred thereto; and let ξρ, ηp, ζp be the coordinates of the particle P whose radius is p. Then the component accelerations for reducing the planet’s centre to rest are - mM∣∕r 2 , - mM 2 ∕r 3 , - wιM s ∕r 3 ; and since these are the differential coefficients with respect to pξ, ρη, pζ of the function ∞p,, ir -^∕( m iS+ m 2 , 1+ m A and since cos z=M 1 ∣ + M 2 ιj + M 3 f, it follows that the potential of the forces by which the planet’s centre is to be reduced to rest is mp f cos z. r 2
Now let us consider the other forces acting on the particle. The Slanet is spheroidal, and therefore does not attract equally in all irections; but in this investigation we may make abstraction of the ellipticity of the planet and of the ellipticity of the ocean due to the planetary rotation. This, which we set aside, is considered in the theories of gravity and of the figures of planets. Outside of its body, then, the planet contributes forces of which the potential is M/p. Next the direct attraction of the satellite contnbutes forces of which the potential is the mass of the satellite divided by the distance between the point P and the satellite; this is— m √(r 2 + p 2 - 2rpcosz}'
To determine the forces from this potential we regard ρ and z as the variables for differentiation, and we may add to this potential any constant we please. As we are seeking to find the forces which urge P relatively to Μ, we add such a constant as will make the whole potential at the planet’s centre zero, and thus we take as the potential of the forces due to the attraction of the satellite— rn m √{r 2 +p 2 - 2rpcosz} r
It is obvious that r is very large compared with ρ, and we may therefore expand this in powers of p∕r. This expansion gives us 5∕⅛ 1+ ⅛> i+ ⅛ +sc Λ , r Λ r 2 2 r 3 ó J , where P 1 = cos z, Z 2 =f cos 2 z-⅜, P 3 =⅛ cos 3 z-⅜ cos z, &c. The reader familiar with spherical harmonic analysis of course recognizes the Legendre’s functions; but the result for a few terms, which is all that is necessary, is easily obtainable by simple algebra.
Now, collecting together the various contributions to the potential, and noticing that · ^P 1 ≈ i ~ cos z, and is therefore equal and opposite to the potential by which the planet’s centre was reduced to rest, we have as the potential of the forces acting on a particle whose coordinates are pξ, ρη, pζ j+ ’ ^f(i cos22 -⅜)+ 7 ^ 3 (∣cos 32 -∣cosz)+ (U-
The first term of (1) is the potential of gravity, and the terms of the series, of which two only are written, constitute the tide-generating potential. In all practical applications this series converges so rapidly that the first term is amply sufficient, and thus we shall generally denote
U=∣Jp 2 (cos 2 z-i) (2)
as the tide-generating potential.
In many mathematical works the tide-generating force is presented as being due to an artificial statical system, which produces nearly the same force as the dynamical system considered above. This statical system is as follows. Stopping all the rotations, we divide the satellite into two equal parts, and place them diametrically opposite to one another in the orbit. Then it is clear that, instead of the term rn m
√{r 2 +p 2 -2rρcosz} r ’ we have ⅜m ⅜∕⅞ m √{r 2 + p 2 -2rρcosz} √]r 2 + p 2 +2rpcosz} r
And this reduces to Jp 2 Λ+>Λ+∙∙∙
The first term is the same as before; hence the statical system produces approximately the same tide-generating force as the true system. The “moon” and “anti-moon,” however, produce rigorously the same force on each side of the planet, whereas the true system only satisfies this condition approximately.
§ 6. Form of Equilibrium.
Let us consider the shape assumed by a layer of fluid of density σ, lying on a globe of mass Μ, when acted on by disturbing forces whose potential is Γ= g>(cos 2 2 - ⅛) (3).
Suppose the layer to be very thin, and that the mean radius of the layer is α, and let the equation to the boundary of the fluid be ρ=α[l + e(cos 2 ≡- ⅜)] (4).
We assume this form, because the theory of harmonic analysis tells us that the departure from sphericity must be represented by a function of the form cos 2 2 - ⅜. That theory also gives us as the potential of a layer of matter of depth rα(cos 2 s- ⅜), and density σ, at an external point the value
⅜ιrσα 2 0) f(cos 2 2 -l).
Hence the whole potential, outside of and up to the fluid layer, is y + f^P 2 (co ≡ 2 z - έ) + ⅜ 7rσα2 (^) e (co ≡ 2 z - ⅛) (5 )∙
The first term of (5) is the potential of the globe, the second that of the disturbing force, and the third the potential due to departure from sphericity.
Now the fluid must stand in a level surface; hence, if we equate this potential to a constant, we must get back to the equation (4), which was assumed to be that of the surface. In other words, if we put p = α[l + t(cos 2 s-⅜)] in (5), the result must be constant, provided the departure from sphericity is small. In effecting the substitution for p, we may put ρ=a in the small terms, but in the first term of (5) we put
^=⅛-ε(cos 2 z-⅛)]∙ p CL
The whole potential (5) can only be constant if, after this substitution, the coefficient of cos 2 z — ⅛ vanishes. Thus we must have Μ Zma? . . n
^α' + ⅛S- + i,r ' ra ' e = 0 ∙
But if <5 be the mean density of the planet M=⅛πa 3 δ, and gravity g=M ∣ a i . Then we easily find that
3mα 1 ...
r- 2<r 3 l-*√δ w '
Thus the equation to the surface is r = o { 1 +∣pr⅛s<∞ β, *-»>} i7 >∙
If σ be small compared with δ, the coefficient is ⅛maf2,gr 3 ; thus we see that 1∕(1 - j<r∕δ) is the coefficient by which the mutual attraction of the fluid augments the deformation of the fluid under the action of the disturbing force. If the density of the fluid be the same as that of the sphere, the augmenting factor becomes f, and we have c — i fmajgr 3 , which gives the form of equilibrium of a fluid sphere under the action of these forces. Since χ-⅛ = -fl-t~V> r 2r i a ∖ 5o∕
it follows that, when the form of equilibrium is p= α[l + r(cos 2 z - ⅜)], the potential of the forces is r =!(1 -⅛) ’ ' ,, < cos, ≈-i> < 8 >∙
More generally, if we neglect the attraction of the fluid on itself, so that σ∕δ is treated as small, and if p=α(l + s) be the equation to the surface òf the fluid, where s is a function of latitude and longitude, then the potential of the forces under which this is an equilibrium form is V=Á (9).
a x
It thus appears that we may specify any tide-generating forces by means of the figure of equilibrium which the fluid would assume under them, and in the theory of the tides it has been found practically convenient to specify the forces in this way.
By means of the principle of “forced vibrations” referred to in the historical sketch, we shall pass from the equilibrium form to the actual oscillations of the sea.
§ 7. Development of Tide-generating Potential in Terms of Hour-Angle and Declination.
We now proceed to develop the tide-generating potential, and shall of course implicitly (§ 6) determine the equation to the equilibrium figure.
We have already seen that, if z be the moon’s zenith distance at the point P on the earth’s surface, whose coordinates referred to A, B, C, axes fixed in the earth, are a£, an, aζ, then cos 2=ξM 1 + ηM 2 + f M 3 , where M 1 , M 2 , M 3 are the moon’s direction cosines referred to the same axes. Then with this value of cos z— cos 2 z - ⅛=2⅛M 1 M 2 +2⅛i m i 8 -M, 2 ÷ 2 , f M 2 M 3 + 2tfM 1 M 3
, 3. ^+^-2Γ 2 M 1 2 + M 2 2 -2M 3 2
^*^2 3 3 μυ ' ’
The axis of C is taken as the polar axis, and AB is the equatorial plane, so that the functions of £, n, ζ are functions of the latitude and longitude of the point P, at which we wish to find the potential·
The functions of M 1 , M 2 , M 3 depend on the moon’s position, and we shall have occasion to develop them in two different ways,— first in terms of her hour-angle and declination, and secondly (§ 23) in terms of her longitude and the elements of the orbit.
Now let A be on the equator in the meridian of P, and B 90° east of A on the equator. Then, if Μ be the moon, the inclination of the plane MC to the plane CA is the moon’s easterly local hour-angle. Let h— local hour-angle of moon and δ=moon’s declination: we have
M 1 =cosδcosΛ, M 2 = cosδsinΛ, M 3 =sin δ, whence
2M 1 M 2 =cos 2 δ sin 2h, M 1 2 - M 2 2 = cos 2 δ cos 2ä, 2M 2 M 3 =2 sin δ cos δ sin h, 2M 1 M 3 =2 sin δ cos δ cos Ä,
M 1 2 + M 2 2 -2M 3 2 1 . , s
— 1 2 ∙ = ⅜ _ sιn 2 δ.
Also, if λ be the latitude of P, ξ=cos λ, η=0, f=sin λ, and
^=0,-^-^=⅜cos 2 λ, ξf=⅜sin2λ, ξι 7 =0, ^ + η 2 -2f 2 )=i-sm 2 λ.
Hence (10) becomes cos 2 z - ⅛=⅛ cos 2 λ cos 2 δ cos 2Ä + sin 2λ sin δ cos δ cos h + ⅜ (⅜-sin 2 δ)(⅜-sin 2 λ) (11).
The angle h, as defined at present, is the eastward local hour-angle, and therefore diminishes with the time. As, however, this function does not change sign with h, it will be more convenient to regard it as the westward local hour-angle. Also, if h 0 be the Greenwich westward hour-angle at the moment under consideration; and l be the west longitude of the place of observation P, we have h=h 0 -l (12).
Hence we have at the point P, whose radius vector is a, V=^i {⅜ cos 2 λ cos 2 δ cos 2(A 0 - l) + sin 2λ sin ð cos δ cos (⅛ 0 - Z) 2z ^ + ⅜ (⅛-sin 2 δ)(⅛-sin 2 λ)} (13).
The tide-generating forces are found by the rates of variation of V for latitude and longitude, and also for radius α, if we care to find the radial disturbing force.
§ 8. Evaluation of Tide-generating Forces, and Lunar Deflexion of Gravity.
The westward component of the tide-generating force at the earth’s surface, where p=a, is dV ∣ a cos λ<ZZ, and the northward component is dV ∣ adλ ; the change of apparent level is the ratio of these to gravity g. Therefore, differentiating (13), changing signs, and writing f° r we have component change of level southward
= ⅛i {sin 2λ cos 2 δ cos 2(Λ 0 - Z) - 2 cos 2λ sin 2δ cos (A 0 - Z)
44Z∖ r∕ + sin 2λ (1 _ 3 sin 2 δ)}; component change of level westward
Y { cos × c0 ≡ 2 5 ≡i n 2 (⅞ ~ 0
2M ∖ r∕ ⅛ sin λ sin 2δ sin (⅛ 0 - Z)} (14).
The westward component is made up of two periodic terms, one going through its variations twice and the other once a day. The southward component has also two similar terms; but it has a third term, which does not oscillate about a zero value. If Δ be a declination such that the mean value of sin 2 δ is equal to sin 2 ∆, then, to determine the southward component so that it shall be a truly periodic function, we must subtract from the above sin 2λ(l - 3 sin 2 Δ), and the last term then becomes
3 sin 2λ(sin 2 Δ - sin 2 δ).
In the case of the moon, Δ varies a little according to the position of the moon’s node, but its mean value is about 16° 31'.
The constant portion of the southward component of force has its effect in causing a constant heaping up of the water at the equator; or, in other words, the moon’s attraction has the effect of causing a small permanent ellipticity of the earth’s mean figure. This augmentation of ellipticity is of course very small, but it is necessary to mention it in order that the meaning to be attributed to lunar deflexion of gravity may be clearly defined.
If we consider the motion of a pendulum-bob during any one day, we see that, in consequence of the semi-diurnal changes of level, it twice describes an ellipse with major axis east and west, with ratio of axes equal to the sine of the latitude, and with linear dimensions proportional to cos 2 δ, and it once describes an ellipse whose north and south axis is proportional to sin 2δ cos 2λ and whose east and west axis is proportional to sin 2δ sin λ. Obviously the latter is circular in latitude 30°. When the moon is on the equator, the maximum deflexion occurs when the moon’s local hour-angle is 45°, and is then equal to
3wι ∕ α∖ 3 .
— ( - 1 cos λ. 2M ∖ rf
At Cambridge in latitude 52° 43' this angle is 0"O216.
An attempt, made by George and Horace Darwin, to measure the lunar deflexion of a pendulum failed on account of incessant variability of level occurring in the supports of the pendulum and arising from unknown terrestrial changes. The work done, therefore, was of no avail for the purposes for which it was instituted, but remained as a contribution to an interesting subject now beginning to be studied, viz., the small changes which are always taking place on the upper strata of the earth.
§ 9. Correction to Equilibrium Theory for Continents.
In the equilibrium theory as worked out by Newton and Bernoulli it is assumed that the figure of the ocean is at each instant one of equilibrium under the action of gravity and of the tidegenerating forces. Sir W. Thomson has, however, reasserted a point which was known to Bernoulli, but has since been overlooked, namely, that this law of rise and fall of water cannot, when portions of the globe are continents, be satisfied by a constant volume of water in the ocean. The law would still hold if water were appropriately supplied to and exhausted from the ocean; and, if in any configuration of the tide-generating body we imagine water to be instantaneously so supplied or exhausted, the level will everywhere rise or fall by the same height. Now the amount of that rise or fall depends on the position of the tide-generating body with reference to the continents, and is different for each such position. Conversely, when the volume of the ocean remains constant, we have to correct Bernoulli’s simple equilibrium theory by an amount which is constant all over the globe at any instant, but which changes in time. Thomson’s solution of this problem has since been reduced to a form which is easier to grasp intelligently than in the shape in which he gave it, and the results have also been reduced to numbers. It appears that there are four points on the earth’s surface at which in the corrected theory the semidiurnal tide is evanescent, and four others where it is doubled. A similar statement holds for the diurnal tide. As to the tides of long period, there are two parallele of latitude of evanescent and two of doubled tide.
Now in Bernoulli’s theory the semi-diurnal tide vanishes at the poles, the diurnal tide at the poles and the equator, and the tides of long period in latitudes 35° 16' north and south. The numerical solution of the corrected theory shows that the points and lines of doubling and evanescence in every case fall close to the points and lines where in the uncorrected theory there is evanescence. When in passing from the uncorrected to the corrected theory we speak of a doubled tide, the tide doubled may be itself nil, so that the result may still be nil. The conclusion, therefore, is that Thomson’s correction, although theoretically interesting, is practically so small that it may be left out of consideration.
III. Dynamical Theory of Tides.
§ 10. Historical Explanation.
The problem of tidal oscillation is essentially a dynamical one. Even when the ocean is taken as covering the whole earth, it presents formidable difficulties, and this is the only case in which it has been hitherto solved. Laplace gives the solution in bks. i. and iv. of the Mécanique Céleste ; but his work is unnecessarily complicated by the inappropriate introduction of spherical harmonic analysis, and it is generally admitted that his investigation is difficult. Airy, in his “Tides and Waves” (in Ency. Metrop.) presents the solution free from that complication, but he has made a criticism of Laplace’s method which we believe to be wrong. Sir W. Thomson has written some interesting papers (in Phü. Mag., 1875) in justification of Laplace, and on these we base the following paragraphs. This portion of the article is given more fully than others, because there exists no complete presentment of the theory free from objections of some kind.
§11. Equations of Motion.
Let r, θ, φ be the radius vector, co-latitude, and east longitude of a point with reference to an origin, a polar axis, and a zero-meridian rotating with a uniform angular velocity n from west to east. Then, if 72, 77, S be the radial, co-latitudinal, and longitudinal accelerations of the point, we have _ (Fr ∕dθ ∖ 2 . 2 .fdφ , ∖ 2 Ί R =dfi-\di ) - r81η ∙ β (s + *)
S=-⅛(r 2 ^) -rsinØcosø(^4-n) - (15).
r dt ∖ dt) ∖ dt J ' ' H= — i ~^λ r * s ^ χ2 ff (⅛f + «)Ί
rsm0c⅛L ∖ dt ∕J >
Now suppose that the point never moves far from a zero position and that its displacements ξ, nsiηθ co·latitudinally and longitudinally are very large compared with its radial displacement p, and that the velocities are so small that their squares and products are negligible compared with n 2 r 2 ; then we have
(Zr dp 11 -r f = f,> a very small quantity; uí dt . a dφ d, . .. rsιηθ dt=dt^ aiηθ ^ dθ~dξ r dt~ df
Hence (15) is approximately
7⅛ ’ = -ra 2 rsin 2 θ ∙x w d 2 ξ η . a a ⅜ 0 =^- 2 y ιn8c ∞ l, Λ L (i β ),
Ae—sin θ ⅛ + 2rt J
With regard to the first equation of (16), we observe that the time has disappeared, and that R has exactly the same form as if the system were rendered statical by introducing a potential ⅜n 2 r 2 sin 2 0 and annulling the rotation of the axes. Since inertia plays no sensible part radially, it follows that, if we apply these expressions to the formation of equations of motion for the ocean, the radial motion need not he considered. We are left, therefore, with only the last two equations of (16).
We now have to consider the forces by which an element of the ocean is urged in the direction of co-latitude and longitude. These forces are those due to the external disturbing forces and to the pressure of the surrounding fluid, the attraction of the fluid on itself being supposed negligible. We have seen in (9) that, if fluid on a sphere of radius a be under the action of disturbing forces whose potential is Ur i , and if r=a + t) be the equation to the surface, then must pfj = Ua 2 . Hence, if í be the equilibrium height of tide, the potential of the disturbing force is gv ∣ a ∣ a 2 . But, if the elevation be fj, the potential under which it would be in equilibrium is gtyf ∣ a 2 . Therefore, if ]) be the elevation of the tide in our dynamical problem, the forces due to hydrostatic pressure on an element of the ocean are the same as would be caused by a potential - gty a ∣ a 2 . Hence it follows that the whole forces on the element are those due to a potential - <∕(lj - f)ι a [a 2 . Therefore from (16) we see that the equations of motion are
⅛ - 2nsinØcosØ ⅛= - ~⅛⅛ - e) )
dt dt cl du I (17)
sin 0 ¾ + 2n cos 0 d J = dt i dt a sin 0 dφ j J
It remains to find the equation of continuity. This maybe deduced geometrically from the consideration that the volume of an element of the fluid remains constant; but a shorter way is to derive it from the equation of continuity as it occurs in ordinary hydrodynamical investigations. If V be a velocity potential, the equation of continuity for incompressible fluid is δr^ ( r i -j~ siηθ δθ δφ ∖ + δθ^-(r sinØ -↑ a δrδφ ∖ dr ∖ dr J dθ ∖ rdθ )
÷ δ ≠⅛(r ^^S δr δ0 ) = °- dφ ∖ r sin 0 dφ ∕
The element referred to in this equation is defined by r, θ, φ, r + δr, θ + δθ, φ + δφ. The co-latitudinal and longitudinal velocities are the same for all the elementary prism defined by 0, φ, θ + δθ, φ+δφ, send the sea bottom. Then —∙^^ λ^ 7 =
rdθ df rsιηθdφ sin 0—; and, since the radial velocity is d↑) ∣ dt at the surface of the ocean, where r=a + y, and is zero at the sea bottom, where r=a, , dV r-adh ττ . , ,. ... , , . we have -y~= -√. Hence, integrating with respect to r from GfΓ *y (X∕C r=a+y to r=a, and again with respect to t from the time t to the time when lj, ξ, n all vanish, and treating y and f) as small compared with a, we have i)a sin 0 + ^ a (yξ sin 0) + ~(yn sin 0) = 0 (18).
du dφ
This is the equation of continuity, and, together with (17), it forms the system which must be integrated in the general problem of the tides. The difficulties in the way of a solution are so great that none has hitherto been found, except on the supposition that y, the depth of the ocean, is only a function of latitude. In this case (18) becomes ⅛- + ⅛e⅛w= to ^ + >⅛= 0 < 19 >∙
§ 12. Adaptation to Forced Oscillations.
Since we may suppose that the free oscillations are annulled by friction, the solution required is that corresponding to forced oscillations. Now we have seen from (13) that ε (which is proportional to V) has terms of three kinds, the first depending on twice the moon’s (or sun’s) hour-angle, the second on the hour-angle, and the third independent thereof. The coefficients of the first and second terms vary slowly, and the whole of the third varies slowly. Hence c has a semi-diurnal, a diurnal, and a long period term. We shall see later that these terms may be expanded in a series of approximately semi-diurnal, diurnal, and slowly varying terms, each of which is a strictly harmonic function of the time. Thus we may assume for c a form e cos (2n∕Z + kφ + a), where f and k are numbers, and where e is only a function of co-latitude and of the elements of the orbit of the disturbing body. According to the usual method of treating oscillating systems, we may therefore make the following assumption for the form of solution
í=e cos (2nft + kφ + α) 1
fj = hcos(2n∕ι! + ⅛≠ + a) f ∕ 20 ∖
ξ = xcos{2nft + kφ + a') l · '* ’ ' h η=y sin (2nft + kφ + a) ' where e, h, x, y are functions of co-latitude 0 only. Substituting these values in (19), we have 8 ⅛⅛< 1,xsini > +lγy+hα = 0 (2I) ·
Then, if we write u for h - e, and^>ut m=n-a ∣ g, substitution from (20) in (17) leads at once to
V 2 +y∕sin0cos0 = A ∣
4τzι I (22)
y∕^ 2 sin 0+ x∕cos 0= Γ
j ∙ , j Am sin θ J
Solving (22) for x and y, we have , λ β λ × 1 fdvι k cos 0∖ λ
x(∕ 2 - cos 2 0) = —1 -73 +- i ,u -.— 5 ) I
v ' Am ∖ dθ f sin 0/ I z93).
• λ zλ o λ , -l∕cos0<fu ⅛u ∖ f " λ- y sin θ (f i - cos 2 0) — — ( — 7 - - i χ + -—ñ I I j v , Arn ∖ f dθ sin 0/7
Then substituting from (23) in (21), we have ι d k ∖ cοs -^ j-
1 d Γ√ sin0 ⅛n + > ucos0∏ 7 ' f <Z0 + sin0 Si∏<⅞L ∕-oo<M -J^⅛<V 1 -
+ 4ma(u + e)=0 (24).
This is Laplace’s equation for tidal oscillations in än ocean whose depth is only a function of latitude. When u is found from this equation, its value substituted in (23) will give x and y.
§ 13. Preparation for Solution.
The ocean which is considered in this case is not like that on the earth’s surface, and therefore it does not seem desirable to pursue the integration of (24) except in certain typical cases.
In (13) we have the expansion of the disturbing potential and implicitly of the disturbing forces in three terms, the first of which is variable in half a day, the second in a day, and the third in half the period of revolution of the tide-raising body. Forestalling the results of chapter iv.—each of these terms may be expressed as the sum of a series of strictly harmonic functions of the time; the first set of these have all approximately semi-diurnal periods, the second approximately diurnal periods, and the third vary slowly in dependence on the periodic time of the tide-generating body. The first set involve twice the terrestrial longitude, the second the longitude, and the third set are independent of the longitude of the place of observation. From these statements compared with (13) we see that in the semi-diurnal terms f is approximately unity, k=2, and e=E sin 2 0; in the diurnal terms f is approximately ⅜, ⅛=1, and i = E sin 0 cos 0; in the terms of long period f is a small fraction (for the fortnightly tide about ⅛), k=Q, e=E(⅛ - cos 2 0). The departure from exactness in the relation ∕=1 for the semi-diurnal, and ∕=⅛ for the diurnal terms is generally (except for certain critical depths of ocean) not such as to greatly change the nature of the results from those obtained when ∕=1 and ⅜ rigorously. Hence the integration of (24) will be pursued on these three hypotheses, giving Laplace’s three kinds of oscillation. The hypothesis which wall be made with regard to y is that y = Z(l - q cos 2 0), and in the case of the semi-diurnal tides we shall be compelled by mathematical difficulties to suppose q to be either unity or zero. The tides of zonal seas may be worked out, and more complex laws of depth may be assumed; but for the discussion of such cases the reader is referred to Thomson’s papers in Phil. Mag., 1875.
