§ 1. A GENERAL statement of the principles of the undulatory theory, with elementary explanations, has already been given under Light, and in the article on Ether the arguments which point to the existence of an all-pervading medium, susceptible in its various parts of an alternating change of state, have been traced by a master hand; but the subject is of such great importance, and is so intimately involved in recent optical investigation and discovery, that a more detailed exposition of the theory, with application to the leading phenomena, was reserved for a special article. That the subject is one of difficulty may be at once admitted. Even in the theory of sound, as conveyed by aerial vibrations, where we are well acquainted with the nature and properties of the vehicle, the fundamental conceptions are not very easy to grasp, and their development makes heavy demands upon our mathematical resources. That the situation is not improved when the medium is hypothetical will be easily understood. For, although the evidence is overwhelming in favour of the conclusion that light is propagated as a vibration, we are almost entirely in the dark as to what it is that vibrates and the manner of vibration. This ignorance entails an appearance of vagueness even in those parts of the subject the treatment of which would not really be modified by the acquisition of a more precise knowledge, e.g., the theory of the colours of thin plates, and of the resolving power of optical instruments. But in other parts of the subject, such as the explanation of the laws of double refraction and of the intensity of light reflected at the surface of a transparent medium, the vagueness is not merely one of language; and if we wish to reach definite results by the a prion road we must admit a hypothetical element, for which little justification can be given. The distinction here indicated should be borne clearly in mind. Many optical phenomena must necessarily agree with any kind of wave theory that can be proposed; others may agree or disagree with a particular form of it. In the latter case we may regard the special form as disproved, but the undulatory theory in the proper wider sense remains untouched.

Of such special forms of the wave theory the most famous is that which assimilates light to the transverse vibrations of an elastic solid. Transverse they must be in order to give room for the phenomena of polarization. This theory is a great help to the imagination, and allows of the deduction of definite results which are at any rate mechanically possible. An isotropic solid has in general two elastic properties—one relating to the recovery from an alteration of volume, and the other to the recovery from a state of shear, in which the strata are caused to slide over one another. It has been shown by Green that it would be necessary to suppose the luminiferous medium to be incompressible, and thus the only admissible differences between one isotropic medium and another are those of rigidity and of density. Between these we are in the first instance free to choose. The slower propagation of light in glass than in air may be equally well explained by supposing the rigidity the same in both cases while the density is greater in glass, or by supposing that the density is the same in both cases while the rigidity is greater in air. Indeed there is nothing, so far, to exclude a more complicated condition of things, in which both the density and rigidity vary in passing from one medium to another, subject to the one condition only of making the ratio of velocities of propagation equal to the known refractive index between the media.

When we come to apply this theory to investigate the intensity of light reflected from (say) a glass surface, and to the diffraction of light by very small particles (as in the sky), we find that a reasonable agreement with the facts can be brought about only upon the supposition that the rigidity is the same (approximately, at any rate) in various media, and that the density alone varies. At the same time we have to suppose that the vibration is perpendicular to the plane of polarization.

Up to this point the accordance may be regarded as fairly satisfactory; but, when we extend the investigation to crystalline media in the hope of explaining the observed laws of double refraction, we find that the suppositions which would suit best here are inconsistent with the conclusions we have already arrived at. In the first place, and so long as we hold strictly to the analogy of an elastic solid, we can only explain double refraction as depending upon anisotropic rigidity, and this can hardly be reconciled with the view that the rigidity is the same in different isotropic media. And if we pass over this difficulty, and inquire what kind of double refraction a crystalline solid would admit of, we find no such correspondence with observation as would lead us to think that we are upon the right track. The theory of anisotropic solids, with its twenty-one elastic constants, seems to be too wide for optical double refraction, which is of a much simpler character.

For these and other reasons, especially the awkwardness with which it lends itself to the explanation of dispersion, the elastic solid theory, valuable as a piece of purely dynamical reasoning, and probably not without mathematical analogy to the truth, can in optics be regarded only as an illustration.