There might be reason to conjecture that the form of u would be similar to that of e, and this is in fact the case for the diurnal tides for any value of q and for the semi-diurnal tides when q is unity. Before proceeding further it will be convenient to exhibit two purely analytical transformations of the first two terms of (24) which hold true for certain values of k and f and when u has such a form as that suggested. If we put ⅛=1,∕≈⅜, 7=Z(l -q cos 2 0), then, if v=A sin 0 cos 0, it will be found on substitution that
, 7fsin 0^ + 2vcosøì 2 cot 0⅜+ .∖ a
1 d ' ∖ dθ ) dθ sιn 2 0 o1 , a ,×
pτs¼— *— r-ctfo = - 8 ⅛ , -< 25 >∙
Again, if we put⅛=2,∕=l, ç=l, 7=Z(l - cos 2 0) = Z sin 2 0, and if v—A sin 2 0,
, 7(sin 0⅛ + 2fl cos ø) +
1 d ∖ dθ ∕ o dθ sιn 2 0 o7 sin0<Z0 l-cos 2 0 2γ l-cos 2 0 -
Another general property of (24) is derived from the supposition that u is expressed in a senes proceeding by powers of l ; thus
"=⅝+¾+⅞^+ (27).
Let r 0 , ι> 1 , ¾ &c., be so chosen that, when u is substituted in (24), the coefficient of each power of l vanishes independently; then the term independent 01 l obviously gives v 0 = -e, and the connexion between successive ν , s is r- ∕ d ν k ∖ - ∣ f cosθdv m ι kν m ∖ _1_ d 7(sin 0-^ 5 + →∙ m cosø) _ V ∕ rfø + sin0∕ sin0 dθ _— y∙2 -co ^0 y sin 0 (∕ 2 - cos 2 0)
+ 4τn⅛, +1 = 0 (28).
We shall suppose below that u is expansible in the form(27), and shall use (28) in conjunction with (25) or (26) for finding the successive values of the v’s.
§ 14. Diurnal Tide.
Let us first consider the diumal tides. We have e = E sin 0 cos 0, k= 1, and ∕=⅜; then v 0 = - E sin 0 cos 0. Hence by (28) and (25)
-8⅞τ o + 4nιfo 1 =0 (29), and therefore ν 1 = ⅛∙ 0 , Applying the same theorem a second time, v 2 — (¾M> 1 , and so on; therefore u = r 0 [l + 2lq∕ma + (2lqjmaf+...] - 1 - 2lη ∣ ma~ 1 - 2lqfma
Butu=h-e; hence . 2lq ∣ ma ~ l-2lqjma
It appears, therefore, that the tide is “inverted,” giving low water where the equilibrium tide gives high water. If q= 0, so that the ocean is of uniform depth, the tide vanishes.
§ 15. Semi-Diumal Tide, with Variable Depth.
Next let us consider the semi-diurnal tide in the case where q= 1, so that y=l sin 2 0. Then e = E sin 2 0, k=2, f=l ; also v 0 = — e= - E sin 2 0. Hence by (28) and (26) - 8Z-υ 0 + Anlι ∖ = 0, whence -ι⅛ = 2 ∣ mv 0 , Applying the same theorem a second time, v 2 =(2∕mγv o , and so on; therefore u=v 0 [l + 2l ∣ ma + (2Z∕mα) 2 +...]
_ ⅞ _ e
1 - 2l ∣ nιa 1 - 2l ∣ ma'
Hence h=u + e= - 1 2 ⅞→ (32).
1 - 2l ∣ ma
If 2l ∣ ma= A, the height of tide is equal to the equilibrium height; but it is inverted, giving low water where the equilibrium theory gives high water. In the case of the earth m∙= 1/289, and therefore this relation is satisfied if Z=α∕1156. Hence in a sea 3000 fathoms deep at the equator, and shallowing to the poles, we have inverted semi-diurnal tides of the equilibrium height.
§ 16. Semi-Diurnal Tide, with Uniform Depth.
The method of development used above, w here we proceed by powers of the depth of the ocean, is not applicable where the depth is uniform, because it leads to a divergent series. We have therefore to resume equation (24). In the case of the semi-diurnal tides we have for the depth y=l (a constant by hypothesis), ⅛=2,∕=1 approximately, and e=Esin 2 0. Now for brevity let β=Arnaf, ν = sin 0, so that e = Ev 2 . Then we find that on development (24) becomes j u p 2(l _ v ifLΞ _ f2 - (8 - 2x 2 - ∕3√*)u = - j3Ev 6 (33).
Let us now assume as the solution of this equation u= (K 2 - E> 2 + K i v i + K 6 1> β + ... + A" 2i v 2i + (34).
Substituting from (34) in (33), and equating to zero the coefficients of the successive powers of v, we find K„ = E, K i apparently indeterminate, and
2f(2ι + 6)A<>i+4 ^^ 2i(2i+3)A⅛∙+2 ^t ßFa = 0 (35 )∙
Since 2Γ o =0, this equation of condition may be held to apply for all positive integral values of i, beginning with i=0. It is obvious that K 6 is determinable in terms of K i and K 2 , K 8 in terms of A' β and K i , &c., so that all the K’s are to be found in terms of Ä 2 , which is known, and of K 4 , which is apparently indeterminate.
The condition for the convergency of the series (34) for u and for the series dxx ∣ dv is that K ii j r dK∙ 2 i shall tend to a limit less than unity. The equation (35) may be written ľ-n _ 2i+3 ß Kg ⅛⅛,∙ +2 2i + 6 2i(2i+6) Kn +2 k ’ '
Now K2i+2 ∣ K ü tends to be either infinitely small or not infinitely small. If it be not infinitely small in the limit, the second term on the right of (36) becomes evanescent when i is very great, and we have in the limit when i is very large—
2i±_3 Γ 1 3_Ί 2= Γ 1 _ 3Ί 2 ‰ 2 f 2 H- 2 2í+6 L 2(i + 3)J L 2τJ
But the ratio of successive terms of √(1 - v 2 ) tends to become (1 - ∣∕r> 2 . Henc e, if W a+2 ∕A 2i does not tend to become infinitely small, u=A + B∖∕l -v*, where A and B are finite for all values of v. Again, under the same circumstances we have in the limit when i is very large—
(2i + 4‰ μ ^ , 2f + 4 2i + 3 2 ∕, , 1 ∖Λ 8_\ 2
(2i+2‰. a ∙' 2i + 1 ^2i + 2'2i+6 ~V i + l∕∖ 2(i+3)Λ
=(l-l∕2i> 2 .
But the ratio of successive terms of (1 — p 2 ) - ⅛ tends to (1 - ⅜∕i)p 2 . Hence, if AΓ<M +2 ∕2Γ 2i does not tend to become infinitely small, i∕u∕dv=C + D(l -r 2 )~1, where C and D are finite for all values of v.
Now ⅛=⅛ √Γ≡7 r = C√fV√ 2 +D.
dθ dv
Therefore at the equator, where v=l, du∕dθ = D, a finite quantity. Hence the hypothesis that A' 2 ,∙ +2 ∕A7 2ι tends to be not infinitely small leads to the conclusion that u and dx ∖∣ dθ are finite at the equator. But on account of the symmetry of the system the co-latitudinal displacement ξ must vanish at the equator, and therefore x also. By (23), when∕=l, ⅛=2, p = sin 0, r 2 x=~( ⅛ + 2u cos 0 Y 4m ∖ dθ ∕
But we have just seen that this hypothesis makes u finite when v=l or 0=90°, and therefore at the equator 1 rftt c ∙x t ∙* x = τ =, a finite quantity.
4m dθ 1 j
Now symmetry necessitates a vanishing value of d ∖ x ∣ dθ at the equator. Thus the hypothesis that AΓ 2 i+ 2 ∕A 2 i tends to be not infinitely small is negatived, and we conclude that, on account of the symmetry of the motion, it is infinitely small for infinitely great values of i. This being established, let us write (36) in the form ìß (36 a).
F2i 2i 2 + 3i — (2i 2 + 6i)K2i+4 ∣ F2i+2
Hence by repeated application of (36a) we have r⅛ 2 ìß . ∕Λ 4. o,∙∖o ¾ ~ 2C ‘ + äÌ -2ÌÎ I I? l ¾i 1 11 t(i+l) 1 ÷S(<+l)lβ 4t + l) +0(1 + 1) 2(i+2) 2 + 3(i + 2)-&c. (37).
And we know that this is a continuous approximation to K2i+2 ∣ Fn, which must hold in order that the latitudinal velocity may vanish at the equator. Writing A r i = A 2i+2 ∕ A^ 2i ∙, all the N’s may be computed from the continued fraction (37). Then
A 2 =E, A 4 ∕E = A 1 , A 6 ∕E = A⅛ W 8 ∕E = W 1 2V 2 2V 3 , &c.
We cannot compute K a from K i , K s from K 6 , and so on; for, if we do, then, short of infinite accuracy in the numerical values, we shall be gradually led to successive values of the K's which tend to equality.
This process was followed by Laplace without explanation. It was attacked by Airy in his “Tides and Waves” (in Ency. Metrop. ) and by Ferrel in his Tidal Researches (U.S. Coast Survey, 1873), but was justified by Sir W. Thomson in the Phil. Mag. (1875, p. 230). The investigation given here is substantially Thomson’s.
Laplace gives numerical solutions for three different depths of the sea, τ ⅛ y , 7 ⅛ 5 , ∙∏⅛ of the earth’s radius. Since w∙ = ⅛r, these correspond respectively to the cases of /3=40, 10, 5, and the solutions are
/3=40, h=E{x 2 +20 ’ 1862j' 4 +10∙1164v 6 - 13∙1047v 8 -15*4488√ 10
-7∙458K 12 - 2∙1975p u - 0·450L· 1 « - 0'0687r 18
-0∙0082κ 2 °- 0O008χ 22 - 0Ό001» 24 ...}
/3=10, h = E{v 2 + 6∙1960j> 4 + 3 , 2474√ i + 07238j' 8 + 0O919p 1 ° + 0O076p 12 + 0O004p 14 ...}
/3= 5, h = E{χ 2 + 0∙7504^ + 0∙1566κ 6 + 0O157>' 8 + 0O009i' lo + ...}
Since h vanishes when v = 0, there is no rise and fall of water at the poles. When v = 1 at the equator, we find
/3=40, h= -7∙434E /3=10, h = 11∙267E /8= 5, h= l·924E.
The negative sign in the first case shows that the tide is inverted at the equator, giving low water when the disturbing body is on the meridian. Near the pole, however, that is, for small values of v, the tides are direct. In latitude 18° (approximately) there is a nodal line of evanescent semi-diurnal tide. In the second and third cases the tides are everywhere direct, increasing in magnitude from pole to equator. As /3 diminishes the tides tend to assume their equilibrium value, because all the terms, save the first, become evanescent. When /3=1 (depth 7 ⅛ of radius) the tide at the equator still exceeds its equilibrium value by 11 per cent As /3 diminishes from 40 to 10 the nodal line of evanescent tide contracts round the Çjle, and when it is infinitely small the tides are infinitely great, he particular value of /3 for which this occurs is that where the free oscillation of the ocean has the same period as the forced oscillation. The values chosen by Laplace were not well adapted for the illustration of the results, because in the cases of /3=40 and /3 = 10 the depth of the ocean is not much different from that value which would give infinite semi-diurnal tide. For values of ß greater than 40 we should find other nodal lines dividing the sphere into regions of direct and inverted tides. We refer the reader to Sir W. Thomson’s papers for further details on this interesting point.
§ 17. Tides of Long Period; Laplace's Argιιment from Friction.
In treating these oscillations ‰place remarks that a very small amount of friction will be sufficient to cause the surface of the ocean to assume at each instant its form of equilibrium, and he adduces in proof of his conclusion the considerations given below. The friction here contemplated is such that the integral effect is represented by a retarding force proportional to-the velocity of the water relatively to the bottom. Although proportionality to the square of the velocity would probably be nearer to the truth, yet Laplace’s hypothesis suffices for the present discussion.
In oscillations of this class the water moves for half a period north, and then for half a period south. In oscillating systems, where the resistances are proportional to the velocities, it is usual to specify the resistance by a modulus of decay, namely, that period in which a velocity is reduced to e^^ 1 of its initial value by friction. Now the friction contemplated by Laplace is such that the modulus of decay is short compared with the semi-period of oscillation. The quickest of the important tides of long period is the fortnightly (see chapter iv.); hence, for the applicability of Laplace’s conclusion, the modulus of decay must be short compared with a week. Now it seems practically certain that the friction of the bed of the ocean would not materially affect the velocity of a slow ocean current in a day or two. Hence we cannot accept Laplace’s discussion as satisfactory. However this may be, we now give λvhat is substantially his argument.
Let us write S for the reciprocal of the modulus of decay. Then the frictional forces introduced on the left-hand side of (17) are + Gdξ ∣ dt in the first and sin ΘSdη ∣ dt in the second. Laplace’s hypothesis with regard to the magnitude of the frictional forces enables us to neglect the terms d 1 Cdt- and sin 0 drr ∣∣ dP compared with the frictional forces. Then, if we observe that in oscillations of this class the motion is entirely latitudinal, equations (17) and (19) become Rξ η . a a dη g d . q Sf t - 2n sin 0 cos 0 √ = - j √h - r) dt dt adθ' j j sin ΘS⅛+2n cos0^∣=0 - (38).
i)a sin 0 + ^(γ⅜ sin 0) = 0
From the first two of these we easily obtain
(⅛ + ⅛ 2 cos 2 0)g=-^ rf (b- t ) (39).
As a first approximation we treat dξ ∣ dt as zero, and obtain fj=e, or the height of water satisfies the equilibrium theory. In these tides (see chap, iv.) C = E (⅜-cos 2 0) cos it, so that from the third equation of (38) we can obtain a first approximation to ξ; then, substituting in (39), we obtain on integration a second approximation to lj. Laplace, however, considers as adequate the first approximation, which is simply the conclusion of the equilibrium theory.
§ 18. Tides of Long Period in an Ocean of Uniform Depth.
As it seems certain that these tides do not satisfy even approximately the equilibrium law, we now proceed to find the solution where there is no friction. In the case of these tides k= 0,∕ a small fraction, and e=E (⅜-cos 2 0). The equation (24) then becomes s ι⅛ ⅛fc⅞) +<≡ (·>+«)=» i y 2 -cos 2 0 z or, writing μ for cos θ and e=E (⅜ - μ τ }, ½-C⅛⅛ ⅞ u ) = 4w*(u+e) (40).
dμ ∖ μ^-f 2 dμj
We shall confine the investigation to the case where y = l, a constant, and where the sea covers the whole surface of the globe. 'Γhe symmetry of the motion in this case demands that u when expanded in a series of powers of μ shall only involve even powers. Let us assume, therefore, that Tμ= B1A + b ≠ + *” + + (41) · τhel1 ⅛≡7≡ Tμ = + (B * " B1>S + · · + (Ba+1 “ δ2i - 1 ) μ * i+1 + * * · H⅛^∣)= b > +s < b ∙- b >> ,+ ∙∙∙
+ (2Ï+1) (B a+1 - B 2i - 1 > 2i + (42).
Again, g= -∕ 2 B 1 μ+(B 1 -∕ 2 B 3 > 3 + ... + (B 2i -ι -∕¾ +1 > 2i + 1 + (43), u=C-V s Bpχ 2 + 4(B 1 -√ 2 B 3 ><+ ... +^(B 2i -j-∕ 2 B 24 -ι> 2i + (44), where C is a constant. Then, writing ß for ima ∣ l, as in the case of the semi-diurnal tide, substituting fιom (42), (43), and (44) in (40), and equating to zero the successive coefficients of the powers of μ, we find C = - ⅞E + B 1 ∕ ß -
B s -B 1 (l-⅛^) + ⅛E=0 ∣
I⅛ t l - ‰-ι(l - 2 i(2i + J) ) - 2if+ 1) b ≈- ’ = 0
Thus the constant C and B 3 , B δ , &c., are all expressible in terms of B 1 , and B 1 is apparently indeterminate. We may remark that, if
-⅛. 1 =⅛∕3E, or B- 1 =-2E, z.o the equation of condition (45) may be held to apply for all values of i, from one to infinity. Let us write (45) in the form
⅛±l-] _ —J f2ß . ß β 2∙-3 ∕ 4 βX
B a -ι^ 2i(2i+l∕ 2ι(2i+1) I⅛-ι λ
λVhen i is large Boj+i/Bm-i either tends to become infinitely small or it does not do so. Let us suppose that it does not tend to become infinitely small. Then it is obvious that the successive B’s tend to become equal to one another, and so also do the values of (B 24 - 2 -∕ 2 B 9i - 1 ) ∕2ι and the coefficients of dvι ∣ dμ, Hence we have dvjdμ= L + JP∕(l-μ 2 ), for all values of μ, where L and Μ are finite. Hence this hypothesis gives infinite velocity to the fluid at the pole, where μ=l. But with a water-covered globe this infinite velocity is impossible, and therefore the hypothesis is negatived, and B2∣+1∕B2i-1 must tend to become infinitely small. This being established, let us write (46) in the form ß I⅛-1 _ 2z'(2ι + l) ∕ 4 ⅛∖
B2i-3 1 _ f i β _ ¾+ι
2ι(2i + l) B 2i -ι
By repeated applications of (47), we have in the form of a continued fraction ß
B 2 ,= 1 2i(2i + l) ß
B j - 3 ~ 1 _ f 2 ß + (23 + 2)(2i + 3) ß
2i(2ι + l) 1 f-ß (2i+4)(2i+5)
(2t+2)(2¾ + 3) + Pß ι 1 (2i+4)(2ï + 5) + & c ∙(48 )∙ And we know that this is a continuous approximation, which must hold in order to satisfy the condition that the water covers the whole globe. Let us denote this continued fraction by - A 7 i . Then, if we remember that B -1 = - 2E, we have
B 1 =2EA 7 1 , B 3 ∕B 1 = —-ZV 2 , B 5 ∕B 3 = — A γ 3 , B 7 ∕B 5 = — A 4 , &c., so that
B 3 = -2EA 1 A 7 2 , B 5 = 2EΛ' 1 Λ t √V 3 , B 7 = - 2EA τ W 2 A 3 A∙ 4 , &c., and C=-⅛E + 2EA√∕3.
Then h=u + e
= C + ⅜E-(E + ⅜∕ 2 B 1 W + ⅛(B 1 -∕≡B 3 )μ 4 + ⅛(B 3 -∕ 2 B 5 )∕4 6 + ...
= E {2N 1 ∕β - (1 +∕ 2 A' 1 )μ 2 + ⅜ W 1 (l +f 2 N 2 )μ* -iA 1 A s (l+∕∙W 3 )M≡+...} (49).
Now we find that, when /3=40, which makes the depth of the sea 3000 fathoms or - 5 - s ⅛-∏- of the radius of the earth, and with ∕=‘0365012, which is the value for the fortnightly tide (see chap, iv.),
Λ 7 1 = 3∙040692, Λ r 2 =l∙20137, A 7 3 = ∙66744, A 7 4 = ∙42819, W 5 =∙29819, Λ 7 6 =∙21950, A 7 7 =∙16814, W β =∙13287, Λ t 9 =∙107, A r 10 =∙l.
These values give 2A γ j ∕∕3=∙15203, 1+∕ 2 A 1 = 1O041, ⅛N 1 (1 +∕ 2 V 2 ) = 1'5228,
⅜ AΓ√V,(1 +∕ 2 W 3 ) = 1∙2187, ⅛A∙ 1 Λ γ 2 A^ 3 (1 +∕ 2 A 4 ) = ∙60988, ⅜W 1 ... W 4 (l +∕ 2 A t 5 ) = ∙20888, ⅞AΓ 1 ... Λ τ 5 (l +f 2 N β ) = ∙05190, jN 1 .. . A r β (l + ∕ 2 Λ γ 7 ) = ∙00976, ⅛N 1 .. . A 7 (l +∕ i Λ 7 8 ) = *0014,
⅜W 1 ... W 8 (l +HN 9 ) = Ό0017.
So that h∕e= {∙1520 - lO041μ a + l∙5228μ 4 - l'2187g β + ∙6099μ 8 - ∙2089μ w + O519√ 2 - ∙0098μ w + - 0014μ lδ - ∙0002μ 18 } ÷(⅜ - μ 2 ) (50).
At the pole, where μ=l, h= -Ex - 1037=e × ∙1556 ì
and at the equator, where μ —Q, h=+E× ∙1520 = e × ’ 4561 ∫ ' ’’
Now let us take a second case, where /3=10, which was also one of those solved for the case of the semi-diurnal tide by Laplace, and we find h∕E= ∙2363-lO016μ 2 + ∙5910μ 4 - ∙1627∕χ β + ∙0258g 8 - ∙0026∕χ 1 ° + ∙0002√ 2 .
At the pole, where g=l, we find h = - Ex ’ 3137 = ex ’471, and at the equator h=+E× ’ 2363 = e× '709. With a deeper ocean we should soon arrive at the equilibrium value for the tide, for Λ r 1 , A γ 2 , &c., become very small, and 2N 1 ∕β becomes equal to ⅜. In this case, with such oceans as those with which we have to deal, the tides of long period are considerably smaller than the equilibrium value.
§ 19. Stability of the Ocean.
Imagine a globe of density δ, surrounded by a spherical layer of water of density <r. Then, still maintaining the spherical figure, and with water still covering the nucleus, let the layer be displaced sideways. The force on any part of the water distant P from the centre of the water and r from the centre of the nucleus is ⅜ιrσr ’ towards the centre of the fluid sphere and ⅜π(δ - σ)r towards the centre of the nucleus. If δ be greater than σ there is a force tending to carry the water from places where it is deeper to places where it is shallower; and therefore the equilibrium, thus arbitrarily disturbed, is stable. If, however, δ is less than <r (or the nucleus lighter than water) the force is such that it tends to carry the water from where it is shallower to where it is deeper, and therefore the equilibrium of a layer of fluid distributed over a nucleus lighter than itself is unstable. As Sir William Thomson has remarked, if the nucleus is lighter than the ocean, it will float in the ocean with part of its surface dry. Suppose, again, that the fluid layer be disturbed, so that its equation is r=α(l +s i ), where s i is a surface harmonic of degree i ; then the potential due to this deformation is -∖ 7r0 ^ ‰X ι s i , and the whole potential is
2ι + l r*+l ’ 1
4τrδα 3 4τr<r aí÷3
8r ^^2r + l ri+l s *'
If, therefore, σ∕(2i + l) is greater than ⅛δ, the potential of the forces due to deformation is greater than that due to the nucleus. But we have seen that a deformation tends to increase itself by mutual attraction, and therefore the forces are such as to increase the deformation. If, therefore, o∙=∣(2t + l)δ, all the deformations up to the zth are unstable, but the ï + lth is stable. If, however, σ be less than δ, then all the deformations of any order are such that there are positive forces of restitution. For our present purpose it suffices that this equilibrium is stable when the fluid is lighter than the nucleus.
§ 20. Precession and Nutation.
Suppose we have a planet covered with a shallow ocean t and that the ocean is set into oscillation. Then, if there are no external disturbing forces, so that the oscillations are “free,” not “forced,” the resultant moment of momentum of the planet and ocean remains constant. And, since each particle of the ocean executes periodic oscillations about a mean position, it follows that the oscillation of the ocean imparts to the solid earth oscillations such that the resultant moment of momentum of the whole system remains constant. But the mass of the ocean being very small compared with that of the planet, the component angular velocities of the planet necessary to counterbalance the moment of momentum of the oscillations of the sea are very small compared with the component angular velocities of the sea, and therefore the disturbance of planetary rotation due to oceanic reaction is negligible. If now an externa! disturbing force, such as that of the moon, acts on the system, the resultant moment of momentum of sea and earth is unaffected by the interaction between them, and the processional and nutational couples are the same as if sea and earth were rigidly connected together. Therefore the additions to these couples on account of tidal oscillation are the couples due to the attraction of the moon on the excess or deficiency of λvater above or below mean sea-level. The tidal oscillations are very small in height compared with the equatorial protuberance of the earth, and the density of water is ⁴∕₁₁ths of that of surface rock; hence the additional couples are very small compared with the couples due to the moon’s action on the solid equatorial protuberance. Therefore precession and nutation take place sensibly as though the sea were congealed in its mean position. If the ocean be regarded as frictionless, the principles of energy show us that these insensible additional couples must be periodic in time, and thus the corrections to nutation must consist of semi-diurnal, diurnal, and fortnightly nutations of absolutely insensible magnitude. We shall have much to say below on the results of the introduction of friction into the conception of tidal oscillations as a branch of speculative astronomy.
§ 21. Some Phenomena of Tides in Rivers.
In § 2 we have given a description of some of the phenomena of the tide-wave in rivers. As a considerable part of our practical knowledge of tides is derived from observations in estuaries and rivers, we give an investigation of two of the most important features of the tide-wave in these cases. It must be premised that when the profile of a wave does not present the simple harmonic form it is convenient to analyse its shape into a series of partial waves superposed on a fundamental wave; and generally the principle of harmonic analysis is adopted, in which the actual wave is regarded as the sum of a number of simple harmonic waves.
The tide-wave in a river is a “long” wave in which the vertical motion of the water is very small compared with the horizontal, the river very shallow compared with the wave-length, and the water which is at any moment in a vertical plane always remains so throughout the oscillation.
Suppose that the water is contained in a straight and shallow canal of uniform depth; then take an origin of coordinates at the bottom, with the x axis horizontal in the direction of the canal, and the y axis vertical; let h be the undisturbed depth of water; let h + η be the ordinate of the surface corresponding to that fluid whose undisturbed abscissa is x and disturbed abscissa x + ξ ; and let g be gravity. The equations of motion and continuity are d⅜ d 2 ifdx 2 Λ dt 3 9 ∖ l+⅛ ∣ dxf ∖ ~-hdifdx f (52 )∙
η l+dξ∕dx J For brevity we shall write v i =gh and u = vt - x. Since for “long” waves dìfdx is small, the equations (52) become approximately dp-r d A l 6 dx) I , 58i 3--^ 4 √^V f h~ dx' ∖ dx) J
For finding a first approximation we neglect the second term on the right of each of (53). The solution is obviously ξ=acosmipt-x)=acosmu' ∖ ∙η= -mahsiηmu j ' 0 h
(54) gives the height of the water whose undisturbed abscissa is x, and since ξ is small this is approximately the height at the point on the bank whose abscissa is x. But now suppose that at the origin (the mouth of the river) the canal communicates with a basin in which there is a forced oscillation of water-level given by η=H sin nt (55).
the over-tide is τ ⅛th of the fundamental and has a range of 2 feet. If the river shallows very gradually, the formula will still hold, and we see that the height of over-tide varies as (depth) - !.
Fig. 1 read from left to right exhibits the progressive change of shape. The steepness of the advancing crest shows that it is a shorter time from low to high water than vice versa. The law of the ebb and flow of currents mentioned in § 2 may also be easily determined from the above investigation. We leave the reader to determine the effect of friction, which is given by inserting a term - μdξ ∣ dt on the right-hand side of (57).
(ii.) Compound Tides (see § 24).—We shall now consider the mutual influence of two waves of different periods travelling up the river together. In the first approximation they are quite independent, and we may assume ξ=a cos m{vt -x) + b cos [n(vt - x) + e] (61).
In proceeding to the second approximation, we only take notice of those terms which result from the interaction of the two, and omit all others, writing for the sake of brevity {m-nf =(m-n)(vt-x) - €, {m + n} ≈(rn + n)(νt-x) + e.
With the value of ξ assumed in (61), we find, on substituting in (53) and only retaining terms depending on mutual influence, that the equations for the second approximation are
This represents the oceanic tide, and n is that which we call below (§ 23) the speed of the tide. Then obviously m=n ∣ v, so that at any point x up the river η=Hsin n(t--5-Λ (56).
∖ ∖ ∕ gh∕
(56) gives the first approximation to the forced tide-wave, and it is clear that any number of oscillations may be propagated independently up the river with the velocity ∖ Jgh due to the depth of the river. In passing to the second approximation we must separate the investigation into two branches.
(i.) Over-Tides (see § 24).—We now suppose that the tide at the river mouth is simply (55). On substituting the approximate values (54) in (53) our equations become
â = ^â +|r2aWsin2wiM ì _ r (5 ').
h = ^^ dz∙ + ⅛ m,2β2 - ⅛ m2α2 cos % mu J
We have now to assume an appropriate form for the solution of (57), such as ξ=a cos mu + Ax cos 2mu + B sin 2mu (58).
We have here in effect assumed that the second and third terms of (58) are small compared with the first. It is clear, however, that at a distance from the origin the term in A will become large. This difficulty may be eluded by taking the canal of finite length, and supposing that, where the canal debouches into a second basin, a second appropriate forced oscillation is maintained. The length of the canal remains arbitrary, save that the second term of (58) shall still be small compared with the first. On substituting from (58) in (57) we have B indeterminate and A= - gα 2 jn 2 ; hence η ∣ h = ⅛m 2 a 2 -masiηmu + %m 3 a 2 xsin2mu + (2?nB- ⅛w 2 α 2 )cos2mw (59). This gives the elevation of the water whose undisturbed abscissa is x, that is to say, at the point whose abscissa along the bank is X=x + ξ. If we put x =X - ξ in the largest term of (59), and treat ⅛ as small, and put x=X in the small terms, (59) becomes ∙η ∣ h — - ma sin m(pt - X) + lm 3 a 2 X sin 2m(pt - X) + (2wιB - ∣τn 2 α 2 ) cos 2m(vt - X).
But at the origin (55) holds true, therefore B = 1 ⅞mα 2 , -rnah = II, and mν=n. Thus the solution is y=H sin n( t - 7?) + fΓf= i X sin 2n(t- ( 60 )∙
X χ∕gh' 4h ∖ fgh ∖ y/ght
From (60) we can see what the proper forced oscillation at the further end of the canal must be; but this matter has no present interest. The first term of (60) being called the fundamental, the second gives what is called the first over-tide; and by further approximation we can get the second, third, &c. The over-tide travels up the river at the same rate as the fundamental, but it has double frequency or “speed,” and the ratio of its amplitude to that
,., , . ... 3Ae nX of the fundamental is --,- —=.
As a numerical example, let the range of tide at the river mouth be 20 feet and the depth of river 50 feet. The “speed” of the semi-diurnal tide is about 1/1 "9 radians per hour; *fgh=27 miles per hour; hence ‰ — =·=õth V. Therefore 34 miles up the river 4h . Λ>λ 342
=Ιr 2 ^ + ¾v 2 α57nn[(wι + n)sin{7n + n} -(m-π)sin{τn-n}]
η /h = - abmn ∖ cos {m + η} - cos {τn - η} ] - dξ ∣ dx J Now let us assume as the solution ξ=acosm{vt-x) + Axcos{m+n} +Bsin{m + n} ] fβ3 , )∙
+ ά cos [τι(rt - ar) + e] + Car cos - n} +Dsin{m-n} J ’ "' ‘ ’
and let us elude the difficulty about the increasing magnitude of the second term in the same way as before. Substituting in the equation of motion, we have for all time,
2(rnι + n)A sin {m + n} + 2(m - n)C sin {m - n} + ^abmn[im + n) sin {m+n} - {m - n) sin {m - n} ] = 0.
This gives A = - ^abmn and C = + %abmn. B and D remain arbitrary as before, and will be dropped, because they are to be determined by the condition that at the origin the terms of dCdx in cos {wι + n}, cos{τn-n} are to vanish, whence ∙η ∣ h = - am sin m{vt -x}-bη sin [n(vt - x) + e]
+ ⅜abmn ∖ [m +w)aj sin {rn + n} - (m - n}x sin {m - n} ]
+ terms in cos {m + n} and cos {m - n}.
Then we pass from ar to A" as in the last section, and make the terms in cos {m + n} and cos {m — n} vanish by proper values of B and D, and we have η = amhsiηrn(pt - X) - bnhsin[n( ι vt - x) + e] + f abmηX[(m + n) sin {m + n} -{mn) sin {m - n} ] (64). Now at the river’s mouth, where x= 0, suppose that the oceanic tide is represented by n = 27 1 sin n 1 i + Ae sin in.ø + e).
Then — am=-Hfh , — bn = H 2 fh, habmn=H 1 H 2 fh, n 1 ±n 2 mv=n 1 , nv≈n 2 , v=fgh, m ÷ n= ’ so that (64) becomes
,=ι>⅛(l -i)t i i⅛[⅝(l .i) + .]
3∕7ι-∕∕o 4 - v ∙ · Γ∕ ∖Z ∖
+-⅛ ⅛- rs, "L t,,ι+ ^v-¼i) + 'J ⅛¾∙⅛γ< i ⅜->j(<-- ⅛)-∙1 - < 65 >∙
4Ä √gA L ∖ ∖ lgh∕
As a numerical example, suppose at the mouth of a river 50 feet deep that the solar semi-diurnal tide has a range 27Z 1 = 4 feet, and the lunar 2M 2 = 12 feet; then n 1 + n 2 =⅜⅜ radians per hour, n 1 - n 2 —-⅛ radians per hour, and as before ∖ fgh=2Ι miles per hour. With these figures ⅛H 1 Η 2 ra 1 +w 2 „ _ 1
4⅛ ≠⅛170'
Thus 15 miles up the river the quater-diurnal tide (in § 24 below, called MS) has a semi-range of an inch. But the luni-solar fortnightly tide (called MSf in § 24) would have a semi-range, of √ σ th of an inch. Where the two interacting tides are of nearly equal speed, the summation compound tide is very large compared with the differential tide. As before, when the river shallows gradually this formula will still hold.
It is interesting to note the kind of effect produced by these compound tides. When the primary tides are in the same phase n 1 <=n 2 Z + e.
Then (n 1 + n 2 )(i - + e=2π√-(n l + n 2 )^;
(n 1 -n 2 )(i-^)-e= -(π 1 -n 2 )ς^∙, and
3-fijlZ 2 7⅛ ~ t⅛ , {nι~ η j) γ 4A sm √ 9 Λ
Hence the front slope of the tide-wave is steeper at spring than at neap tide, and the compound tide shows itself in the form of an augmentation of the first over-tide; and the converse statements hold of neap tide. Also mean-water mark is lower and higher alternately up the river at spring tide, and higher and lower alternately at neap tide, by a small amount which depends on the differential tide. With the river which we were considering, the alternation would be so long that it would in actuality be either all lower or all higher.
IV. The Harmonic Analysis.
§ 22. Methods of applying Theory to Practice.
The comparison between tidal observations and tidal theories, and the formation of tables predicting the tidal oscillations of the sea, have been carried out in two different ways, which may be called the “synthetic” and the “analytic.”
The semi-diurnal rise and fall of tide with the weekly alternation of spring and neap would naturally suggest to the investigator to make his formula conform to the apparent simplicity of the phenomenon. He would seek to represent the height of water by either one or two periodic functions with a variable amplitude; such a representation is the aim of the synthetic method. That method has been followed by all the great investigators of the past, —Newton, Bernoulli, Maclaurin, Laplace, Lubbock, Whewell, Airy. Since at European ports the two tides which follow one another on any one day are nearly equal, or, in other words, there is scarcely a sensible diurnal tide, these investigators bestowed comparatively little attention to the diurnal tides. If these are neglected, the synthetic method is simple, for a single function suffices to represent the tide. In non-European ports, however, the diurnal tide is sometimes so large as to mask the semi-diurnal, and to make only a single instead of a double high water in twenty-four hours. To represent this diurnal tide in the synthetic method we are compelled to introduce at least one more function. There should also be a third function representing the tides of long period; but until the last few years these tides have scarcely been considered, and therefore we shall have little to say of them in explaining the synthetic method. The expression for the tide-generating forces due to either sun or moon consists of three terms, involving the declinations and hour-angles of the planet. One of these terms for each goes through its period approximately twice a day, a second once a day, and the third varies slowly (§ 7). The mathematical basis of the synthetic method consists of a synthesis of the mathematical formulae. The semi-diurnal term for the moon is fused with that for the sun, and the same process is carried out for the diurnal and slowly varying terms. A mass of tidal observation at a place where the diurnal tide is small, even if, as in all the older observations, it consists merely of heights and times of high and low water, soon shows that the fusion of two simple harmonic or periodic functions is insufficient to represent the state of tide; and the height and time of high water are found to need corrections for the variations of declination, of motion in right ascension, and of the parallaxes of both bodies.
But when continuous tide-gauges were set up far more extended data than those of the older observations became accessible to the investigator, and more and more corrections were found to be expedient to adapt the formulae to the facts. A systematic method of utilizing all the data became also a desideratum. This state of matters led. Sir W. Thomson to suggest the analytic method. It is true that the dynamical foundations of that method have always lain below the surface of the synthetic method, and have constantly been appealed to for the theoretical determination of corrections; nevertheless, we must regard the explicit adoption of the analytic method as a great advance. In this method we conceive the tidal forces or potential due to each disturbing body to be developed into a series of terms each consisting of a constant (determined by the elements of the planet’s orbit and the obliquity of the ecliptic) multiplied by a simple harmonic function of the time. Thus in place of the terms of the synthetic method for the three classes of tides we have an indefinitely long series of terms for each of the three classes. The loss of simplicity in the expression for the forces is far more than counterbalanced by the gain of facility for the discussion of the oscillations of the water. This facility arises from the great dynamical principle of forced oscillations, which we have explained in the historical sketch. Applying this principle, we see that each individual term of the harmonic development of the tidegenerating forces corresponds to an oscillation of the sea of the same period, but the amplitude and phase of that oscillation must depend on a network of causes of almost inextricable complication. The analytic method, then, represents the tide at any port by a series of simple harmonic terms whose period is determined from theoretical considerations, but whose amplitude and phase are found from observation. Fortunately the series representing the tidal forces converges with sufficient rapidity to permit us to consider only a moderate number of harmonic terms in the series.
Now it seems likely that the corrections which have been applied in the use of the synthetic method might have been clothed in a more satisfactory and succinct mathematical form had investigators first carried out the harmonic development. In this article we shall therefore invert history and come back on the synthetic method from the analytic, and shall show how the formulae of correction stated in harmonic language may be made comparable with them in synthetic language. One explanation is expedient before proceeding with the harmonic development. There are certain terms in the tide-generating forces of the moon, depending on the longitude of the moon’s nodes, which complete their revolution in 18’6 years. Now it has been found practically convenient, in the application of the harmonic method, to follow the synthetic plan to the extent of classifying together terms whose speed differs only in consequence of the movement of the moon’s node, and at the same time to conceive that there is a small variability in the intensity of the generating forces.
§ 23. Development of Equilibrium Theory of Tides in Terms of the Elements of the Orbits.
Within the limits at our disposal we cannot do more than indicate the processes to be followed in this development.
We have already seen in (3) that the expression for the moon’s tide-generating potential is
V = g i > 2 (COS 2 Z-⅜), and in (10) that cos 2 z - i = 2frM 1 M 3 +2⅛ 2 · ⅛⅛ + 2tfM 2 M 3 + 2∏M 1 M 3
3ξ 2 + √ ι -2^ Mi 2 +M 2 2 -2M 3 2 + 2 3 ' 3 ’
where M∣,M 2 ,M 3 are the direction cosines of the moon referred to axes fixed in the earth. We require to find the functions MjM 2 , ⅜(M 1 2 - M 2 2 ), &c., of the moon’s direction cosines. Let A, B, C (fig. 2) be the axes fixed in the earth, C being the north pole and AB the equator; let X, Y, Z be a second set of axes, XY being the plane of the moon’s orbit; Μ the projection of the moon in her orbit; Z=ZC, the obliquity of the lunar orbit to the equator; χ = AX = BCYj Z=MX, the moon’s longitude in her orbit measured from X, the intersection of the equator with the lunar orbit, hereafter called the “intersection.” Then
M 1 ≈ cos Z cos χ + sin Z sin χ cos Z1
M 2 = - cos Z sin χ + sin Z cos χ cos I j- (66).
M 3 = sinZsinZ J
Writing for brevity j>=cos⅜Z, j=sin⅛Z. (67), we find that
M 1 2 - M 2 2 ≈p i cos 2(χ - Z) + 2p 2 j 2 cos 2χ + q i cos 2(χ + Z) '
- 2M 1 M 2 =the same with sines in place of cosines
M 2 M 3 = -∕3'Cθs(χ-2Z)+jog'(^ 2 -<z 2 )cosχ+^7 3 cos(χ + 2Z) >(68). M 1 M s = the same with .sines in place of cosines
⅜ - M 3 2 = ⅜ (p i - 4p¾ 2 + q i ) + 2p 2 q 2 cos 2Z ,
These are the required functions of M 1 , M 2 , M 3 .
Now let c be the moon’s mean distance, e the eccentricity of the moon’s orbit, and let τ≈ z ~ (69).
Then, putting
(70), we have v ÷(Υ¾^= 2 ^ xy+2 ⅛^⅞ ς - 2 + 2 ^ yz + 2 ^ xz 3∣ 2 +, 2 -2^X 2 + Y 2 -2Z 2 , yn + 2 3 3 V ’ '
Corresponding to the definition of a simple tide given in § 1, the expression for each term of the tide-generating potential should consist of a solid spherical harmonic, multiplied by a simple timeharmonic. In (71) p¾, p 2 (f 2 -η 2 ), &c., are solid spherical harmonics, and in order to complete the expression for V it is necessary to develop the five functions of X, Y, Z in a series of simple timeharmonics. But (71) may be simplified in such a way that the five functions are reduced to three. The axes fixed in the earth may be taken, as in § 7, to have their extremities as follows :—the axis C the north pole, the axis B 90° E. of A on the equator, and the axis A on the equator in the meridian of the place of observation. Thus, if λ be the latitude of that place, we have
ξ=cosλ, η=Q, f=sinλ,
Then, writing a for the earth’s radius at the place of observation, (71) becomes
V = (Γ⅞≠ i co ≡ 2 λ < χ2 “γ8 >+ sin 2λ xz l e +⅜(⅜-sin 2 λ)⅜(X 2 +Y 2 -2Z 2 )] (71a).
The process of developing the three functions of X, Y, Z consists in the introduction of the formulae of elliptic motion into (66) and (70), the subsequent development of the X-Y-Z functions in a series of trigonometrical terms, and the rejection of terms which appear numerically to be negligible. The terms depending on the principal lunar inequalities—evection and variation— are also introduced. Finally, the three X-Y-Z functions are obtained as a series of simple time-harmonics, with the arguments of the sines and cosines linear functions of the earth’s rotation, the moon’s mean motion, and the longitude of the moon’s perigee. The next step is to pass, according to the principle of forced oscillations, from the potential to the height of tide generated by the forces corresponding to that potential The X-Y-Z functions being simple time-harmonics, the principle of forced vibrations allows us to conclude that the forces corresponding to V in (71a) will generate oscillations in the ocean of the same periods and types as the terms in N, but of unknown amplitudes and phases. Now let X 2 -∣∣ 2 , ≡2, ⅛(≡ 2 + g 2 - 2Z 2 ) be three functions having respectively similar forms to those of
X 2 -Y 2 XZ , 1(X 2 + Y 2 -2Z 2 ) (1-e 2 ) 3 ’ (1 - e 2 ) 3 ’ 3 (1 - e 2 ) 3 ’
but differing from them in that the argument of each of the simple time-harmonics has some angle subtracted from it, and that the term is multiplied by a numerical factor. Then, if g be gravity and h the height of tide at the place of observation, we must have h = y ⅛ cos 2 λ(≡ 2 -g 2 ) + sin 2λ 3iZ + ⅜(⅛ -sin 2 λ)⅛(≡ 2 + g 2 - 22 2 )] (72). The factor τa 2 ∕g may be more conveniently written 1^")% where JZis the earth’s mass. It has been so chosen that, if the equilibrium theory of tides were fulfilled, with water covering the whole earth, the numerical factors in the J-g-Z functions would be each unity and the alterations of phase would be zero. The terms in ⅛(≡ 2 + g 2 - 22 2 ) require special consideration. The function of the latitude being ⅜-sin 2 λ, it follows that, when in the northern hemisphere it is high water north of a certain critical latitude, it is low water on the opposite side of that parallel; and the same is true of the southern hemisphere. It is best to adopt a uniform system for the whole earth, and to regard high tide and high water as consentaneous in the equatorial belt, and of opposite meanings outside of the critical latitudes. We here conceive the function always to be written ⅜ - sin 2 λ, so that outside of the critical latitudes high tide is low water. We may in continuing the development write the X-g-2í functions in the form appropriate to the equilibrium theory, with water covering the whole earth; for the actual case it is only then necessary to multiply by the reducing factor, and to subtract the phase alteration κ. As these are unknown constants for each place, they would only occur in the development as symbols of quantities to be deduced from observation. It will be understood, therefore, that in the following schedules the “argument” is that part of the argument which is derived from theory, the true complete argument being the “argument” - κ, where κ is derived from observation.
Up to this point we have supposed the moon’s longitude and the earth’s position to be measured from the intersection; but in order to pass to the ordinary astronomical formulae we must measure the longitude and the earth’s position from the vernal equinox. Hence we determine the longitude and right ascension of the intersection in terms of the longitude of the moon’s node and the inclination of the lunar orbit, and introduce them into our formulae for the ï-g-Z functions. The expressions for the functions corresponding to solar tides may be written down by symmetry, and in this case the intersection is actually the vernal equinox.
The final result of the process sketched is to obtain a series of terms each of which is a function of the elements of the moon’s or sun’s orbit, and a function of the terrestrial latitude of the place of observation, multiplied by the cosine of an angle which increases uniformly with the time. We shall now write down the result in the form of a schedule; but we must first state the notation employed :— e, e l = eccentricities of lunar and solar orbits; Z, ω= obliquities of equator to lunar orbit and ecliptic; p,p l == longitudes of lunar and solar perigees; σ,σ, = hourly increments of p, p l ; s, h— moon’s and sun’s mean longitudes; σ, ∙η= hourly increments of s, h; t= local mean solar time reduced to angle; y - η = 15° per hour; λ = latitude of place of observation; ξ, v =longitude in lunar orbit, and R.A. of the intersection; N= longitude of moon’s node; i= inclination of lunar orbit. The speed of any tide is defined as the rate of increase of its -argument, and is expressible, therefore, as a linear function of γ, η, σ, σ; for wé may neglect σ, as being very small.
The following schedules, then, give h the height of tide. The arrangement is as follows. First, there is a universal coefficient which multiplies every term of all the schedules. Secondly, there are general coefficients, one for each schedule, viz., cos 2 λ for the semi-diurnal terms, sin 2λ for the diurnal, and ⅜ - ⅜ sin 2 λ for the terms of long period. In each schedule the third column, headed “coefficient,” gives the functions of Zand e (and in two cases also of p). In the fourth column is given the mean semi-range of the corresponding term in numbers, which is approximately the value of the coefficient in the first column when Z=ω; but we pass over the explanation of the mode of computing the values. The fifth column contains arguments, linear functions of t, h, s,p, v, ξ. In [A, i.] 2t + 2(h-v') and in [A, ii.] t + (h-v) are common to all the arguments. The arguments are grouped in a manner convenient for subsequent computation. Lastly, the sixth is a column of speeds, being the hourly increase of the arguments in the preceding column, estimated in degrees per hour. It has been found practically convenient to denote each of these partial tides by an initial letter, arbitrarily chosen. In the first column we give a descriptive name for the tide, and in the second the arbitrarily chosen initial. In some cases no initial has been chosen, and here we indicate the tide by the analytical expression for its speed, or hourly increase of argument.
The schedule for the solar tides is drawn up in precisely the same manner, the only difference being that the coefficients are absolute constants. The eccentricity of the solar orbit is so small that the elliptic tides may be omitted, except the larger elliptic semidiurnal tide. In order that the comparison of the importance of the solar tides with the lunar may be complete, the same universal coefficient a is retained, and the special coefficient for each term is made to involve the factor —. Here τ z =⅜- 3 , m i being the sun’s mass. With
-=81∙5, — = ’ 46035 = o - .
m τ 2Ί7226
To write down any term, take the universal coefficient, the general coefficient for the class of tides, the special coefficient, and multiply by the cosine of the argument. The result, taken with the positive sign, is a term in the equilibrium tide, with water covering the whole earth. The transition to the actual case by the introduction of a factor and a delay of phase (to be derived from observation) has been already explained. The sum of all the terms is the complete expression for the height of tide h.
Schedule of Lunar Tides.
[A, i.]—Universal Coefficient = iit ∙
Semi-diurnal Tides; General Coefficient=cos 2 λ. [table]
[A, ii.]—Diurnal Tides; General Coefficient = sin 2λ. [table]
[A, iii. ]—Long Period Tides; General Coefficient ⅜ - ⅞ sin 2 λ. [table]
[B.]— Schedule of Solar Tides.
Solar Tides; Universal Coefficient a. 2m ∖ cJ [table]
From the fourth columns we see that the coefficients in de-Scale of scending order of magnitude are M 2 , K 1 (both combined), S 2 , import-θ, Ki (lunar), N, P, K 1 (solar), K 2 (both combined), K 2 (lunar), Mf, ance of Q, Mm, K 2 (solar), Ssa, v, M 1 , J, L, T, 2N, μ., 00, 3<τ-σ, tides, γ- 3σ- σ + 2i;, γ- 4<r + 2cτ, σ-2η + τs, 2(σ-ιj), λ.
The tides depending on the fourth power of the moon’s parallax arise from the potential V=^-p 3 (⅞ cos 3 z - % cos 2). They give rise to a small diurnal tide M 1 , and to a small ter-diurnal tide M 3 ; but we shall not give the analytical development.
§ 24. Meteorological Tides, Over-Tides, and Compound Tides.
All tides whose period is an exact multiple or submultiple of a mean solar day, or of a tropical year, are affected by meteorological conditions. Thus all the tides of the principal solar astronomical series S, with speeds γ-17, 2(γ-τ∕), 8(7-17), &c., are subject to more or less meteorological perturbation. An annual inequality in the diurnal meteorological tide S 1 will also give rise to a tide*γ - 2η, and this will be fused with and indistinguishable from the astronomical P; it will also give rise to a tide with speed 7, which will be indistinguishable from the astronomical part of K 1 . Similarly the astronomical tide K 2 may be perturbed by a semi-annual inequality in the semi-diurnal astronomical tide of speed 2(7-17). Although the diurnal elliptic tide S 1 or 7 -17 and the semi-annual and annual tides of speeds 2η and 17 are all probably quite insensible as arising from astronomical causes, yet they have been found of sufficient importance to be considered. The annual and semiannual tides are of enormous importance in some rivers, representing in fact the yearly flooding in the rainy season. In the reduction of these tides the arguments of the S series are t, 2t, 3f, &c., and of the annual, semi-annual, ter-annual tides h, 2h, Sh. As far as can be foreseen, the magnitudes of these tides are constant from year to year.
We have in § 21 considered the dynamical theory of over-tides. The only tides of this kind in which it has hitherto been thought necessary to represent the change of form in shallow water belong to the principal lunar and principal solar series. Thus, besides the fundamental astronomical tides M 2 and S 2 , the over-tides M 4 , M β , M 8 , and S 4 , S 6 have been deduced by harmonic analysis. The height of the fundamental tide M 2 varies from year to year, according to the variation in the obliquity of the lunar orbit, and this variability is represented by the coefficient cos 4 ⅜Z. It is probable that the variability of M 4 , M 6 , M 8 will be represented by the square, cube, and fourth power of that coefficient, and theory (§ 21) indicates that we should make the argument of the over-tide a multiple of the argument of the fundamental, with a constant subtracted.
Compound tides have been also considered dynamically in § 21. By combining the speeds of the important tides, it will be found that there is in many cases a compound tide which has itself a speed identical with that of an astronomical or meteorological tide. We thus find that the tides 0, K 1 , Mm, P, M 2 , Mf, Q, M 1 , L are liable to perturbation in shallow water. If either or both the component tides are of lunar origin, the height of the compound tide will change from year to year, and will probably vary proportionally to the product of the coefficients of the component tides. For the purpose of properly reducing the numerical value of the compound tides, we require not merely the speed, but also the argument. The following schedule gives the adopted initials, argument, and speed of the principal compound tides. The coefficients are the products of those of the two tides to be compounded.
[C.]— Schedule of Compound Tides. [table]
§ 25. On the F o rm o f Presentation of R e sults of Tid a l Obs e rv a ti o n s.
Supposing n to be the speed of any tide in degrees per mean solar hour, and t to be mean solar time elapsing since 0 h of the first day of (say) a year of continuous observation, then the immediate result of harmonic analysis is to obtain A and B, two heights (estimated in feet and tenths) such that the height of this tide at the time t is given by A cos nt + B sin nt. If we put R=√(A 2 + B 2 ) and tan ζ = B∕A, then the tide is represented by
R cos (nt - ζ).
In this form R is the semi-range of the tide in British feet, and ζ is an angle such that ζ ∣ n is the time elapsing after 0 h of the first day until it is high water of this particular tide. It is obvious that ζ may have any value from 0 i to 360°, and that the results of the analysis of successive years of observation will not be comparable with one another when presented in this form.
But let us suppose that the results of the analysis are presented in a number of terms of the form fHcos(V+u-κ), where V is a linear function of the moon’s and sun’s mean longitudes, the mean longitude of the moon’s and sun’s perigees, and the local mean solar time at the place of observation, reduced to angle at 15° per hour. V increases uniformly with the time, and its rate of increase per mean solar hour is the n of the first method, and is called the speed of the tide. It is supposed that u stands for a certain function of the longitude of the node of the lunar orbit at an epoch half a year later than 0 h of the first day. Strictly speaking, u should be taken as the same function of the longitude of the moon’s node, varying as the node moves; but, as the variation is but small in the course of a year, u may be treated as a constant and put equal to an average value for the year, which average value is taken as the true value of u at exactly mid year. Together V+ u constitute that function which has been tabulated as the “argument” in the schedules of § 23. Since V+ u are together the whole argument according to the equilibrium theory of tides, with sea covering the whole earth, it follows that κ∕n is the lagging of the tide which arises from kinetic action, friction of the water, imperfect elasticity of the earth, and the distribution of land. It is supposed that H is the mean value in British feet of the semi-range of the particular tide in question; f is a numerical factor of augmentation or diminution, due to the variability of the obliquity of the lunar orbit. The value of f is the ratio of the “coefficient” in the third column of the preceding schedules to the mean value of the same term. For example, for all the solar tides f is unity, and for the principal lunar tide M 2 it is equal to cos 4 ⅜Z÷cos 4 ⅜ω cos 4 ⅜i; for the mean value of this term has a coefficient cos 4 ⅜ω cos 4 ⅜i. It is obvious, then, that, if the tidal observations are consistent from year to year, H and κ should come out the same from each year’s reductions. It is only when the results are presented in such a form as this that it will be possible to judge whether the harmonic analysis is yielding satisfactory results. This mode of giving the tidal results is also essential for the use of a tide-predicting machine (see § 38).
We must now show how to determine H and κ from R and ζ. It is clear that H = R∕f, and the determination of f from the schedules depends on the evaluation of the mean value of each of the terms in the schedules, into which we shall not enter. If V o be the value of V at 0 h of the first day, then clearly ~ζ= V 0 +u-κ, so that κ=ζ+ V 0 +u.
Thus the rule for the determination of κ is: Add to the value of f the value of the argument at O⅛ of the first day.
The results of harmonic analysis are usually tabulated by giving Η, κ under the initial letter of each tide; the results are thus comparable from year to year. For the purpose of using the tide-predicting machine the process of determining H and κ from R and ζ has simply to be reversed, with the difference that the instant of time to which to refer the argument is 0 h of the first day of the new year, and we must take note of the different value of u and f for the new year. Tables have been computed for f and u for all longitudes of the moon’s node and for each kind of tide, and the mean longitudes of moon, sun, and lunar perigee may be extracted from any ephemeris. Thus when the mean semi-range H and retardation κ of any tide are known its height may be computed for any instant. The sum of the heights for all the principal tides of course gives the actual height of water.
§ 26. Numerical Harmonic Analysis for Tides of Short Period.
The tide-gauge (described below, § 36) furnishes us with a continuous graphical record of the height of the water above some known datum mark for every instant of time. The first operation performed on the tidal record is the measurement in feet and decimals of the height of water above the datum at every mean solar hour. The period chosen for analysis is about one year and the first measurement corresponds to noon.
If T be the period of any one of the diurnal tides, or the double period of any one of the semi-diurnal tides, it approximates more or less nearly to 24 m. s. hours, and, if we divide it into twenty-four equal parts, we may speak of each as a T-hour. We shall for brevity refer to mean solar time as S-time. Suppose, now, that we have two clocks, each marked with 360°, or 24 hours, and that the hand of the first, or S-clock, goes round once in 24 S-hours, and that of the second, or T-clock, goes round once in twenty-four T-hours, and suppose that the two clocks are started at 0° or 0 h at noon of the initial day. For the sake of distinctness, let us imagine that a T-hour is longer than an S-hour, so that the T-clock goes slower than the S-clock. The measurements of the tide curve give us the height of water exactly at each S-hour; and it is required from these data to determine the height of water at each T-hour. For this end we are, in fact, instructed to count T-time, but are only allowed to do so by reference to S-time, and, moreover, the time is always to be specified as an integral number of hours. Commencing with 0 h of the first day, we begin counting 0, 1, 2, &c., as the T-hand comes up to its hour-marks. But, as the S-hand gains on the T-hand, there will come a time when, the T-hand being exactly at the p hour-mark, the S-hand is nearly as far as p + ⅛. When, however, the T-hand has advanced to the p+ 1 hour-mark, the S-hand will be a little beyond p +1 + ⅜,—that is to say, a little less than half an hour before p + 2. Counting, then, in T-time by reference to S-time, we jump from p to p + 2. The counting will go on continuously for a number of hours nearly equal to 2jo, and then another number will be dropped, and so on throughout the whole year. If it had been the T-hand which went faster than the S-hand, it is obvious that one number would be repeated at two successive hours instead of one being dropped. We may describe each such process as a “change.”
Now, if we have a sheet marked for entry of heights of water according to T-hours from results measured at S-hours, we must enter the S-measurements continuously up to p, and we then come to a change; dropping one of the S-series, we go on again continuously until another change, when another is dropped; and so on. Since a change occurs at the time when a T-hour falls almost exactly half-way between two S-hours, it will be more accurate at a change to insert the two S-entries which fall on each side of the truth. If this be done the whole of the S series of measurements is entered on the T-sheet. Similarly, if it be the T-hand which goes faster than the S-hand, we may leave a gap in the T-series instead of duplicating an entry. For the analysis of the T-tide there is therefore prepared a sheet arranged in rows and columns; each row corresponds to one T-day, and the columns are marked 0 h , l h ,... 23 h ; the 0’s may be called T-noons. A dot is put in each space for entry, and where there is a change two dots are put if there is to be a double entry, and a bar if there is to be no entry. The numbers entered in each column are summed; the results are then divided, each by the proper divisor for its column, and thus the mean value for that column is obtained. In this way 24 numbers are found which give the mean height of water at each of the 24 special hours. It is obvious that if this process were continued over a very long time we should in the end extract the tide under analysis from amongst all the others; but, as the process only extends over about a year, the elimination of the others is not quite complete. The elimination of the effects of the other tides may be improved by choosing the period for analysis not exactly equal to one year.
Let us now return to our general notation, and consider the 24 mean values, each pertaining to the 24 T-hours. We suppose that all the tides except the T-tide are adequately eliminated, and, in fact, a computation of the necessary corrections for the absence of complete elimination, which is given in the Tidal Report to the British Association in 1872, shows that this is the case. It is obvious that any one of the 24 values does not give the true height of the T-tide at that T-hour, but gives the average height of the water, as due to the T-tide, estimated over half a T-hour before and naif a T-hour after that hour. A consideration of this point shows that certain augmenting factors, differing slightly from unity, must be applied. In the reduction of the S-series of tides, the numbers treated are the actual heights of the water exactly at the S-hours, and therefore no augmenting factor is requisite.
We must now explain how the harmonic analysis, which the use of these factors presupposes, is carried out
If t denotes T-time expressed in T-hours, and n is 15°, we express the height h, as given by the averaging process above explained, by the formula h=A 0 + A 1 cosπf+ B 1 sinni + A 2 cos 2nf + B 2 sin 2nt + ..., where t is 0, 1, 2, .. . 23. Then, if Σ denotes summation of the series of 24 terms found by attributing to t its 24 values, it is obvious that
A o = 2 h ∑h ' f A] = ∑hcos nt j B∣ = ^ 2 ∙ Σ hsιn nt; A 2 = t ⅛ ∑hcos 2nt; B 2 =∙ i ⅛ ∑hsin 2nt ; &c., &c.
Since n is 15° and t is an integer, it follows that all the cosines and sines involved in these series are equal to one of the following, viz., 0, ÷sin 15°, ±sin 30°, ÷sin 45°, ÷sin 60°, ±sin 75°, ±1. It is found convenient to denote these sines by 0, ±S j ,±S 2 ,÷S 3 ,±S 4 , ±S 5 ,÷1. The multiplication of the 24 h’s by the various S’s and the subsequent additions may be arranged in a very neat tabular form, like that given in a Report to the British Association in 1883. The A’s and B’s having been thus deduced, we have R= √(A 2 + B 2 ). R must then be multiplied by the augmenting factor. We thus have the augmented R. Next the angle whose tangent is B/A gives ζ. The addition to ζ of the appropriate V o + u gives κ, and the multiplication of R by the appropriate 1/f gives H. The reduction is then complete. An actual numerical example of harmonic analysis is given in the Admiralty Scientific Manual (1886) In the article “Tides”; but the process there employed is slightly different from the above, because the series of observations is supposed to be a short one.
§ 27. Harmonic Analysis for Tides of Long Period.
For the purpose of determining the tides of long period we have to eliminate the oscillations of water-level arising from the tides of short period. As the quickest of these tides has a period of many days, the height of mean water at one instant for each day gives sufficient data. Thus there will in a year’s observations be 365 heights to be submitted to harmonic analysis. To find the daily mean for any day we take the arithmetic mean of 24 consecutive hourly values, beginning with the height at noon. This height will then apply to the middle instant of the period from 0 h to 23 h , —that is to say, to ll h 30 m at night. The formation of a daily mean does not obliterate the tidal oscillations of short period, because none of the tides, except those of the principal solar series, have commensurable periods in mean solar time. A small correction, or “clearance of the daily mean,” has therefore to be applied for all the important tides of short period, except for the solar tides. Passing by this clearance, we next take the 365 daily means, and find their mean value. This gives the mean height of water for the year. We next subtract the mean height from each of the 365 values, and find 365 quantities δh, giving the daily height of water above the mean height. These quantities are to be the subject of the harmonic analysis, and the tides chosen for evaluation are those which have been denoted above as Mm, Mf, MSf, Sa, Ssa.
Let δh = A cos (<r-σ)i + B sin (σ∙-σ)U
+ C cos 2σt + D sin 2σt
+ C'cos2(ff-ι∕)i + D'sin2(<r-i7)i (73),
+ E cos ηt + F sin ∙ηt
+ G cos 2∙ηt +H sin2τ∕i where t is time measured from the first ll h 30 m . If we multiply the 365 δh ’ s by 365 values of cos (<r - τs)t and effect the summation, the coefficients of B,C,D, &c., are very small, and that of A is nearly 182⅜. Similarly, multiplying by sin (<τ - σ)i, cos 2σt, &c., we obtain 10 equations for A,B,C, &c., in each of which one coefficient is nearly 182⅜ and the rest small. These equations are easily solved by successive approximation. In this way A,B,C, &c., are found, and afterwards the clearance to which we have alluded is applied. Finally the cleared A,B,C, &c., are treated exactly as were the components of the tides of short period. Special forms and tables have been prepared for facilitating these operations.
V. Synthetic Method.
§ 28. On the Method and Notation.
The general nature of the synthetic method has been already explained; we now propose to develop the expressions for the tide from the result as expressed in the harmonic notation. If it should be desired to make a comparison of the results of tidal observation as expressed in the synthetic method with those of the harmonic method, or the converse, or to compute a tide-table from the harmonic constants by reference to the moon’s transits and from the declinations and parallaxes of sun and moon, the analytical expressions of the following sections are necessary.
In chapter iv. the mean semi-range and angle of retardation or lag of any one of the tides have been denoted by H and κ. We shall here, however, require to introduce several of the H’s and κ's into the same expression, and they must therefore be distinguished from one another. This may in general be conveniently done by writing as a subscript letter the initial of the corresponding tide; for example H wι , κ m will be taken to denote the H and κ of the principal lunar tide M 2 . This notation does not suit the K 2 and K 1 tides, and we shall therefore write H", κ" for the semi-diurnal K 2 , and H', κ' for the diurnal K 1 tide. These two tides proceed according to sidereal time and arise from the sun and moon jointly, and a synthesis of the two parts of each is effected in the harmonic method, although that synthesis is not explained in chapter iv. The ratio of the solar to the lunar part of the total K 2 tide is ∙46407; hence ’ 683 H" is the lunar portion of the total K 2 . There will be no occasion to separate the two portions of K 1 , and we shall retain the synthesis which is effected in the harmonic method.
§ 29. Semi-Diurnal Tides.
The process adopted is to replace the mean longitudes and elements of the orbit in each term of the harmonic development of the schedules of § 23 by hour-angles, declinations, and parallaxes.
At the time t (mean solar time of port reduced to angle) let a, δ, ψ be S’s R.A., declination, and hour-angle, and l J’s longitude measured from the ‘ ‘ intersection.” These and other symbols when written with subscript accent are to apply to the sun. Then v being the R.A. of the intersection, we nave from the right-angled spherical triangle of which the sides are l, δ, a-v the relations tan(α-v)=cosZtanζ sin δ=sin I sin l (74).
Now s-ξ is the J’s mean longitude measured from the intersection and s-p is the mean anomaly; hence approximately l=s-l∙ + 2esnι{s-ρ) (75).
From (74) and (75) we have approximately a=s+(v - £) + 2e sin (s -p) - tan 2 ⅛Z sin 2(s - ξ).
Now, h being the ©’s mean longitude, t + h is the sidereal hour-angle, and ψ = t + h-a.
Hence t + h - s - (y - ξ ) = ψ + 2e sin (s - p) - tan 2 ∣Zsin 2(s - ξ) (76).
Again, if we put cos 2 ∆=1 - ⅜ sin 2 Z (77), we have approximately from (74) and (75)
cos 2 δ - cos 2 ∆ n , ∙∖
whence ≡⅛¾- i, g=≡iπ2(s -{) f ’·
<rsur∆ dt J
Obviously Δ is such a declination that sin 2 Δ is the mean value of sin 2 δ during a lunar month. Again, if Τ’ be the ratio of the J’s parallax to her mean parallax, the equation to the ellipse described gives -(-P - l)=cos (s~ρ y ) ∣
. 1 dP . , . f (79)∙
Now it appears in schedule A of § 23 that the arguments of all the lunar semi-diurnal tides are of the form 2(i + A-v)±2(s-ξ) or ±(s-p). It is clear, therefore, that the cosines of such angles may by the relations (76), (78), (79) be expressed in terms of hour-angles, declinations, and parallaxes. Also by means of (77) we may introduce Δ in place of I in the coefficients of each term. An approximate formula for Δ is 16°'51 + 3°∙44 cos W-0° - 19 cos 2N. In the Report to the British Association for 1885, the details of the processes indicated are given.
Before giving the formula it must be remarked that the result is expressed more succinctly by the introduction of the symbol δ' to denote the J’s declination at a time earlier than that of observation by an interval which may be called the “age of the declina-tional inequality,” and is computed from the formula tan (κ"- κ m )∕2<τ or 52 h- 2tan {κ''- κ m ). Similarly, it is convenient to introduce P to denote the value of P at a time earlier than that of observation by the “age of the parallactic inequality,” to be computed from tan (κ m -κ n )∕(σ-ct) θr 105 h- 3 tan (κ m -κ n )- These two “ages” probably do not differ in general much from a third period, computed from {κ, — κ m )∕2(σ — η), which is called the “age of the tide.”
The similar series of transformations when applied to the solar tides leads to simpler results, because Δ, is a constant, being 16°∙33, and the “ages” may be treated as zero; besides the terms depending on dδjdt and dPfdt are negligible. If now we denote by h 2 the height of water with reference to mean water-mark, in so far as the height is affected by the harmonic tides M 2 , S 2 , K 2 , N, L, T, R, the harmonic expression is transformed into h 2 = H m cos (2≠ - κm) + H, cos (2≠, - κ,)
cos 2 'δ'-cos 2 ∆ , 6g3 h ,, cos ( _ k ,, sιn 2 ∆, cos 2 a - cosL δ,. 317h ,, cos ^φ t -κ-) τ sιn 2 ∆, τ, '
sinδcosδZδΓ - 683H" „ 2 Ί . . n , .
- van; ¾ L∞.(√'-,.j ^ h - un, ∙ i J«"<2≠-⅛) + ^(P--l) H · coa X H ' cos¾ eo s (2»-,) cos 2 ∆ z ecose x τ ,
+ (P, - 1) h *~-⅞ cos (2≠, - r,) i ∕
cos 2 Δ dPlddΓ ra H n H, -J. . nt woλx 4 -=-τ- ,—~—< 4H m e 7 î ; ∣sιn(2≠ - k, λ ) (80), ecos 2 ∆, (<rσ)L cos(κ m -κ β ) cosκ l -κ m )J r m h where e is an auxiliary angle defined by t3-∏ € —— Ί r ’ ττ f 81 )∙
H n cos κ n - ⅛ cos κι x ,
The first two terms are the principal tides, and the physical origin of the remaining small terms is indicated by their involving δ', δ,, dδ ∣ dt, P, P„ dP ∣ dL The terms in dδ∕dt and dP ∣ dt are generally smaller than the others.
The approximation may easily be carried further. But the above is in some respects a closer approximation than the expression from which it is derived, since the hour-angles, declinations, and parallaxes necessarily involve all the lunar and solar inequalities.
§ 30. Syntheses of Solar and of Lunar Portions of the Semi-Diurnal Tide.
Let us write
, r cos 2 ∆ ττ . cos 2 δ'- cos 2 Δ ββnττβ , « , m ⅛δ∕ h ∙∙ + — ⅛>a, ∙6S3H-<∞(√-⅛)
+ S2ζA (í ._ j, H s ∞⅛ ς ⅛∞ 5 , cm ( , _ , cos 2 Δ, ' e cos e ' ’ ’
ι cos 2 δ'-cos 2 ∆ βooττw . , „ .
z a = κ m + - i - τ - · 683 H sin (κ* - κ m )
SIU iΛ f ι cos 2 ∆ . τv ,. cos k_ - Hj cos κ l . , .
+ — 5 -τ-(P-l) — 3 ≡ sin (e-κ m ) cos 2 ∆, κ ' ecose x '
sinδcosδZδΓ ∙683H" ττ . 2 .∏
-√ap∆7 λ L cos(< ,, -⅛)~ · n -J
cos 2 ∆ dP∕dt Γ „ H n H t ~l e cos 2 Δ, ∣r-σ L m cos(κ m - κ n )”cos(κ l -κ m ) J ’
M l = H, + c ° s2δ ' 7^° s2δ ' ∙317H"+(P,-1) ⅛- Hr ;
' sιn 2 ∆, ' * ' e z
2/4, =κ, ...............................................................(82).
Since observation and theory agree in showing that κ β is generally very nearly equal to κ,, we are justified in substituting κ, for κ" in the small solar declinational term of (80) involving ‘317 H ff . Then, using (82) in (80), h 2 =Μ cos 2{ψ - ∕ι) + Μ, cos 2(≠, - μ l ) (83).
If the equilibrium theory of tides were true, each H would be proportional to the corresponding term in the harmonically developed potential. This proportionality holds nearly between tides of almost the same speed; hence, using the expressions in the column of coefficients in schedule [B, L], § 23 (with the additional tide R there omitted, but having a coefficient (τ,∕τ)⅜.⅜e, cos 4 ⅜ω, found by symmetry with the lunar tide L), and introducing Δ, in place of ω in the solar tides, we may assume the truth of the proportion g⅛ Wir- 1 ≡⅞S-H,
With this assumption, Μ, reduces to
Hence M,=P, 3 -^L∆h, (84).
' ' cos 2 ∆, v '
This is the law which we should have derived directly from the equilibrium theory, with the hypothesis that all solar semi-diurnal tides suffer nearly equal retardation. Save for meteorological influences, this must certainly be true.
. A similar synthesis of Μ cannot be carried out, because the considerable diversity of speed amongst the lunar tides makes a similar appeal to the equilibrium theory incorrect. It may be seen, however, that it would be more correct to write cos 2 δ' instead of cos 2 Δ in the coefficient of the parallactic terms in Μ and 2/4.
The three terms of Μ in (82) give the height of lunar tide with its declinational and parallactic corrections, and similarly the formula for μ in (82) gives its value and corrections.
If now τ denotes the mean solar time elapsing since the moon’s upper transit and γ the angular velocity of the earth’s rotation, it is clear that the moon’s hour-angle ψ=(y-dajdt}r', and, since Mcos2(¾i'-μ) is a maximum when φ=μ or differs from μ by 180°, it follows that μ ∣ (y-da ∣ dP) is the “interval” from the moon’s upper or lower transit to high water of the lunar tide. Since τ is necessarily less than 12 h , we may during the interval from transit to high water take as an approximation da ∣ dt=σ, the moon’s mean motion. Hence that interval is ∕4∕(7 — σ), or ⅛u hours nearly, when μ is expressed in degrees. Thus (82) for μ gives by its first term the mean interval for the lunar tide, and by the subsequent teπns the declinational and parallactic corrections.
We have said that the synthesis of Μ cannot be carried out as in the case of M,, but the partial synthesis below will give fairly good results. The proposed formula is m =^≡⅛ h -+ 52 ⅛⅛^∙ 683 h "∞≡; 2∕4=k ot +^^∞ s2δ .∙683 H"sin(κ"- κ m );
M,=P,⅛H, i
‘ cos 2 Δ, * 2/4, = κ, (85).
These formulae have been used in the example of the computation of a tide-table given in the Admiralty Scientific Manual (1886).
§ 31. Synthesis of Lunar and Solar Semi-Diurnal Tides.
Let A be the excess of P’s over Θ ’ s R. A., so that A = α-α,, ]
≠, = ≠ + A, j- (86).
and h 2 =M cos 2(γ-∕ι) + M, cos 2(≠ + A-∕c,) J
The synthesis is then completed by writing
H cos 2(∕4 - φ) = Μ + Μ, cos 2( A - /4, + /4), Hsin2(μ-≠) = Μ, sin2(A-μ,+μ), so that h a =H cos 2(≠- φ) (87).
Then H is the height of the total semi-diurnal tide and ≠∕(γ - da ∣ dt} or ≠∕(γ - <r) or ⅛ φ, when φ is given in degrees, is the “interval” from the moon’s transit to high water.
The formulae for H and φ may be written
H = √ {M 2 +M, 2 + 2MM, cos 2(A - /4,+/4)} λ
tan 2(u - ⅛) = M ' 8ip 2(A ~ μ -∙ F < 88 >∙
ιanz l ∕4 φ i M ÷ cos 2 ( A _ + jlt ) J
They may be reduced to a form adapted for logarithmic calculation. Since A goes through its period in a lunation, it follows that H and φ have inequalities with a period of half a lunation. These are called the “fortnightly or semi-menstrual inequalities” in the height and interval.
Spring tide obviously occurs when A=μ l - μ. Since the mean value of A is s - h (the difference of the mean longitudes), and since the mean values of μ and μ, are ⅜κ m , ⅜κ,, it follows that the mean value of the period elapsing after full moon and change of moon up to spring tide is (κ, - κ m )∕2(<r - if). The association of spring tide with full and change is obvious, and a fiction has been adopted by which it is held that spring tide is generated in those configurations of the moon and sun, but takes some time to reach the port of observation. Accordingly (κ, - κ m )∕2(<r - if) has been called the “age of the tide.” The average age is about 36 hours as far as observations have yet been made. The age of the tide appears not in general to differ veιy much from the ages of the declinational and parallactic inequalities.
In computing a tide-table it is found practically convenient not to use A, which is the difference of R. A.’s at the unknown time of high water, but to refer the tide to A o , the difference of R·A.’s at the time of the moon’s transit. It is clear that A o is the apparent time of the moon’s transit reduced to angle at 15° per hour. We have already remarked that φfiy - da ∣ dt} is the interval from transit to high water, and hence at high water
- w.
As an approximation we may attribute to all the quantities in the second term their mean values, and we then have σ-n
A = A 0 + -A
u y-σ and A-∕4,+∕4=A 0 -∕4,+^-^∕4=A 0 -∕4, + ∣g∕4 (90).
This approximate formula (90) may be used in computing from (88) the fortnightly inequality in the “height” and “interval.”
In this investigation we have supposed that the declinational and parallactic corrections are applied to the lunar and solar tides before their synthesis; but it is obvious that the process might be reversed, and that we may form a table of the fortnightly inequality based on mean values H m and H s , and afterwards apply corrections. Tlιis is the process usually adopted, but it is less exact. The labour of computing the fortnightly inequality, especially by graphical methods, is not great, and the plan here suggested seems preferable.
§ 32. Diurnal Tides.
These tides have not been usually treated with completeness in the synthetic method. In the tide-tables of the British Admiralty we find that the tides at some ports are “affected by diurnal inequality”; such a statement may be interpreted as meaning that the tides are not to be predicted by the information given in the so-called tide-table. The diurnal tides are indeed complex, and do not lend themselves easily to a complete synthesis. In the harmonic notation the three important tides are K 1 ,O,P, and the lunar portion of K 1 is nearly equal to O in height, whilst the solar portion is nearly equal to P. A complete synthesis may be carried out on the lines adopted in treating the semi-diurnal tides, but the advantage of the plan is lost in consequence of large oscillations of the amplitude through the value zero, so that the tide is often represented by a negative quantity multiplied by a circular function. It is best, then, only to attempt a partial synthesis, and to admit the existence of two diurnal tides.
We see from schedules [A, ii.] and [B, L], § 23, that the principal diurnal tides are those lettered O, P, K 1 . Of these K 1 occurs both for the moon and the sun. The synthesis of the two parts of K 1 is effected without difficulty, and the result is a formula for the total K 1 tide like that in [A, ii.], but with the v which occurs in the argument replaced by a different angle denoted as v'. If, then, we write
V 0 =i + Λ- 2s-v + 2ξ + ⅛ιr ∖ ,∩-∩
V'=i + Λ-√-⅜τr ∫ (° 1 )»
the three tides 0,K 1 ,P are written as follows :—
0 =f 0 H o cos (V 0 -Kø), K x =f H'cos(V'-√), P =-H j ,cos[V'-k , -(2A-√) + √-x j ,)] (92).
The last two tides have very nearly the same speed, so that we may assume κ' = κ p , and that H i > has the same ratio to H' as in the equilibrium theory. Now, in schedules [A, ii.], [B, ii.], § 23, the coefficient of K i , viz., H' (the sum of the lunar and solar parts), is •26522, and the coefficient of P, viz., H p , is Ό8775, so that H' — 3’023 H p , or say=3H p , Hence we have
K 1 + P=H' [f - ⅜ cos (2Λ - √)] cos (V' - √)
- H' ⅜ sin(2Λ - √) sin (V ’- κ').
If, therefore, we put
R' cos ≠=H' [f - ⅛ cos (2Λ- √)] ì , 93 .
R' sin ≠=⅜ H' sin (2A-√) J <- ydλ
K 1 + P=R ’ cos ( V'+ ψ - κ').
It is clear that ψ and R' have a semi-annual inequality, and therefore for several weeks together R' and ψ may be treated as constant.
Now suppose that we compute V o and V' at the epoch—that is, at the initial noon of the period during which we wish to predict the tides—and with these values put f 0 =κ 0 - V o at epoch, ζ ’ ≈κ ’- V' at epoch - ý.
Then the speed of Y o is γ-2<r, or 13° , 94303 per hour, or 360°-25°'3673 per day; and the speed of V' is γ, or 15°O410686 per hour, or 360° - 9856 per day. Hence, if t be the mean solar time on the (n+l)th day since the initial moment or epoch,
V o - Ko =360°n +13°∙943 t - ζ 0 - 25°·367n,
V' + ≠-√=360° τι + 15°∙041 t-f'+ 0°∙986‰
Therefore the diumal tides at time t of the (n + l)th day are given by 0=foH 0 cos[13°∙943t-f 0 -25°∙367n] ì , 0 , λ K 1 +P=R' cos [15°∙0411 - f+ 0°∙986n] J
If we substitute for t the time of high or low water as computed simply from the semi-diurnal tide, it is clear that the sum of these two expressions will give the diumal correction for height of tide at high or low water, provided the diurnal tides are not very large. If we consider the maximum of a function
A cos 2(t - α) + B cos n(t - β), where B is small compared with A and n is nearly unity, we see that the time of maximum is given approximately by t = α, with a correction δt determined from
-2A sin (2δt)-nB sin n(a-β)=0;
, x 180° nB . . ox or ot — . --. sin η(a - β). π 4A ' r,
In this way we find that the corrections to the time of high water from 0 and K 1 +P are
δt 0 = - 0 h ∙988ζ 1 sin [13°∙943t- f 0 - 25°·367n] )
δt'= -0 h ∙988( 1+—- ì gsin[15°∙041t-f + 0°∙986n] )
H denoting the height and t the time of high water as computed from the semi-diurnal tide. If t next denotes the time of low water the same corrections with opposite sign give the corrections for low water.
If the diumal tides are large a second approximation will be necessary. These formulae have been used in computing a tide-table in the example given in the Admiralty Scientific Manual (1886).
§ 33. Explanation of Tidal Terms in common use; Datum Levels.
The mean height at spring tide between high and low water is called the spring rise, and is equal to 2(H m + H,). The height between mean high-water mark of neap tide and mean low-water mark at spring tide is called the neap rise, and is equal to 2H m . The mean height at neap tide between high and low water is called the neap range ; this is equal to 2(H m - H,). Neap range is usually about one-third of spring range. The mean period between full or change of moon and spring tide is called the age of the tide ; this is equal to (x, - κ m )∕2(<r - η), or, if κ, - κ m be expressed in degrees, 0 h- 984 x (κ, - κ m ); κ, - κ m is commonly about 36°, and the age about 36 h . The period elapsing from the moon’s upper or lower transit until it is high water is called the interval or the lunitidal interval. The interval at full moon or change of moon is called the establishment of the port or the vulgar establishment. The interval at spring tide is called the corrected or mean establishment.
The mean establishment may be found from the vulgar establishment by means of the spring and neap rise and the age of the tide, as follows.
Let α be the age of the tide reduced to angle at the rate of 1°∙016 to the hour. Then the mean establishment in hours is equal to the vulgar establishment in hours, diminished by a period expressed in hours numerically equal to ⅛ of the angle whose tangent is H i sin α∕(H m +H i cos a), expressed in degrees. Also H,∕H m is equal to the ratio of the excess of spring rise over neap rise to neap rise.
The French have called a quantity which appears to be identical with H m + H p or half the spring rise, the unit of height, and then define the height of any other tide by a tidal coefficient.
The practice of the British Admiralty is to refer their soundings and tide-tables to “mean low-water mark of ordinary spring tides.” This datum is found by taking the mean of the low-water marks of such observations at spring tide as are available, or, if the observations are very extensive, by excluding from the mean such spring tides as appear to be abnormal, owing to the largeness of the moon’s parallax at the time or any other cause. The Admiralty datum is not, then, susceptible of exact scientific definition; but, when it has once been fixed with reference to a bench mark ashore, it is expedient to adhere to it, by whatever process it was first fixed.
It is now proposed to adopt for any new Indian tidal stations a low-water datum for the tide-table to be called “Indian low-water mark,” and to be defined as H m + H, + H' + H 0 below mean-water level. Although such a datum is not chosen from any precise scientific considerations, it is susceptible of exact definition, is low enough to exclude almost all negative entries from the table (a sine qua non for a good datum), and will differ but little from the Admiralty datum, however that may be determined. A valuable list of datum levels is given by Mr J. Shoolbred in a Report to the British Association in 1879.
§ 34. On the Reduction of Observations of High and Low Waler.
A continuous register of the tide or observation at fixed intervals of time, such as each hour, is certainly the best; but for the adequate use of such a record some plan analogous to harmonic analysis is necessary. Observations of high and low water only have, at least until recently, been more usual. Some care has to be taken with respect to these observations, for about high and low water an irregularity in the rise and fall becomes very noticeable, especially if the place of observation is badly chosen. Observations should therefore be taken every five or ten minutes for half an hour or an hour, embracing the time of high and low water. The time and height of high and low water should then be found by plotting down a curve of heights, and by taking as the true tide-curve a line which presents a sweeping curvature and smoothes away the minor irregularities. A similar but less elaborate process would render hourly observations more perfect. In the reduction the immediate object is to connect the times and heights of high and low water with the moon’s transits by means of the establishment, age, and fortnightly inequality in the interval and height. The reference of the tide -to the establishment is not, however, scientifically desirable, and it is better to determine the mean establishment, which is the mean interval from the moon’s transit to high water at spring tide, and the age of the tide, which is the mean period from full moon and change of moon to spring tide.
For these purposes the observations may be conveniently treated graphically. An equally divided horizontal scale is taken to represent the twelve hours of the clock of civil time, regulated to the time of the port, or—more accurately—arranged always to show apparent time by being fast or slow by the equation of time; this time-scale represents the time-of-clock of the moon’s transit, either upper or lower. The scale is perhaps most conveniently arranged in the order V, VI, . . ., XII, I . . . ∏∏. Then each interval of time from transit to high water is set off as an ordinate above the corresponding time-of-clock of the moon’s transit. A sweeping curve is drawn nearly through the tops of the ordinates, so as to cut off minor irregularities. Next along the same ordinates are set off lengths corresponding to the height of water at each high water. A second similar figure may be made for the interval and height at low water. In the curve of high-water intervals the ordinate corresponding to XII is the establishment, since it gives the time of high water at full moon and change of moon. That ordinate of high-water intervals which is coincident with the greatest ordinate of high-water heights gives the mean establishment. Since the moon’s transit falls about fifty minutes later on each day, in setting off a fortnight’s observations there will be about five days for each four times-of-clock of the upper transit. Hence in these figures we may regard each division of the time-scale I to II, II to III, &c., as representing twenty-five hours instead of one hour. Then the distance from the greatest ordinate of high-water heights to XII is called the age of the tide. From these two figures the times and heights of high and low water may in general be predicted with Trans., 1831 sq.) improved the method of Laplace by taking into account all the observed tides, and not merely those appertaining to certain configurations. He divided the observations into a number of classes. First, the tides are separated into parcels, one for each month; then each parcel is sorted according to the hour of the moon’s transit. Another classification is made according to declination; another according to parallax; and a last for the diurnal inequalities. This plan was followed in treating the tides of London, Brest, St Helena, Plymouth, Portsmouth, and Sheemess. Whewell (Phil. Trans., 1834 sq.) did much to reduce Lubbock’s results to a mathematical form, and made a highly important advance by the introduction of graphical methods by means of curves. The method explained above is due to him. Airy remarks of WhewelΓs papers that they appear to be “the best specimens of reduction of new observations that we have ever seen.”
VI. Tidal Instruments and Tidal Prediction.
§ 35. General Remarks.
Practical tidal work is divisible into the three stages of observa-vation, reduction of observations, and prediction.
The simplest observation is that of the height of water on a fair approximation. We find the time-of-clock of the moon’s upper or lower transit on the day, correct by the equation of time, read off the corresponding heights of high and low water from the figures, and the intervals being also read off are added to the time of the moon’s transit and give the times of high and low water. At all ports there is, however, an irregularity of heights and intervals between successive tides, and in consequence of this the curves present more or less of a zigzag appearance. Where the zigzag is perceptible to the eye, the curves must be smoothed by drawing them so as to bisect the zigzags, because these diurnal inequalities will not present themselves similarly in the future. When, as in some equatorial ports, the diurnal tides are large, this method of tidal prediction fails.
This method of working out observations of high and low water was not the earliest. In the Mécanique Celeste, bks. i and v., Laplace treats a large mass of tidal observations by dividing them into classes depending on the configurations of the tide-generating bodies. Thus he separates the two syzygial tides at full moon and change of moon and divides them into equinoctial and solstitial tides. He takes into consideration the tides of several days embracing these configurations. He goes through the tides at quadratures on the same general plan. The effects of declination and parallax and the diurnal inequalities are similarly treated. Lubbocκ (Phil. graduated staff fixed in the sea, with such allowance as is possible made for wave-motion. It is far better, however, to sink a tube into the sea, into which the water penetrates through small holes. The wave-motion is thus annulled. In this calm water there lies a float, to which is attached a cord passing over a pulley and counterpoised at the end. The motion of the counterpoise against a scale is observed. In either case the observations may be made every hour, which is preferable, or the times and heights of high and low water may be noted. We have explained in § 34 the methods of reducing the latter kind of observation. Although more appropriate for rough observations, this method is susceptible of great accuracy when carefully used. It has been largely superseded by the harmonic method, but is still adhered to by the British Admiralty. In more careful observations than those of which we are speaking the tidal record is automatic and continuous; the reduction may be, and probably at some future time will be, mechanical; and the prediction is so already. We shall therefore devote some space to general descriptions of the three classes of instrument. The harmonic reductions are at present (1887) actually done numerically, and in chapter iv. we have indicated the nature of the arithmetical processes.
§ 36. The Tide-Gauge.
The site for the erection of a tide-gauge depenαs on local circumstances. It should be placed so as to present a fair representation of the tidal oscillations of the surrounding area. A tank is generally provided, communicating by a channel with the sea at about 10 feet (more or less according to the prevalent surf) below the lowest low-water mark. In many cases on open coasts and frequently in estuaries the tank may be dispensed with. At any rate we suppose that water is provided rising and falling with the tide, without much wave-motion. The nature of the installation depends entirely on the circumstances of the case. A vertical pipe is fixed in the water in such a way as to admit it only through holes small enough to annul wave-motion and large enough to make no sensible retardation of its rise and fall in the pipe. The diameter of the pipe differs greatly in different instruments: sometimes that which we have described as the tank serves as the pipe, and sometimes the pipe alone dips into the sea. A cylindrical float, usually a hollow metallic box or a block of green-heart wood, hangs and floats in the pipe, and is of such density as just to sink without support. In Sir W. Thomson’s gauge the float hangs by a fine platinum wire, in Newman’s (used in India) by a metallic ribbon. In the latter a chain hangs at the bottom of the float of such weight that, whether the water be high or low, there is the same upward force on the float. It is necessary that the pull on the float should be constant, otherwise a systematic error is introduced between rising and falling water. The suspension wire is wrapped round a wheel, and imparts to it rotation proportional to the rise and fall of tide. By a simple gearing this wheel drives another, by which the range is reduced to any convenient extent. A fine wire wound on the final wheel of the train drags a pencil or pen up and down or to and fro proportionately to the tidal oscillations. The pencil is lightly pressed against a drum, which is driven by clockwork so as to make one revolution per day. The pen leaves its trace or tide-curve on paper wrapped round the drum. Generally, however, the paper is fixed to the drum, and the record of a fortnight may be taken without change of paper. An example of a tide-curve for Apollo Bunder, Bombay, from 1st to 15th January 1884, is shown in fig. 3. Sometimes the paper is in a long band, which the drum picks off from one coil and delivers on to another. The contact of the pen must be such that the work done in dragging it over the paper is small, otherwise a varying tension is thrown on to the float wire. Hence, if the friction is considerable, the float must be large.
The conditions necessary for a good tide-gauge appear to be better satisfied by Sir W. Thomson’s than by any other; but, as his instrument is recent, other forms have been much more extensively used, and have worked well. The peculiarity of Thomson’s tidegauge is that, by giving the drum an inclination to the vertical, the pressure of the pen on the paper and on its guides is very delicately regulated to the minimum necessary for effecting the purpose. In other gauges the drum has been either vertical or horizontal, and the amount of friction has necessarily been considerably greater.
§ 37. The Harmonic Analyser.
If a function H be expressed as a series of harmonic terms, and if one pair of these terms be A cos nt + B sin nt, then, if T be a multiple of the complete neriod 2τr∕π, we have
2 PΤ 9 Pτ h-=- 7j -, ∕ Hcosηtdt,P> = - ∕ 27^sin nt dt. dj 0 1 -∕o
Thus a machine which will effect these integrations will give A and B. Such a machine has been invented by Prof. James Thomson and perfected by Sir W. Thomson. In fig. 4 let TT' be a circular table, capable of rotation about the inclined shaft s. Let S be a sphere touching the table anywhere along its horizontal diameter. Let C be a cylinder, of somewhat smaller diameter than the sphere, capable of rotation about a horizontal axis parallel to the table, and touching the sphere so that CS is parallel to TT'. Suppose that the point of contact of the sphere with the table is distant x from the centre of the table, and nearer to us than the shaft; then, when the shaft s and the table TT' turn in such a direction that T rises from the paper and T' goes below it, the sphere will turn in the direction of its arrow. If the radius of the sphere is a, and that of the cylinder b, then, when the table turns through a small angle δθ, the sphere turns through xδθ ∣ a and the cylinder through xbθ ∣ b. This angle vanishes if S touches the table at the centre, and is reversed if the sphere be moved across to the other side of the centre. Also whilst the table is turning the sphere may be rolled backwards and forwards without rubbing, and thus transmits motion from the table to the cylinder without slipping. Now suppose the turning of the table is so constrained that 5θ=kcosψdψ, whilst x is constrained to be equal to the arbitrarily varying quantity H. Then the total angle turned through by the cylinder, as the machine runs, is proportional to/H cosfdψ. If we impart to the table a simple harmonic oscillatory motion, with a period proportional to the lunar half-day, whilst the sphere moves, relatively to the centre of the table, proportionately to the tide-heights on the same time-scale, then, at the end of a sufficient number of lunar days, we shall find that the total angle turned through by the cylinder is proportional to either the A or B component of the lunar semi-diurnal tide. An index, which points to a dial, may be fixed to the cylinder, so that the required result may be read off.
In the harmonic analyser the tide-curve diagram is wrapped on a drum, which is turned by one hand, whilst with the other a pointer is guided to follow the tide-curve. As the drum turns proportionately to mean solar time, appropriate gearing causes two tables to execute harmonic oscillations in phases at right angles, with lunar semi-diurnal period. At the same time a fork attached to the pointer guides the two spheres so that their distances from the centres of their tables are equal to the tide-height in the diagram. The indexes attached to the two cylinders give the two components of the lunar semi-diurnal tide, and the approximation improves the longer the tide-curve which is passed through the machine. Corresponding to each of the principal lunar and solar tides there are a pair of tables, spheres with guiding forks, and cylinders similarly geared, and there is another sphere and another table, which last always turns the same way and at the same rate as the drum, from which the mean height of water is determined. Such an instrument has been constructed under the supervision of Sir W. Thomson, but has not yet been put into practical use, so that we cannot say how it will compete with the arithmetical harmonic analysis. A similar, but less complex machine for the analysis of meteorological observations is in constant use in the Meteorological· Office in London, and is found to work well.
§ 38. The Tide-Predicting Instrument.
The first suggestion for instrumental prediction of tides was given, we believe, by Sir W. Thomson in 1872, and the instruments since made have been founded on the principles which he then laid down. Mr Edward Roberts bore a very important part in the first practical realization of such a machine, and a tide-predicter was constructed by Légé for the Indian Government under his direction. Thomson’s is the only instrument in Europe as yet in regular practical use for navigational purposes. It requires much skill and care in manipulation, and it has been ably worked by Mr Roberts for the production of the Indian tide-tables ever since its completion. We refer the reader to Sir W. Thomson’s paper on “Tidal Instruments,” in Inst. C.E., vol. lxv., and to the subsequent discussion, for a full account of the several instruments, and for details of the share borne by the various persons concerned in the realization of the idea.
Fig. 5 illustrates diagrammatically the nature of the instrument. A cord passes over and under a succession of pulleys, being fixed at one end and having at the other a pen which touches a revolving drum. If all the pulleys but one be fixed, and if that one executes a simple harmonic motion up and down, the pen will execute the same motion with half amplitude. If a second pulley be now given an harmonic motion, the pen takes it up also with half amplitude. The same is true if all the pulleys are in harmonic motion. Thus the pen sums them all up, and leaves a trace on the revolving drum. When the drum and pulleys are so geared that the angular motion of the drum is proportional to mean solar time, whilst the harmonic motions of the pulleys correspond in range and phase to all the important lunar and solar tides, the trace on the drum is a tide-curve, from which a tide-table may be constructed. The harmonic motion of the pulley is given by an arrangement indicated only in the case of the lower pulley in the figure. The pulley frame has attached to its vertical portion a horizontal slot, in which slides a pin fixed to a wheel. Suppose that whilst the drum turns through 15" the wheel turns through 28° ’ 984. Now a lunar day is 24'842 mean solar hours; hence as the drum turns through 15° × 24’842 the wheel turns through 24'842 × 28° - 984 or 720°. Thus, if the drum turns with an angular velocity proportional to solar time, the wheel turns with twice the angular velocity proportional to lunar time, and the pulley geared to the wheel executes lunar semi-diurnal harmonic oscillations. When the throw of the pin and its angular position on its wheel are adjusted so as to correspond with the range and phase of the observed lunar semidiurnal tide, the oscillation of the pulley remains rigorously accurate for that tide for all future time, if the gearing be rigorously accurate, and with all needful accuracy for some ten years of tide with gearing as practically constructed. The upper pulleys have to be carefully counterpoised as indicated. It has not been found that any appreciable disturbance is caused by the inertia of the moving parts, even when the speed of working is high, lhe pre-dicter of the India Office takes about four hours to run off a year’s tides, but greater speed seems attainable by modification in the gearing. The Indian instrument, in the store department at Lambeth, has pulleys for the following tides (see chap, iv.) :—M 2 , M 4 , M β , K 1 , S 1 , S 2 , O, N, P, K 2 , Q, v, J, L, λ, 2MS, 2SM, MS, Ssa, Sa.
§ 39. Numerical Harmonic Analysis and Prediction.
In chapter iv. we have discussed the application of the numerical harmonic method to a long series of hourly observations. An actual numerical example of this analysis, with modifications to render it applicable to a short series, such as a fortnight, is given in the Admiralty Scientific Manual, 1886, where also an example of the numerical and graphical prediction of the tides may be found. The formulae used are those given in chapter v.
VII. Progress of the Tide-Wave over the Sea, and the Tides of the British Seas.
§ 40. Meaning of Cotidal Lines.
Sufficient tidal data would of course give the state of the tide at everv part of the world at the same instant of time, and if we were where μ is in degrees. Therefore, if we draw over the ocean a succession of lines defined by equidistant integral values of the Greenwich time of high water, and if we neglect the separation of the moon from the sun in longitude in twelve hours, the successive lines will give the motion of the semi-diurnal tide-wave in one hour.
§ 41. Cotidal Lines of the World.
No recent révisai of cotidal lines has been made with the aid of the great mass of tidal data which is now being accumulated, and we therefore reproduce (fig. 6) the chart of the world prepared by Sir George Airy for his article on “Tides and Waves.” The parts of the world for which data are wanting are omitted. The Roman numerals upon the cotidal lines denote the hour in Greenwich time of high water on the day of new or full moon. Airy remarks (§§ 575-584) that the cotidal lines of the North Atlantic are accurately drawn, that those of the South Atlantic are doubtful, and in the Pacific east of New Zealand are almost conjectural. The embodiment of recent observations in a cotidal chart would necessitate some modification of these statements.
When a free wave runs into shallow water it travels with less velocity and its height is increased. This is observable in the flexure and crowding of the cotidal lines near continents and oceanic islands, as, for example, about the Azores, the Bermudas, and the coast of South America. The velocity of the tide-wave gives good information as to the depth of the sea. In the North Sea it appears to travel at about 45 miles an hour, which corresponds to a depth to follow the successive changes we should be able to picture mentally the motion of the wave over the ocean and the successive changes" in its height. The data are, however, as yet very incomplete and only a rough scheme is possible. A map purporting to give the progress of the tide-wave is called a map of cotidal lines. For a perfect representation three series of maps would be required, one for the semi-diurnal tides, a second for the diurnal tides, and. a third for the tides of long period. Each class of map would then show the progress of the wave for each configuration of the tidegenerators. But as yet the only cotidal maps made are those for the mean semi-diurnal tide, and only for the configuration of new and full moon. The knowledge of the tides is not very accurate throughout the world, and therefore in the maps which we give it is assumed that the same interval elapses at all places between new and full moon and spring tide.
At spring tide, as we have seen in (87) and (88), h 2 =(M+M,) cos 2 (ψ- μ), since A - μ, becomes then equal to - μ. As a rough approximation spring tide occurs when the moon’s transit is at one o’clock at night or in the day. We only assume, however, that it occurs simultaneously everywhere. Now let τ be the Greenwich mean time of high water, and I the E. long, in hours of the place of observation, then, the local time of high water being the time of the moon’s transit plus the interval, and local time being Greenwich time plus E. long., we have τ=μ∕(γ-σ)-Z=- j ⅛μ-Z + l h , of 140 feet, and we know that the depth along the deeper channel is greater and along the sides less than this. In the Atlantic the wave passes over 90° of latitude, from the southern to the northern one o’clock line, in twelve hours, that is at the rate of 520 miles an hour. If the Atlantic tide could be considered as a free wave generated by the Pacific tide, this velocity would correspond to a depth of 18,000 feet. Airy considers, however, that the Atlantic forms too large a basin to permit the neglect of the direct tidal action, and thinks that the tides of this ocean derive extremely little of their character from the Pacific.
“There is another consideration,” he says, “which must not be left out of sight. It is that, supposing the cotidal lines to be accurately what they profess to be—namely, the lines connecting all the points at which high water is simultaneous—they may, nevertheless, with a compound series of tide-waves, not at all represent the ridge of the tide-wave which actually runs over the ocean. Thus an eye at a great distance, capable of observing the swells of the tide-waves, might see one huge longitudinal ridge extending from the mouth of the Amazon to the sea beyond Iceland, making high water at one time from Cape de Verde to the North Cape, and at another time from Florida to Greenland, and another ridge transversal to the former, travelling from the coast of Guiana to the northern sea; and the cotidal lines which we have traced may depend simply on the combination of these waves. It does not appear likely that we can ever ascertain whether it is so or not; but it is certainly possible that the original waves may have these or similar forms; and if so it is vain for us to attempt entirety to explain the tides of the Atlantic.”
He sums up the discussion of the chart by saying :—
“Upon the whole, therefore, we are driven to the conclusion that we cannot at all explain the cause of the form of the cotidal lines in the ocean, so far as they have been traced with any probability. And, supposing us to know with tolerable certainty those corresponding to the semi-diurnal tide, we cannot at all predict those which should hold for the diurnal tide.”
§ 42. Cotidal Lines of the British Seas.
Fig. 7 shows the cotidal lines in the seas surrounding the British Islands. Here the lines refer to full moon and change of moon and not to spring tide. The small figures along different parts of the coast denote the extreme range of the tide in yards. This figure is from the same source as the preceding one, and we again reproduce a portion of Airy’s remarks.
“The tides in the English Channel claim notice as having been the subject of careful examination by many persons, English and
French. It appears that in the upper part of the Channel the water flows up the Channel nearly three hours after high water and runs down nearly three hours after low water (this continuance of the current after high water, if it last three hours, is called by sailors tide-aηd-half-tide ; if it last one hour and a half, it is called lide-and-quarter-ticle). On the English side of the Channel, especially opposite the entrance of bays, the directions of the currents turn in twelve hours in the same direction as the hands of a watch; on the French side they turn in the opposite direction. This is entirely in conformity with theory. The same laws are recognized as holding in the British [Bristol ?] Channel, and in the German or North Sea near the Scotch and English coasts.
“With regard to the Irish Channel we have only to remark that there is a very great difference in the height of the tide on the different sides, the tide on the east side being considerably the greater. They are also greater in the northern part (north of Wicklow on one side, and of Bardsey Island on the other side) than in the southern part. Between Wexford and Wicklow they are γery small.
“The tides of the German Sea present a very remarkable peculiarity. Along the eastern coast of England, as far as the mouth of the Thames, the tide-wave, coming from the Atlantic round the Orkney Islands, flows towards the south. Thus, on a certain day, it is high water in the Murray [Moray] Firth at eleven o’clock, at Berwick at two o’clock, at Flamborough Head at five o’clock, and so on to the entrance of the Thames. But on the Belgian and Dutch coasts immediately opposite, the tide-wave flows from the south towards the north. Thus, on the day that we have supposed, it will be high water off the Thames at eleven o’clock (the tide having travelled in twelve hours from the Murray Firth) and at Calais nearly at the same time; but at Ostend it will be at twelve, off The Hague at two, off the Helder at six, and so on.
“We believe that a complete explanation may be found in the arrangement of the great shoals of the North Sea. It must be remarked that (except within a very small distance of Norway) the North Sea is considerably deeper on the English side than on the German side; so much so that the tide-wave coming from the north runs into a deep bay of deep water, bounded on the west side by the Scotch and English coasts as far as Newcastle, and on the east side by the great Dogger Bank. As far as the latitude of Hull, the English side is still the deep one; and, though a species of channel through the shoal there allows an opening to the east, yet immediately on the south of it is the Wells Bank, which again contracts the deep channel to the English side. After this (that is, in the latitude of Yarmouth) the deep channel expands equally to both sides. It seems reasonable to conclude from this that the great set of north tide is on the English side of the North Sea, both between the Dogger Bank and England and between the Wells Bank and England (a branch stream of tide having been given off to the east between these two banks), and that any passage of tidewave over these banks may be neglected. Now this view is supported in a remarkable degree by the tidal observations on two dangerous shoals called the Ower and Leman, lying between Cromer and the Wells Bank, but nearer to the latter. It appears that on these shoals the direction of the tide-current revolves in the same manner as the hands of a watch, proving conclusively that the Ower and Leman are on the left hand of the main stream of tide (supposing the face turned in the direction in which the tide proceeds), or are on its eastern border, and therefore that the central stream is still nearer to the coast of Norfolk. From a point not far south of this we may suppose the tide to diverge in a fan-shaped form over the uniformly deep Belgian Sea. Along the English coast the wave will flow to the south; but it will reach the whole of the Belgian and Dutch coast at the same instant; and, if this tide alone existed, we doubt not that the time of high water would be sensibly the same along the whole of that coast.
“But there is another tide of great magnitude, namely, that which comes from the English Channel through the Straits of Dover. This also diverges, we conceive, in a fan form, affecting the whole Belgian Sea: the western part turns into the estuary of the Thames; the eastern part runs along the Dutch coast, producing at successive times high water (even as combined with the North Sea tide) along successive points of that coast from Calais towards the Helder. And this we believe to be the complete explanation of the apparently opposite tide-currents. The branch tide of the North Sea running between the Dogger Bank and the Wells Bank will assist in propagating the tide along the German coast from the Helder towards the mouth of the Elbe. We have gone into some detail in this explanation for the purpose of showing the importance of considering the form of the bottom in explanations of specific tides.
“A set of observations has been made by Captain Hewett (at the point A, fig. 7) on the depth and motion of the water at a strictly definite point a few miles south of the Wells Bank, near the middle of the Belgian Sea. The result was that the change of elevation of the surface was insensible, but that there was a considerable stream of tide alternately north-east and south-west (magnetic). The point in question corresponds pretty well to the intersection of the cotidal lines of nine o’clock of the North Sea tide and three o’clock of the English Channel tide (ordinary establishment), and these tides would therefore wholly or partially destroy each other as regards elevation. As regards the compound tide-stream, the greatest positive current from one tide will be combined with the greatest negative current of the other, and this will produce a stream whose direction agrees well with observation. At 3 h on the day of new moon, the North Sea tide would be running north (magnetic), and the English Channel tide would be running east, and therefore the compound current would be running north-east; at 9 h it would be running south-west. Both currents, and consequently the compound current, would cease at about 0 h , 6 h , &c., on the day of new moon; and, as the observations were made rather more than a day before new moon, the slack water would occur an hour or more before noon. The whole of this agrees well with the observations.”
VIII. Tidal Deformation of the Solid Earth.
§ 43. Elastic Tides.
The tide-generating potential varies as the square of the distance from the earth’s centre, and the corresponding forces act at every point throughout its mass. No solid matter possesses the property of absolute rigidity, and we must therefore admit the probable existence of tidal elastic deformation of the solid earth. The problem of finding the state of strain of an elastic sphere under given stresses was first solved by Lamé; he made, however, but few physical deductions from his solution. An independent solution was found by Sir W. Thomson, who drew some interesting conclusions concerning the earth.
His problem, in as far as it is now material, is as follows. Let a sphere, of radius a and density w, be made of elastic material whose bulk and rigidity moduli are k and n, and let it be subjected to forces due to a potential wr 2 S 2 per unit volume, where S 2 is a surface spherical harmonic of the second order. Then it is required to find the strain of the sphere. We refer the reader to the original sources for the methods of solution applicable to spherical shells and to solid spheres. In order to write Thomson’s solution we put r, λ, I for radius vector, latitude, and longitude, and p, μ, v for the corresponding displacements. Then the solution is as follows :—
''=im⅛0< 8βi - 3 ^÷ w ¾
(96) ·
For either tidal or rotational stresses ⅜=τ(sin 2 λ- ⅞), in the case of tides τ=fm∕c 3 , m and c being the moon’s mass and distance, and in the case of rotation τ= -⅛ω 2 , ω being the angular velocity about the polar axis. The equation to the surface is found by putting r=a+ρ, where in the expression for ρ we put r=a. Hence from (96) the form of surface is given by r-∙{ 1 + 1 ⅞i[ 1 + Γ⅛⅛ιΙ τ < 8i " 5λ ^ i >} ~< 97 >∙
In most solids the bulk modulus is considerably larger than the rigidity modulus, and in this discussion it is sufficient to neglect n compared with ⅛. With this approximation, the ellipticity e of the surface becomes 5wa 2 e= l9n T
Now suppose the sphere to be endued with the power of gravitation, and write _lVn = 2g_ f99 ι
5wα 2 * ® 5 a ∖ n where g is gravity at the surface of the globe. Then, if there were no elasticity, the ellipticity would be given by e=τ∕g, and without gravitation bye=τ∕r. And it may be proved in several ways that, gravity and elasticity co-operating, e=—=-∙-A- (100).
r + S fl 1+r/g
If n be the rigidity of steel, and if the globe have the size and mean density of the earth, r∕g=2, and with the rigidity of glass r∕g=⅞. Hence the ellipticity of an earth of steel under tide-generating force would be ⅜ of that of a fluid earth, and the similar fraction for glass would be ⅝. If an ocean be superposed on the globe, then, if the globe rises and falls with the tide as though it were fluid, there will obviously be no tide visible to an observer carried up and down with the solid; and with any degree of rigidity the visible tide will be the excess of the fluid tide above the solid tide. Hence on an earth with rigidity of steel the oceanic tides would be reduced to ⅜, and with rigidity of glass to ⅜ of the tides on a rigid earth.
§ 44. Rigidity of the Earth.
Although the computation of oceanic tides is as yet impossible, it cannot be admitted that perfect rigidity in the earth would augment the tides in the proportion of 5 to 2, although they might perhaps be augmented in the proportion 4 to 3. Thus Thomson concludes that the earth’s mass must have an effective rigidity at least as great as that of steel. If it were true, as was held until recently, that the earth is a fluid ball coated with a crust, that crust must be of fabulous rigidity to resist the tidal surgings of subjacent fluid. Hence we are led to the conclusion that far the larger portion of the earth’s mass, if not all of it, is a solid of great rigidity. Up to the present time the argument by which the tides of long period were proved to have approximately their equilibrium height has generally been accepted without much doubt, but we have (§ 17) shown good cause for rejecting Laplace’s argument, at least for a fortnightly tide. It appeared formerly that, from numerical data as to the heights of the tides of long period, we should be able to compute the actual effective rigidity of the earth’s mass. But from § 18 we see that, although these tides remain incalculable, yet with such oceans as ours the tides of long period must conform much more nearly to the equilibrium laws than do the tides of short period. Thus a comparison of the observed heights of the tides of long period with the equilibrium law still remains of interest, although the evaluation of the earth’s rigidity appears with present data unattainable. Acting on the old belief, Mr G. H. Darwin has compared the lunar fortnightly and monthly tides, as observed for thirty-three years at various Indian and European ports, with the equilibrium theory, and has found that the tideheights were about two-thirds of the equilibrium height. From this the conclusion was drawn that the effective rigidity of the earth was as great as that of steel. Whilst, then, this precise com parison with the rigidity of steel falls to the ground, the investigation remains as an important confirmation of Thomson’s conclusion as to the great effective rigidity of the earth. When extensive and accurate knowledge of the tides has been attained, the attempted evaluation of the rigidity may conceivably be possible, because there is a minute tide with a period of 18 - 6 years (§ 23, schedule [A, iii.]) of which Laplace’s argument must hold good. Great accuracy will, however, be necessary, because the height of the tide at the equator only amounts to one-third of an inch, and a preliminary inquiry seems to show that there are other relatively considerable variations of sea-level arising from unexplained causes.
Sir W. Thomson’s solution of the strain of an elastic sphere has been also used to determine what degree of strength the materials of the earth must have in order that the great continental plateaus and mountains may not sink in. In another investigation it has been shown that local elastic yielding on the coast-lines of continents may produce an augmentation of apparent tide in certain places on account of the flexure of the upper strata, when a great weight of water is added and subtracted from the adjacent oceanic area at high and low tide. There is reason to believe that such flexure has actually been observed by a delicate form of level on the coast of the Bay of Biscay.
§ 45. Viscous and Elastico- Viscous Tides.
It might be supposed that the earth is composed of a viscous fluid of great stiffness, or that it possesses an elasticity which breaks down under continued stress. Both these hypotheses have been considered, and the results are confirmatory of the conclusion that the earth is made of very stiff material. These problems appear to have been worthy of attack, although the existence of measurable oceanic tides of long period negatives the adoption of the hypothesis of true viscosity, at least under stresses comparable with tide-generating forces.
If a sphere of radius a, density w, viscosity modulus υ, be under the action of forces due to a potential per unit volume wr s S 2 cos nt, so that n is the speed of the tide, j⅛e solution of the problem shows that the tide of the sphere is expressed by cos e cos (nt - e) (101),
, . t 19υ „ σ
where tane=n-, t= 7 — 5 , ff=⅜ -.
g’ i>wa i, a ° a
Thus the tides of the viscous globe are to the equilibrium tides of a fluid globe as cos e to unity, and there is a retardation e/n of the time of high tide after the passage of the tide-generator over the meridian. Further, by arguments similar to that applied in the case of elastic tides, it is found that oceanic tides are reduced by the yielding in the proportion of sin e to unity, and that there is an apparent acceleration of the time of high water by (⅛τr-e)∕n. It appears by numerical calculation that, in order that the oceanic semi-diurnal tide may have a value equal to two-thirds of the full amount on a rigid globe, the stiffness of the globe must be about twenty thousand times as great as that of pitch at freezing temperature, when it is hard and brittle. We must here pass by the results of the hypothesis of an elasticity degrading under the influence of continued stress.
IX. Tidal Friction.
§ 46. General Explanation.
The investigation of the tides of a viscous sphere has led us to the consideration of a frictionally retarded tide. The effects of tidal friction are of such general interest that we give a sketch of the principal results without the aid of mathematical symbols. In fig. 8 the paper is supposed to be the plane of the orbit of a satellite Μ revolving in the direction of the arrow about the planet C, which rotates in the direction of the arrow about an axis perpendicular to the paper. The rotation of the planet is supposed to be more rapid than that of the satellite, so that the day is shorter than the month. Let us suppose that the planet is either entirely fluid, or has an ocean of such depth that it is high water under or nearly under the satellite. When there is no friction, with the satellite at m, the planet is elongated into the ellipsoidal shape shown, cutting the mean sphere, which is dotted. But, when there is friction in the fluid motion, the tide is retarded, and high tide occurs after the satellite has passed the meridian. Then, if we keep the same figure to represent the tidal elongation, the satellite must be at Μ, instead of at m. If we number the four quadrants as shown, the satellite must be in quadrant 1. The protuberance P is nearer to the satellite than P ’, and the deficiency Q is further away than the deficiency Q'. Hence the resultant action of the planet on the satellite must be in some such direction as MN. The action of the satellite on the planet is equal and opposite, and the force in NM, not being through the planet’s centre, must produce a retarding couple on the planet’s rotation, the magnitude of which depends on the length of the arm CN. This tidal frictional couple varies as the height of the tide, and also depends on the satellite’s distance; its intensity in fact varies as the square of the tide-generating force, and therefore as the inverse sixth power of the satellite’s distance. Thus tidal friction must retard the planetary rotation. Let us now consider its effect on the satellite. If the force acting on Μ be resolved along and perpendicular to the direction CM, the perpendicular component tends to accelerate the satellite’s velocity. It alone would carry the satellite further from C than it would be dragged back by the central force towards C. The satellite would describe a spiral, the coils of which would be very nearly circular and very nearly coincident. If now we resolve the central component force along CM tangentially and perpendicular to the spiral, the tangential component tends to retard the velocity of the satellite, whereas the disturbing force, already considered, tends to accelerate it. With the gravitational law of force between the two bodies the retardation must prevail over the acceleration. The moment of momentum of the whole system remains unchanged, and that of the planetary rotation diminishes, so that the orbital moment of momentum must increase; now orbital moment of momentum increases with increasing distance and diminishing linear and angular velocity of the satellite. The action of tidal friction may appear somewhat paradoxical, but it is the exact converse of the acceleration of the linear and angular velocity and the diminution of distance of a satellite moving through a resisting medium. The latter result is generally more familiar than the action of tidal friction, and it may help the reader to realize the result in the present case. Tidal friction then diminishes planetary rotation, increases the satellite’s distance, and diminishes the orbital angular velocity. The comparative rate of diminution of the two angular velocities is generally very different. If the satellite be close to the planet the rate, of increase of the satellite’s periodic time or month is large compared with the rate of increase of the period of planetary rotation or day; but if the satellite is far off the converse is true. Hence, if the satellite starts very near the planet, with the month a little longer than the day, as the satellite recedes the month soon increases, so that it contains many days. The number of days in the month attains a maximum and then diminishes. Finally the two angular velocities subside to a second identity, the day and month being identical and both very long.
We have supposed that the ocean is of such depth that the tides are direct; if, however, they are inverted, with low water under or nearly under the satellite, friction, instead of retarding, accelerates the tide; and it would be easy by drawing another figure to see that the whole of the above conclusions hold equally true with inverted tides.
§ 47. Exact Investigation of the Secular Effects of Tidal Friction.
The general conclusions of the last section are of such wide interest that we proceed to a rigorous discussion of the principal effects of tidal friction in the elementary case of the circular orbit. In order, however, to abridge the investigation we shall only consider the case when the planetary rotation is more rapid than the satellite’s orbital motion.
Suppose an attractive particle or satellite of mass m to be moving in a circular orbit, with an angular velocity Ω, round a planet of mass Μ, and suppose the planet to be rotating about an axis perpendicular to the plane of the orbit, with an angular velocity n; suppose, also, the mass of the planet to be partially or wholly imperfectly elastic or viscous, or that there are oceans on the surface of the planet; then the attraction of the satellite must produce a relative motion in the parts of the planet, and that motion must be subject to friction, or, in other words, there must be frictional tides of some sort or other. The system must accordingly be losing energy by friction, and its configuration must change in such a way that its whole energy diminishes. Such a system does not differ much from those of actual planets and satellites, and, therefore, the results deduced in this hypothetical case must agree pretty closely with the actual course of evolution, provided that time enough has been and will be given for such changes. Let C be the moment of inertia of the planet about its axis of rotation, r the distance of the satellite from the centre of the planet, h the resultant moment of momentum of the whole system, e the whole energy, both kinetic and potential, of the system. It is assumed that the figure of the planet and the distribution of its internal density are such that the attraction of the satellite causes no couple about any axis perpendicular to that of rotation. A special system of units of mass, length, and time will now be adopted such that the analytical results are reduced to their simplest forms. Let the unit of mass be Mm ∣ {M+m). Let the unit of length 7 be such a distance that the moment of inertia of the planet about its axis of rotation may be equal to the moment of inertia of the planet and satellite, treated as particles, about their centre of inertia, when distant 7 apart from one another. This condition gives ∖ M+mJ ∖ M+mj ’
, iC(JΓ+τn)Ι⅜
whence 7= 4 —¼; r ·
' I Mm J
Let the unit of time τ be the time in which the satellite revolves through 57° ’ 3 about the planet, when the satellite’s radius vector is equal to 7. In this case l∕τ is the satellite’s orbital angular velocity, and by the law of periodic times we have
τ-V=∕√lf+m), where μ is the attraction between unit masses at unit distance. Then by substitution for 7
_ f C 8 (Jf+m) l⅜ τ I μ i {Mmf ∫
This system of units will be found to make the three following functions each equal to unity, viz., μijfm (3∕"+m) - i, μMm, and C. The units are in fact derived from the consideration that these functions are each to be unity. In the case of the earth and moon, if we take the moon’s mass as ⅛d of the earth’s and the earth’s moment of inertia as ⅜ Ma- (as is very nearly the case), it may easily be shown that the unit of mass is ⅛ of the earth’s mass, the unit of length 5 - 26 earth’s radii or 33,506 kilomètres (20,807 miles), and the unit of time 2 hrs. 41 minutes. In these units the present angular velocity of the earth’s diurnal rotation is expressed by •7044, and the moon’s present radius vector by 1T454. The two bodies being supposed to revolve in circles about their common centre of inertia with an angular velocity Ω, the moment of momentum of orbital motion is
Λ(^)’η +Λ (∞L)* b =⅛-λ ∖ M+m) ∖ M+m∕ M+m
Then, by the law of periodic times in a circular orbit, Wτ 3 =μ(M+m); whence Ωr 2 =μ⅛(M+rnfr⅛.
The moment of momentum of orbital motion = μ⅛Mm(M + m)~iri, and in the special units this is equal to rL The moment of momentum of the planet’s rotation is Cn, and C≈l in the special units. Therefore h=n+ri (102).
Since the moon’s present radius vector is 11'454, it follows that the orbital momentum of the moon is 3'384. Adding to this the rotational momentum of the earth, which is - 704, we obtain 4 , 088 for the total moment of momentum of the moon and earth. The ratio of the orbital to the rotational momentum is 4∙80, so that the total moment of momentum of the system would, but for the obliquity of the ecliptic, be 5 - 80 times that of the earth’s rotation. Again, the kinetic energy of orbital motion is ∖ M+mj ∖ M+m∕ 2 M+m τ
The kinetic energy of the planet’s rotation is ⅜C⅛ 2 . The potential energy of the system is - μMm ∣ r. Adding the three energies together, and transforming into the special units, we have
2e=n 2 -l∕r (103).
Now let x=r⅛, y=n i V=2e.
It will be noticed that x, the moment of momentum of orbital motion, is equal to the square root of the satellite’s distance from the planet. Then equations (102) and (103) become h=y + x (104).
Y=↑f- 1∕≈ 2 =(A - a:) 2 - l∕x 2 (105).
(104) is the equation of conservation of moment of momentum, or, shortly, the equation of momentum; (105) is the equation of energy.
Now consider a system started with given positive moment of momentum h ; and we have all sorts of ways in which it· may be started. If the two rotations be of opposite kinds, it is clear that we may start the system with any amount of energy however great, but the true maxima and minima of energy compatible with the given moment of momentum are supplied by dY ∣ dx=Q, or ic-A + l∕x 3 =0, that is to say, ic 4 -Ar 3 + l = 0 (106).
We shall presently see that this quartic has either two real roots and two imaginary, or all imaginary roots. The quartic may be derived from quite a different consideration, viz., by finding the condition under which the satellite may move round the planet so that the planet shall always show the same face to the satellite,—in fact, so that they move as parts of one rigid body. The condition is simply that the satellite’s orbital angular velocity Ω = n, the planet’s angular velocity of rotation, or y= ∖∣ x i , since n=y and ri-Q~l=x, By substituting this value of y in the equation of momentum (104), we get as before ∞ 4 - Λz 3 + 1 = 0
At present we have only obtained one result, viz., that, if with given moment of momentum it is possible to set the satellite and planet moving as a rigid body, it is possible to do so in two ways, and one of these ways requires a maximum amount of energy and the other a minimum; from this it is clear that one must be a rapid rotation with the satellite near the planet and the other a slow one with the satellite remote from the planet. In the three equations h=y+x (107),
K=(A-x) 2 -l∕x 2 (108), se 3 y=l (109),
(107) is the equation of momentum, (108) that of energy, and (109) may be called the equation of rigidity, since it indicates that the two bodies move as though parts of one rigid body. To illustrate these equations geometrically, we may take as abscissa x, which is the moment of momentum of orbital motion, so that the axis of x may be called the axis of orbital momentum. Also, for equations (107) and (109) we may take as ordinate y, which is the moment of momentum of the planet’s rotation, so that the axis of y may be called the axis of rotational momentum. For (108) we may take as ordinate Y, which is twice the energy of the system, so that the axis of Y may be called the axis of energy. Then, as it will be convenient to exhibit all three curves in the same figure, with a parallel axis of x, we must have the axis of energy identical with that of rotational momentum. It will not be necessary to consider the case where the resultant moment of momentum h is negative, because this would only be equivalent to reversing all the rotations; h is therefore to be taken as essentially positive. Then the line of momentum whose equation is (107) is a straight line inclined at 45’ to either axis, having positive intercepts on both axes. The curve of rigidity whose equation is (109) is clearly of the same nature as a rectangular hyperbola, but it has a much more rapid rate of approach to the axis of orbital momentum than to that of rotational momentum. The intersections (if any) of the curve of rigidity with the line of momentum have abscissae which are the two roots of the quartic ae 4 - ha? +1 = 0. The quartic has, therefore, two real roots or all imaginary roots. Then, since x= ∖ Jr, the intersection which is more remote from the origin indicates a configuration where the satellite is remote from the planet; the other gives the configuration where the satellite is closer to the planet. We have already learnt that these two correspond respectively to minimum and maximum energy. When x is very large, the equation to the curve of energy is Y=(hx) 2 , which is the equation to a parabola with a vertical axis parallel to Y and distant h from the origin, so that the axis of the parabola passes through the intersection of the line of momentum with the axis of orbital momentum. When x is very small, the equation becomes Y=-l/a?. Hence the axis of Vis asymptotic on both sides to the curve of energy. Then, if the line of momentum intersects the curve of rigidity, the curve of energy has a maximum vertically underneath the point of intersection nearer the origin and a minimum underneath the point more remote. But, if there are no intersections, it has no maximum or minimum.
Fig. 9 shows these curves when drawn to scale for the case of the earth and moon, that is to say, with Ae=4. The points α and b, which are the maximum and minimum of the curve of energy, are supposed to be on the same ordinates as A and B, the intersections of the curve of rigidity with the line of momentum. The intersection of the line of momentum with the axis of orbital momentum is denoted bv D. but in a fierure of this size it neces sarily remains indistinguishable from B. As the zero of energy is quite arbitrary, the origin for the energy curve is displaced downwards, and this prevents the two curves from crossing one another in a confusing manner. On account of the limitation imposed we neglect the case where the quartic has no real roots. Every point of the line of momentum gives by its abscissa and ordinate the square root of the satellite’s distance and the rotation of the planet, and the ordinate of the energy curve gives the energy corresponding to each distance of the satellite. Part of the figure has no physical meaning, for it is impossible for the satellite to move round the planet at a distance less than the sum of the radii of the planet and satellite. For example, the moon’s diameter being about 2200 miles, and the earth’s about 8000, the moon’s distance cannot be less than 5100 miles. Accordingly a strip is marked off and shaded on each side of the vertical axis within which the figure has no physical meaning. The point P indicates the present configuration of the earth and moon. The curve of rigidity a¾=l is the same for all values of h, and by moving the line of momentum parallel to itself nearer to or further from the origin, we may represent all possible moments of momentum of the whole system. The smallest amount of moment of momentum with which it is possible to set the system moving as a rigid body, with centrifugal force enough to balance the mutual attraction, is when the line of momentum touches the curve of rigidity. The condition for this is clearly that the equation a? - ha? + 1 = 0 should have equal roots. If it has equal roots, each root must be ∣ h, and therefore
(t⅛) 1 -½(tA) 3 + l = 0, whence ^ 4 =4 4 ∕3 3 or Λ = 4∕3^=l - 75. The actual value of h for the moon and earth is about 4; hence, if the moon-earth system were started with less than ⅛ of its actual moment of momentum, it would not be possible for the two bodies to move so that the earth should always show the same face to the moon. Again, if we travel along the line of momentum, there must be some point for which ya? is a maximum, and since ya?=n!O, there must be some point for which the number of planetary rotations is greatest during one revolution of the satellite; or, shortly, there must be some configuration for which there is a maximum number of days in the month. Now ya? is equal to a?fh-x'), and this is a maximum when x=⅜h and the maximum number of days in the month is (∣A) 3 (A-fΛ) or 3 s A 4 ∕4 4 ; if h is equal to 4, as is nearly the case for the earth and moon, this becomes 27. Hence it follows that we now have very nearly the maximum number of days in the month. A more accurate investigation in a paper on the “Precession of a Viscous Spheroid” in Phil,. Trans., part i., 1879, showed that, taking account of solar tidal friction and of the obliquity to the ecliptic, the maximum number of days is about 29, and that we have already passed through the phase of maximum. We will now consider the physical meaning of the figure. It is assumed that the resultant moment of momentum of the whole system corresponds to a positive rotation. Now imagine two points -with the same abscissa, one on the momentum line and the other on the energy curve, and suppose the one on the energy curve to guide that on the momentum line. Then, since we are supposing frictional tides to be raised on the planet, the energy must degrade, and however the two points are set initially the point on the energy curve must always slide down a slope, carrying with it the other point. Looking at the figure, we see that there are four slopes in the energy curve, two running down to the planet and two down to the minimum. There are therefore four ways in which the system may degrade, according to the way it was started; but we shall only consider one, that corresponding to the portion ABδα of the figure. For the part of the line of momentum AB the month is longer than the day, and this is the case with all known satellites except the nearer one of Mars. Now, if a satellite be placed in the condition A—that is to say, moving rapidly round a planet which always shows the same face to the satellite—the condition is clearly dynamically unstable, for the least disturbance will determine whether the system shall degrade down the slopes ac or ab — that is to say, whether it falls into or recedes from the planet. If the equilibrium breaks down by the satellite receding, the recession will go on until the system has reached the state corresponding to B. It is clear that, if the intersection of the edge of the shaded strip with the line of momentum be identical with the point A, which indicates that the satellite is just touching the planet, then the two bodies are in effect parts of a single body in an unstable configuration. If, therefore, the moon was originally part of the earth, we should expect to find this identity. Now in fig. 9, drawn to scale to represent the earth and moon, there is so close an approach between the edge of the shaded band and the intersection of the line of momentum and curve of rigidity that it would be scarcely possible to distinguish them. Hence, there seems a probability that the two bodies once formed parts of a single one, which broke up in consequence of some kina of instability. This view is confirmed by the more detailed consideration of the case in the paper on the “Precession of a Viscous Spheroid,” already referred to, and subsequent papers, in the Philosophical Transactions of the Royal Society.
§ 48. Amount of Tidal Retardation of Earth's Rotation.
With respect to the actual amount of retardation of the earth’s rotation, we quote the following from Thomson and Tait’s Nat. Phil. (1883), § 830.
“In observational astronomy the earth’s rotation serves as a timekeeper, and thus a retardation of terrestrial rotation will appear astronomically as an acceleration of the motion of the heavenly bodies. It is only in the case of the moon’s motion that such an apparent acceleration can be possibly detected. Now, as Laplace first pointed out, there must be a slow variation in the moon’s mean motion arising from the secular changes in the eccentricity of the earth’s orbit around the sun. At the present time, and for several thousand years in the future, the variation in the moon’s motion is and will be an acceleration. Laplace’s theoretical calculation of the amount of that acceleration appeared to agree well with the results which were in his day accepted as representing the facts of observation. But in 1853 Adams showed that Laplace’s reasoning was at fault, and that the numerical results of Damoiseau’s and Plana’s theories with reference to it consequently require to be sensibly altered. Hansen’s theory of the secular acceleration is vitiated by an error of principle similar to that which affects the theories of Damoiseau and Plana; but, the mathematical process which he followed being different from theirs, he arrived at somewhat different results. From the erroneous theory Hansen found the value of 12" - 18 for the coefficient of the term in the moon’s mean longitude depending on the square of the time, the unit of time being a century; in a later computation given in his Darlegung he found the coefficient to be 12"∙56.
“In 1859 Adams communicated to Delaunay his final result, namely, that the coefficient of this term appears from a correctly conducted investigation to be 5" - 7, so that at the end of a century the moon is 5" ’ 7 before the position it would have had at the same time if its mean angular velocity had remained the same as at the beginning of the century. Delaunay verified this result, and added some further small terms which increased the coefficient from 5" - 7 to 6"1.
“Now, according to Airy, Hansen’s value of the ‘advance’ represents very well the circumstances of the eclipses of Agathocles, Larissa, and Thales, but is if anything too small. Newcomb, on the other hand, is inclined from an elaborate discussion of the ancient eclipses to believe Hansen’s value to be too large, and gives two competing values, viz., 8"'4 and 10" ’ 9.
“In any case it follows that the value of the advance as theoretically deduced from all the causes, known up to the present time to be operative, is smaller than that which agrees with observation. In what follows 12" is taken as the observational value of the advance, and 6" as the explained part of this phenomenon. About the beginning of 1866 Delaunay suggested that the true explanation of the discrepancy might be a retardation of the earth’s rotation by tidal friction. Using this hypothesis, and allowing for the consequent retardation of the moon’s mean motion by tidal reaction, Adams, in an estimate which he has communicated to us, founded on the rough assumption that the parts of the earth’s retardation due to solar and lunar tides are as the squares of the respective tide-generating forces, finds 22 sec. as the error by which the earth, regarded as a time-keeper, would in a century get behind a perfect clock rated at the beginning of the century. Thus at the end of a century a meridian of the earth is 330° behind the position in which it would have been if the earth had continued to rotate with the same angular velocity which it had at the beginning of the century. . . .
“Whatever be the value of the retardation of the earth’s rotation it is necessarily the result of several causes, of which tidal friction is almost certainly preponderant. If we accept Adams’s estimate as applicable to the outcome of the various concurring causes, then, if the rate of retardation giving the integral effect were uniform, the earth as a time-keeper would be going slower by ’ 22 of a second per year in the middle, and by ’ 44 of a second per year at the end, than at the beginning of the century. The latter is ¹∕₇₁.₇ ₓ ₁₀⁶ of the present angular velocity; and, if the rate of retardation had been uniform during ten million centuries past, the earth must have been rotating faster by about one-seventh than at present, and the centri fugal force must have been greater in the proportion of 817 2 to 717 2 or of 67 to 51. If the consolidation took place then or earlier, the ellipticity of the upper layers must have been ∙⅛ y instead of about 3⅜ τ , as it is at present. It must necessarily remain uncertain whether the earth would from time to time adjust itself completely to a figure of equilibrium adapted to the rotation. But it is clear that a want of complete adjustment would leave traces in a preponderance of land in equatorial regions. The existence of large continents and the great effective rigidity of the earth’s mass render it improbable that the adjustments, if any, to the appropriate figure of equilibrium would be complete. The fact then that the continents are arranged along meridians rather than in an equatorial belt affords some degree of proof that the consolidation of the earth took place at a time when the diurnal rotation differed but little from its present value. It is probable, therefore, that the date of consolidation is considerably more recent than a thousand million years ago. It is proper, however, to add that Adams lays but little stress on the actual numerical values which have been used in this computation, and is of opinion that the amount of tidal retardation of the earth’s rotation is quite uncertain.”
§ 49. Effects of Tidal Friction on the Elements of the Moon’s Orbit and on the Earth’s Rotation.
It would be impossible within the limits of the present article to discuss completely the effects of tidal friction; we therefore confine ourselves to certain general considerations which throw light on the nature of those effects. We have in the preceding sections supposed that the planet’s axis is perpendicular to the orbit of the satellite, and that the latter is circular; we shall now suppose the orbit to be oblique to the equator and eccentric, and shall also consider some of the effects of the solar perturbation of the moonearth system. For the sake of brevity the planet will be called the earth, and the satellite the moon. The complete investigation was carried out on the hypothesis that the planet was a viscous spheroid, because this was the only theory of frictionally resisted tides which had been worked out. Although the results would be practically the same for any system of frictionally resisted tides, we shall speak below of the planet or earth as a viscous body.
We shall show that if the tidal retardation be small the obliquity of the ecliptic increases, the earth’s rotation is retarded, and the moon’s distance and periodic time are increased. Fig. 10 represents the earth as seen from above the south pole, so that S is the pole and the outer circle the equator. The earth’s rotation is in the direction of the curved arrow at S. The half of the inner circle which is drawn with a full line is a semi-small-circle of south latitude, and the dotted semicircle is a semi-small-circle in the same north latitude. Generally dotted lines indicate parts of the figure which are below the plane of the paper. It will make the explanation somewhat simpler if we suppose the tides to be raised by a moon and anti-moon diametrically opposite to one another. Let Μ and Μ' be the projections of the moon and anti-moon on to the terrestrial sphere. If the fluid in which the tides are raised were perfectly frictionless, or if the earth were a perfect fluid or perfectly elastic, the apices of the tidal spheroid would be at Μ and M'. If, however, there is internal friction, due to any sort of viscosity, the tides will lag, and we may suppose the tidal apices to be at T and T ’. Now suppose the tidal protuberances to be replaced by two equal heavy particles at T and T', which are instantaneously rigidly connected with the earth. Then the attraction of the moon on T is greater than on T', and that of the antimoon on T' is greater than on T. The resultant of these forces is clearly a pair of forces acting on the earth in the direction TM, T ’ M'. These forces clearly cause a couple about the axis in the equator, which lies in the same meridian as the moon and antimoon. The direction of the couple is shown by the curved arrows at L,L'. If the effects of this couple be compounded with the existing rotation of the earth according to the principle of the gyroscope, the south pole S tends to approach Μ and the north pole to approach M'. Hence, supposing the moon to move in the ecliptic, the inclination of the earth’s axis to the ecliptic diminishes, or the obliquity increases. Next the forces TM, T'M' clearly produce, as in the simpler case considered above, a couple about the earth’s polar axis, which tends to retard the diurnal rotation.
This general explanation remains a fair representation of the state of the case so long as the different harmonic constituents of the aggregate tide-wave do not suffer very different amounts of retardation; and this is the case so long as the viscosity is not great. The rigorous result for a viscous planet shows that in general the obliquity will increase, and it appears that, with small viscosity of the planet, if the period of the satellite be longer than two periods of rotation of the planet, the obliquity increases, and vice versa. Hence zero obliquity is only dynamically stable when the period of the satellite is less than two periods of the planet’s rotation.
Suppose the motions of the planet and of its solitary satellite to be referred to the invariable plane of the system. The axis of resultant moment of momentum is normal to this plane, and the component rotations are that of the planet about its axis of figure and the orbital motion of the planet and satellite round their com- . mon centre of inertia; the axis of this latter rotation is clearly the normal to the satellite’s orbit. Hence the normal to the orbit, the axis of resultant moment of momentum, and the planet’s axis of rotation must always lie in one plane. From this it follows that the orbit and the planet’s equator must necessarily have a common node on the invariable plane. If either of the component rotations alters in amount or direction, a corresponding change must take place in the other, such as will keep the resultant moment of momentum constant in direction and magnitude. It has been shown that the effect of tidal friction is to increase the distance of the satellite from the planet, and to transfer moment of momentum from that of planetary rotation to that of orbital motion. If, then, the direction of the planet’s axis of rotation does not change, it follows that the normal to the lunar orbit must approach the axis of resultant moment of momentum. By drawing a series of parallelograms on the same diameter and keeping one side constant in direction, this may be easily seen to be true. This is equivalent to saying that the inclination of the satellite’s orbit will decrease. But this decrease of inclination does not always necessarily take place, for the previous investigations show that another effect of tidal friction may be to increase the obliquity of the planet’s equator to the invariable plane, or, in other words, to increase the inclination of the planet’s axis to the axis of resultant moment of momentum. Now, if a parallelogram be drawn with a constant diameter, it is seen that by increasing the inclination of one of the sides to the diameter (and even decreasing its length) the inclination of the other side to the diameter may also be increased. The most favourable case for such a change is when the side whose inclination is increased is nearly as long as the diameter. From this it follows that the inclination of the satellite’s orbit to the invariable plane may increase, and that it is most likely to increase, when the moment of momentum of planetary rotation is large compared with that of the orbital motion. The analytical solution of the problem agrees with these results, for it shows that if the viscosity of the planet be small the inclination of the orbit always diminishes, but if the viscosity be large, and if the satellite moves with a short periodic time (as estimated in rotations of the planet), the inclination of the orbit will increase. These results convey some idea of the physical causes which may have given rise to the present inclination of the lunar orbit to the ecliptic. For the analytical investigation shows that the inclination of the lunar orbit to a certain plane, which replaces the invariable plane when the solar attraction is introduced, was initially small, that it then increased to a maximum, and that it finally diminished and is still diminishing.
But the laws above referred to would, by themselves, afford a very unsatisfactory explanation of the inclination of the lunar orbit, because the sun’s attraction is a matter of much importance. It has been found that, if the viscosity of the planet be small, the inclination of the orbit of the solitary satellite to the invariable plane will always diminish; but, when solar influence is introduced, the corresponding statement is not true with regard to the inclination of the lunar orbit to the proper plane, for during one part of the moon’s history the inclination to the proper plane would have increased even if the viscosity of the earth had been small.
Consider a satellite revolving about a planet in an elliptic orbit, with a periodic time which is long compared with the period of rotation of the planet; and suppose that frictional tides are raised on the planet. The major axis of the tidal spheroid always points in advance of the satellite, and exercises on it a force which tends to accelerate its linear velocity. When the satellite is in perigee the tides are higher, and this disturbing force is greater than when the satellite is in apogee. The disturbing force may therefore be represented as a constant force, always tending to accelerate the motion of the satellite, and as a periodic force which accelerates in perigee and retards in apogee. The constant force causes a secular increase of the satellite’s mean distance and a retardation of its mean motion. The accelerating force in perigee causes the satellite to swing out further than it would otherwise have done, so that when it comes round to apogee it is more remote from the planet. The retarding force in apogee acts exactly inversely, and diminishes the perigean distance. Thus, the apogean distance increases and the perigean distance diminishes, or, in other words, the eccentricity of the orbit increases. Now consider another case, and suppose the satellite’s periodic time to be identical with that of the planet’s rotation.
Then, when the satellite is in perigee, it is moving faster than the planet rotates, and when in apogee it is moving slower; hence at apogee the tides lag, and at perigee they are accelerated. Now the lagging apogean tides give rise to an accelerating force on the satellite, and increase the perigean distance, whilst the accelerated perigean tides give rise to a retarding force, and decrease the apogean distance. Hence in this case the eccentricity of the orbit will diminish. It follows from these two results that there must be some intermediate periodic time of the satellite for which the eccentricity does not tend to vary.
But the preceding general explanation is in reality somewhat less satisfactory than it seems, because it does not make clear the existence of certain antagonistic influences, to which, however, we shall not refer. The rigorous result, for a viscous planet, shows that in general the eccentricity of the orbit will increase; but, if the obliquity of the planet’s equator be nearly 90°, or if the viscosity be so great as to approach perfect rigidity, or if the periodic time of the satellite (measured in rotations of'the planet) be short, the eccentricity will slowly diminish. When the viscosity is small the law of variation of eccentricity is very simple: if eleven periods of the satellite occupy a longer time than eighteen rotations of the planet, the eccentricity increases, and vice versa. Hence in the case of small viscosity a circular orbit is only dynamically stable if the eleven periods are shorter than the eighteen rotations.
X. Cosmogonic Speculations founded on Tidal Friction.
§ 50. History of the Earth and Moon.
We shall not attempt to discuss the mathematical methods by which the complete history of a planet, attended by one or more satellites, is to be traced. The laws indicated in the preceding sections show that there is such a problem, and that it may be solved, and we refer to Mr Darwin’s papers for details {Phil. Trans., 1879-81). It may be interesting, however, to give the various results of the investigation in the form of a sketch of the possible evolution of the earth and moon, followed by remarks on the other planetary systems and on the solar system as a whole.
We begin with a planet not very much more than 8000 miles in diameter, and probably partly solid, partly fluid, and partly gaseous. It is rotating about an axis inclined at about 11° or 12° to the normal to the ecliptic, with a period of from two to four hours, and is revolving about the sun with a period not much shorter than our present year. The rapidity of the planet’s rotation causes so great a compression of its figure that it cannot continue to exist in an ellipsoidal form with stability; or else it is so nearly unstable that complete instability is induced by the solar tides. The planet then separates into two masses, the larger being the earth and the smaller the moon. It is not attempted to define the mode of separation, or to say whether the moon was initially a chain of meteorites. At any rate it must be assumed that the smaller mass became more or less conglomerated and finally fused into a spheroid, perhaps in consequence of impacts between its constituent meteorites, which were once part of the primeval planet. Up to this point the history is largely speculative, for the conditions of instability of a rotating mass of fluid have not yet been fully investigated. We now have the earth and moon nearly in contact with one another, and rotating nearly as though they were parts of one rigid body. This is the system which was the subject of dynamical investigation. As the two masses are not rigid, the attraction of each distorts the other; and, if they do not move rigorously with the same periodic time, each raises a tide in the other. Also the sun raises tides in both. In consequence of the frictional resistance to these tidal motions, such a system is dynamically unstable. If the moon had moved orbitally a little faster than the earth rotated, she must have fallen back into the earth; thus the existence of the moon compels us to believe that the equilibrium broke down by the moon revolving orbitally a little slower than the earth rotates. In consequence of the tidal friction the periodic times both of the moon (or the month) and of the earth’s rotation (or the day) increase; but the month increases in length at a much greater rate than the day. At some early stage in the history of the system the moon was conglomerated into a spheroidal form, and acquired a rotation about an axis nearly parallel to that of the earth.
The axial rotation of the moon is retarded by the attraction of the earth on the tides raised in the moon, and this retardation takes place at a far greater rate than the similar retardation of the earth’s rotation. As soon as the moon rotates round her axis with twice the angular velocity with which she revolves in her orbit, the position of her axis of rotation (parallel with the earth’s axis) becomes dynamically unstable. The obliquity of the lunar equator to the plane of the orbit increases, attains a maximum, and then diminishes. Meanwhile the lunar axial rotation is being reduced towards identity with the orbital motion. Finally, her equator is nearly coincident with the plane of the orbit, and the attraction of the earth on a tide, which degenerates into a permanent ellipticity of the lunar equator, causes her always to show the same face to the earth.
All this must have taken place early in the history of the earth, to which we now return. As the month increases in length the lunar orbit becomes eccentric, and the eccentricity reaches a maximum when the month occupies about a rotation and a half of the earth. The maximum of eccentricity is probably not large. After this the eccentricity diminishes. The plane of the lunar orbit is at first practically identical with the earth’s equator, but as the moon recedes from the earth the sun’s attraction begins to make itself felt. We must therefore introduce the conception of two ideal planes (here called the proper planes), to which the motion of the earth and moon must be referred. The lunar proper plane is at first inclined at a very small angle to the earth’s proper plane, and the orbit and equator coincide with their respective proper planes. As soon as the earth rotates with twice the angular velocity with which the moon revolves in her orbit, a new instability sets in. The month is then about twelve of our present hours, and the day about six such hours in length. The inclinations of the lunar orbit and of the equator to their respective proper planes increase. That of the lunar orbit to its proper plane increases to a maximum of 6° or 7°, and ever after diminishes, that of the equator to its proper plane increases to a maximum of about 2 i 45', and ever after diminishes. The maximum inclination of the lunar orbit to its proper plane takes place when the day is a little less than nine of our present hours, and the month a little less than six of our present days. The maximum inclination of the equator to its proper plane takes place earlier than this. Whilst these changes have been going on the proper planes have been themselves changing in their positions relatively to one another and to the ecliptic. At first they were nearly coincident with one another and with the earth’s equator, but they then open out, and the inclination of the lunar proper plane to the ecliptic continually diminishes, whilst that of the terrestrial proper plane continually increases. At some stage the earth became more rigid, and oceans were formed, so that oceanic tidal friction probably came to play a more important part than bodily tidal friction. If this be the case, the eccentricity of the orbit, after passing through a stationary phase, begins to increase again. We have now traced the system to a state in which the day and month are increasing, but at unequal rates, the inclination of the lunar proper plane to the ecliptic and of the orbit to the proper plane are diminishing, the inclination of the terrestrial proper plane to the ecliptic is increasing and of the equator to its proper plane is diminishing, and the eccentricity of the orbit is increasing. No new phase now supervenes and at length we have the system in its present configuration. The minimum time in which the changes from first to last can have taken place is 54,000,000 years.
There are other collateral results which must arise from a supposed primitive viscosity or plasticity of the earth’s mass. For during this course of evolution the earth’s mass must have suffered a screwing motion, so that the polar regions have travelled a little from west to east relatively to the equator. This affords a possible explanation of the north and south trend of our great continents. Also a large amount of heat has been generated by friction deep down in the earth; and some very small part of the observed increase of temperature in underground borings may be attributable to this cause. The preceding history might vary a little in detail according to the degree of viscosity which we attribute to the earth’s mass, and according as oceanic tidal friction is or is not, now and in the more recent past, a more powerful cause of change than bodily tidal friction. The argument reposes on the imperfect rigidity of solids and on the internal friction of semi-solids and fluids; these are verae causae. Thus changes of the kind here discussed must be going on, and must have gone on in the past. And for this history of the earth and moon to be true throughout, it is only necessary to postulate a sufficient lapse of time, and that there is not enough matter diffused through space to materially resist the motions of the moon and earth in perhaps 200,000,000 years. It seems hardly too much to say that, granting these two postulates, and the existence of a primeval planet, such as that above described, a system would necessarily be developed which would bear a strong resemblance to our own. A theory, reposing on verae causae, which brings into quantitative correlation the lengths of the present day and month, the obliquity of the ecliptic, and the inclination and eccentricity of the lunar orbit should have claims to acceptance.
§ 51. The Other Planetary Subsystems.
If this has been the evolution of the earth and moon, a similar process must have been going on elsewhere. So far we have only considered a single satellite and the sun, but the theory may of course be extended, with modifications, to planets attended by several satellites. We will now, therefore, consider some of the other members of the solar system. A large planet has much more energy of rotation to be destroyed, and moment of momentum to be redistributed, than a small one, and therefore a large planet ought to proceed in its evolution more slowly than a small one.
Therefore we ought to find the larger planets less advanced than the smaller ones. The masses of such of the planets as have satellites are, in terms of the earth’s mass, as follows:—Mars = f; Jupiter=340; Saturn = 100; Uranus = 17j Neptune = 20.
Mars should therefore be furthest advanced in its evolution, and it is here alone in the whole system that we find a satellite moving orbitally faster than the planet rotates. This will also be the ultimate fate of our moon, because, after its orbital motion has been reduced to identity with that of the earth’s rotation, solar tidal friction will further reduce the earth’s angular velocity; the tidal reaction on the moon will then be reversed, and the moon’s orbital velocity will increase and her distance from the earth diminish. But, since the moon’s mass is very large, she must recede to an enormous distance from the earth before this reversal takes place. Now the satellites of Mars are very small, and therefore they need only recede a very short distance from the planet before the reversal of tidal reaction. The periodic time of the satellite Deimos is 30 h 18 m , and, as the period of rotation of Mars is 24 h 37 m , Deimos must be still receding from Mars, but very slowly. The periodic time of the satellite Phobos is 7 h 39 m ; therefore it must be approaching Mars. It does not seem likely that it has ever been remote from the planet. The eccentricities of the orbits of both satellites are small: that of Deimos is - 0057 and that of Phobos Ό066. If the viscosity of the planet be small, or if oceanic tidal friction be the principal cause of change, both eccentricities are diminishing; but, if the viscosity be large, both are increasing. As we have no means of knowing whether the eccentricities are increasing or diminishing, the larger eccentricity of the orbit of Phobos cannot be a fact of much importance either for or against the present views. But it must be admitted that it is a slightly unfavourable indication. The position of the proper plane of a satellite is determined by the periodic time of the satellite, the oblateness of the planet, and the sun’s distance. The inclination of the orbit of a satellite to the proper plane is not determined by anything in the system. Hence it is only the inclination of the orbit which can afford any argument for or against the theory. The proper planes of both satellites are necessarily nearly coincident with the equator of the planet; but it is in accordance with the theory that the inclinations of the orbits to their respective proper planes should be small. Any change in the obliquity of the equator of Mars to the plane of his orbit must be entirely due to solar tides. The present obliquity is about 30°, and this points also to an advanced stage of evolution, at least if the axis of the planet was primitively at all nearly perpendicular to the ecliptic.
We now come to the system of Jupiter. Thi⅛ enormous planet is still rotating in about ten hours; its axis is nearly perpendicular to the ecliptic; and three of its satellites revolve in seven days or less, whilst the fourth has a period of 16 d 16 h . This system is obviously far less advanced than our own. The inclinations of the proper planes to Jupiter’s equator are necessarily small, but the inclinations of the orbits to the proper planes appear to be very interesting from a theoretical point of view. They are in the case of the first satellite 0° 0' 0", in the case of the second 0° 27' 50", in that of the third 0° 12' 20", and in that of the fourth 0° 14' 58". We have shown above that the orbit of a satellite is first coincident with its proper plane, and that the inclination afterwards rises to a maximum and finally declines. If then we may assume, as seems reasonable, that the satellites are in stages of evolution corresponding to their distances from the planet, these inclinations accord well with the theory. The eccentricities of the orbits of the two inner satellites are insensible, those of the outer two small. This does not tell strongly either for or against the theory, because the history of the eccentricity depends considerably on the nature of the friction to which the tides are subject. Yet it on the whole agrees with the theory that the eccentricity should be greater in the more remote satellites. It appears that the satellites of Jupiter always present the same face to the planet, just as does our moon. This was to be expected.
The case of Saturn is not altogether so favourable to the theory. The extremely rapid rotation, the ring, and the short periodic time of the inner satellites point to an early stage of development; whilst the longer periodic time of the three outer satellites and the high obliquity of the equator indicate a later stage. Perhaps both views may be more or less correct, for successive shedding of satellites would impart a modern appearance to the system. It has probably been previously remarked that the Saturnian system bears a strong analogy to the solar system, Titan being analogous to Jupiter, Hyperion and Iapetus to Uranus and Neptune, and the inner satellites to the inner planets. Thus anything which aids us in forming a theory of the one system will throw light on the other. The details of the Saturnian system seem to be more or less favourable to the theory. The proper planes of the orbits (except that of Iapetus) are nearly in the plane of the ring, and the inclinations of all the orbits thereto annear not to be large. As the result of a careful series of observations made at Washington in 1873, Prof. Asaph Hall finds that the eccentricities of the orbits of Mimas, Enceladus, Tethys, Dione, and Rhea are insensible, that of Titan is ∙0284, of Hyperion ∙1000, and that of Iapetus , 0278. The satellite Iapetus appears always to present the same face to the planet.
Concerning Uranus and Neptune there is not much to be said, as their systems are very little known; but their masses are much larger than that of the earth, and their satellites revolve with a short periodic time. The retrograde motion and high inclination of the satellites of Uranus are very remarkable. The theory of the inclination of the orbit has been based on an assumed smallness of inclination, and it is not very easy to see to what results investigation might lead if the inclination were large. It must be admitted, however, that the Uranian system points to the probability of the existence of a primitive planet, with retrograde rotation, or at least with a very large obliquity of equator.
It appears from this review that the other members of the solar system present some phenomena which are strikingly favourable to the tidal theory of evolution, and none which are absolutely condemnatory. We shall show in the following section that there are reasons why the tidal friction arising in the planetary systems cannot have had so much effect as in the case of the earth and moon. That the indications which we have just noted were not more marked, but yet seemed to exist, agrees well with this conclusion.
§ 52. Influence of Tidal Friction on the Evolution of the Solar System.
According to the nebular hypothesis, the planets and the satellites are portions detached from contracting nebulous masses. In the following discussion that hypothesis will be accepted in its main outline, and we shall examine what modifications are necessitated by the influence of tidal friction. It may be shown that the reaction of the tides raised in the sun by the planets must have had a very small influence in changing the dimensions of the planetary orbits round the sun. From a consideration of numerical data with regard to the solar system and the planetary subsystems, it appears improbable that the planetary orbits have been sensibly enlarged by tidal friction since the origin of the several planets. But it is possible that some very small part of the eccentricities of the planetary orbits is due to this cause. From arguments similar to those advanced with regard to the solar system as a whole, it appears unlikely that the satellites of Mars, Jupiter, and Saturn originated very much nearer the present surfaces of the planets than we now observe them. But, the data being insufficient, we cannot feel sure that the alteration in the dimensions of the orbits of these satellites has not been considerable. It remains, however, nearly certain that they cannot have first originated almost in contact with the present surfaces of the planets, in the same way as in the preceding sketch (§ 50) has been shown to be probable with regard to the moon and earth. Numerical data concerning the distribution of moment of momentum in the several planetary sub-systems exhibit so striking a difference between the terrestrial system and those of the other planets that we should from this alone have grounds for believing that the modes of evolution have been considerably different. The difference appears to lie in the genesis of the moon close to the present surface of the planet, and we shall see below that solar tidal friction may be assigned as a reason to explain how it has happened that the terrestrial planet had contracted to nearly its present dimensions before the genesis of a satellite, but that this was not the case with the exterior planets. The efficiency of solar tidal friction is very much greater in its action on the nearer planets than on the further ones. The time, however, during which solar tidal friction has been operating on the external planets is probably much longer than the period of its efficiency for the interior ones, and a series of numbers proportional to the total amount of rotation destroyed in the several planets would present a far less rapid decrease as we recede from the sun than numbers simply expressive of the efficiency of tidal friction at the several planets. Nevertheless it must be admitted that the effect produced by solar tidal friction on Jupiter and Saturn has not been nearly so great as on the interior planets. And, as already stated, it is very improbable that so large an amount of momentum should have been destroyed as to materially affect the orbits of the planets round the sun.
We will now examine how the differences of distance from the sun would be likely to affect the histories of the several planetary masses. According to the nebular hypothesis, a planetary nebula contracts, and rotates quicker as it contracts. The rapidity of the revolution causes it to become unstable, or perhaps an equatorial belt gradually detaches itself; it is immaterial which of these two really takes place. In either case the separation of that part of the mass which before the change had the greatest angular momentum permits the central portion to resume a planetary shape. The contraction and the increase of rotation proceed continually until another portion is detached, and so on. There thus recur at intervals a series of epochs of instability or of abnormal change. Now tidal friction must diminish the rate of increase of rotation due to contraction, and therefore if tidal friction and contraction are at work together the epochs of instability must recur more rarely than if contraction alone acted. If the tidal retardation is sufficiently great, the increase of rotation due to contraction will be so far counteracted as never to permit an epoch of instability to occur. Since the rate of retardation due to solar tidal friction decreases rapidly as we recede from the sun, these considerations accord with what we observe in the solar system. For Mercury and Venus have no satellites, and there is a progressive increase in the number of satellites as we recede from the sun. Moreover, the number of satellites is not directly connected with the mass of the planet, for Venus has nearly the same mass as the earth and has no satellite, and the earth has relatively by far the largest satellite of the whole system. Whether this be the true cause of the observed distribution of satellites amongst the planets or not, it is remarkable that the same cause also affords an explanation, as we shall now show, of that difference between the earth with the moon and the other planets with their satellites which has caused tidal friction to be the principal agent of change with the former but not with the latter. In the case of the contracting terrestrial mass we may suppose that there was for a long time nearly a balance between the retardation due to solar tidal friction and the acceleration due to contraction, and that it was not until the planetary mass had contracted to nearly its present dimensions that an-epoch of instability could occur. It may also be noted that if there be two equal planetary masses which generate satellites, but under very different conditions as to the degree of condensation of the masses, the two satellites will be likely to differ in mass; we cannot of course tell which of the two planets would generate the larger satellite. Thus, if the genesis of the moon was deferred until a late epoch in the history of the terrestrial mass, the mass of the moon relatively to the earth would be likely to differ from the mass of other satellites relatively to their planets. If the contraction of the planetary mass be almost completed before the genesis of the satellite, tidal friction, due jointly to the satellite and to the sun, will thereafter be the great cause of change in the system; and thus the hypothesis that it is the sole cause of change will give an approximately accurate explanation of the motion of the planet and satellite at any subsequent time. We have already seen that the theory that tidal friction has been the ruling power in the evolution of the earth and moon coordinates the present motions of the two bodies and carries us back to an initial state when the moon first had a separate existence as a satellite; and the initial configuration of the two bodies is such that we are led to believe that the moon is a portion of the primitive earth detached by rapid rotation or other causes. There seems to be some reason to suppose that the earliest form in which the moon had a separate existence was as a ring or chain of meteorites; but this condition precedes that to which the dynamical investigation leads back.
Let us now turn to the other planetary sub-systems. The satellites of the larger planets revolve with short periodic times; this admits of a simple explanation, for the smallness of their masses would have prevented tidal friction from being a very efficient cause of change in the dimensions of their orbits, and the largeness of the planet’s masses would have caused them to proceed slowly in their evolution. If the planets be formed from chains of meteorites or of nebulous matter, their rotation has arisen from the excess of orbital momentum of the exterior over that of the interior matter. As we have no means of knowing how broad the chain may have been in any case, nor how much it may have closed in on the sun in course of concentration, we are unable to compute the primitive angular momentum of a planet. A rigorous method of comparison of the primitive rotations of the several planets is thus wanting. If, however, the planets were formed under similar conditions, then we should expect to find the exterior planets now rotating more rapidly than the interior ones. On making allowance for the different degrees of concentration of the planets, this is the case. That the inner satellite of Mars revolves with a period of less than a third of the planet’s rotation is perhaps the most remarkable fact in the solar system. The theory of tidal friction explains this perfectly; and this will be the ultimate fate of all satellites, because the solar tidal friction retards the planetary rotation without directly affecting the satellite’s orbital motion. Numerical comparison shows that the efficiency of solar tidal friction in retarding the terrestrial and martian rotations is of about the same degree of importance, notwithstanding the much greater distance of the planet Mars. In the above discussion it will have been apparent that the earth and moon do actually differ from the other planets to such an extent as to permit tidal friction to have been the most important factor in their history.
By an examination of the probable effects of solar tidal friction on a contracting planetary mass, we have been led to assign a cause for the observed distribution of satellites in the solar system, and this again has itself afforded an explanation of how it happened that the moon so originated that the tidal friction of the lunar tides in the earth should have been able to exercise so large an influence. We have endeavoured not only to set forth the influence which tidal friction may have, and probably has, had in the history of the system, if sufficient time be granted, but also to point out what effects it cannot have produced. These investigations afford no grounds for the rejection of the nebular hypothesis; but, while they present evidence in favour of the main outlines of that theory, they introduce modifications of considerable importance. Tidal friction is a cause of change of which Laplace’s theory took no account; and, although the activity of that cause may be regarded as mainly belonging to a later period than the events described in the nebular hypothesis, yet it seems that its influence has been of great, and in one instance of even paramount importance in determining the present condition of the planets and their satellites. Throughout the whole of this discussion it has been supposed that sufficient time is at our disposal; Sir W. Thomson and others have, however, adduced reasoning which goes to show that the history of the solar system must be comprised within a period considerably less than a hundred million years. It would perhaps be premature to accept this as the final and definite conclusion of science. If, however, it be confirmed, we shall only be permitted to accept the doctrine that tidal friction has effected considerable modification in the configuration of the moon and earth, and must reject the earlier portion of the history sketched above. (G. H. D.)
Table of Contents. Chap. 1.— On the Nature of Tides.
§ 1. Definition of tide.
§ 2. General description of tidal phenomena.
§ 3. General explanation of the cause of tides.
§ 4. Historical sketch.
Chap. II.— Tide-Generating Forces.
§ 5. Investigation of tide-generating potential and forces.
§ 6. Form of equilibrium.
§ 7. Development of tide-generating potential in terms of hour-angle and declination.
5 8. Evaluation of tide-generating forces and lunar deflexion of gravity.
$ 9. Correction to equilibrium theory for continents.
Chap. III.— Dynamical Theory of Tides.
§ 10. Historical explanation.
§ 11. Equations of motion.
§ 12. Adaptation to forced oscillations.
§ 13. Preparation for solution.
§ 14. Diurnal tide.
§ 15. Semi-diurnal tide with variable depth.
§ 16. Semi-diurnal tide with uniform depth.
§ 17. Tides of long period; Laplace’s argument from friction.
§ 18. Tides of long period in an ocean of uniform depth.
§ 19. Stability of the ocean.
§ 20. Precession and nutation.
§21. Some phenomena of tides in rivers.
Chap. IV.— Harmonic Analysis.
§ 22. Methods of applying theory to practice.
§ 23. Development of equilibrium theory of tides in terms of the elements of the orbits.
§ 24. Meteorological tides, over-tides, and compound tides.
§ 25. On the form of presentation of results of tidal observation.
§ 26. Numerical harmonic analysis for tides of short period.
§ 27. Harmonic analysis for tides of long period.
Chap. V.— Synthetic Method.
§ 28. On the method and notation.
§ 29. Semi-diurnal tides.
§ 30. Synthesis of solar and of lunar portions of the semi-diurnal tide.
§ 31. Synthesis of lunar and solar semidiurnal tides.
§ 32. Diurnal tides.
§ 33. Explanation of tidal terms in common use; datum levels.
§ 34. On reduction of observations of high and low water.
Chap. VI. —Tidal Instruments and Tidal Prediction.
§ 35. General remarks.
§ 36. The tide gauge.
§ 37. The harmonic analyser.
§ 38. The tide-predicting instrument.
§ 39. Numerical harmonic analysis and prediction.
Chap. VII. — Progress of the Tide wave over the Sea and the Tides of the British Seas.
§ 40. Meaning of cotidal lines.
§ 41. Cotidal lines of the world.
§ 42. Cotidal lines of the British seas.
Chap. VIII.— Tidal Deformation of the Solid Earth.
§ 43. Elastic tides.
§ 44. Rigidity of the earth.
§ 45. Viscous and elastico - viscous tides.
Chap. IX.— Tidal Friction.
§ 46. General explanation.
§ 47. Exact investigation of the secular effects of tidal friction.
§ 48. Amount of tidal retardation of earth’s rotation.
§ 49. Effects of tidal friction on the elements of the moon’s orbit and on the earth’s rotation.
Chap. X,— Cosmogonic Speculations FOUNDED ON TlDAL FRICTION.
§ 50. History of the earth and moon.
§51. The other planetary sub-systems.
§ 52. Influence of tidal friction on the evolution of the solar system.