In recent years a theory has been received with much favour in which light is regarded as an electromagnetic phenomenon. The dielectric medium is conceived to be subject to a rapidly periodic “electric displacement,” the variations of which have the magnetic properties of an electric current. On the basis of purely electrical observations Maxwell calculated the velocity of propagation of such disturbances, and obtained a value not certainly distinguishable from the velocity of light. Such an agreement is very striking; and a further deduction from the theory, that the specific inductive capacity of a transparent medium is equal to the square of the refractive index, is supported to some extent by observation. The foundations of the electrical theory are not as yet quite cleared of more or less arbitrary hypothesis; but, when it becomes certain that a dielectric medium is susceptible of vibrations propagated with the velocity of light, there will be no hesitation in accepting the identity of such vibrations with those to which optical phenomena are due. In the meantime, and apart altogether from the question of its probable truth, the electromagnetic theory is very instructive, in showing us how careful we must be to avoid limiting our ideas too much as to the nature of the luminous vibrations.

§ 2. Plane Waves of Simple Type.

Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is Acos j y(Vi-α) + α J (1), in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and must therefore here be left an open question. The phase of the wave at a given time and place is represented by a. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period is τ = λ ∕y (2)<

In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).

There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wavelength, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wavelengths in harmonical progression.

A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same wave-length, are equivalent to a single train of the same type. Thus SAcos ^(Vi-αr) + α

= SA cos a. cos -f ( Vt - x) - SA sin «. sin— (Vt-x) A λ

= Pcos j y^(VZ-≈) + 0 I ..... (3), where P 2 = (SAcosa) 2 + S(Asina) 2 .... (4),

S(Asina)

tan φ = -½ ( 5 ).

S(Acosa)

An important particular case is that of two component trains only. Acos ~(Υt~x') + a y +A'cos ^(Vi-iB) + α' ∣

= Pcos j ~(Vt~x) + φ , where P 2 =≈A 2 ÷A' 2 + 2AA'cos(a-a') .... (6).

The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. Λ group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.

According to the phase-relation, determined by (α-α'), the amplitude of the resultant may vary from (A - A') to (A +A'). If A' and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is usually spoken of as the interference of light. From a purely dynamical point of view the word is not very appropriate, the vibrations being simply superposed with as little interference as can be imagined.

§ 3. Intensity.

The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to the square of the amplitude.

Practical photometry is usually founded upon the law of inverse squares (Light, vol. xiv. p. 583); and it should be remembered that the method involves essentially the use of a diffusing screen, the illumination of which, seen in a certain direction, is assumed to be independent of the precise direction in which the light falls upon it; for the distance of a candle, for example, cannot be altered without introducing at the same time a change in the apparent magnitude, and therefore in the incidence of some part at any rate of the light.

With this objection is connected another which is often of greater importance, the necessary enfeeblement of the light by the process of diffusion. Aud, if to maintain the brilliancy we substitute regular reflectors for diffusing screens, the method breaks down altogether by the apparent illumination becoming independent of the distance of the source of light.

The use of a revolving disk with transparent and opaque sectors in order to control the brightness, as proposed by Fox Talbot, may often be recommended in scientific photometry, when a great loss of light is inadmissible. The law that, when the frequency of intermittence is sufficient to give a steady appearance, the brightness is proportional to the angular magnitude of the open sectors appears to be well established.

§ 4. Resultant of a Large Humber of Vibrations of Arbitrary Phase.

We have seen that the resultant of two vibrations of equal amplitude is wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distributed, and may vary from n 2 to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit?

The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to two opposite phases. We may then conveniently discard the idea of phase, and regard the amplitudes as at random positive or negative. If all the signs are the same, the intensity is n 2 ; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from 0 to n 2 , the smaller values are much more probable than the greater.

The simplest part of the problem relates to what is called in the theory of probabilities the “expectation” of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations are positive is 2 -n , and thus the expectation of intensity corresponding to this contingency is 2 -n . n 2 . In like manner the expectation corresponding to the number of positive vibrations being (n-1) is 2 -".n.(π-2) 2 , and so on. The whole expectation of intensity is thus j l.B≈+n.(a-2r+⅛L≤∖ n -4)≈ ÷-⅜2⅞- l ι ,t - 6 > 1 ÷ · · · J · · · (»· Now the sum of the (n + l) terms of this series is simply n, as may be proved by comparison of coefficients of x 2 in the equivalent forms

(e≈+e-≈) n =2 n (l + ⅜x 2 + . . .) n =β"≈+jιe(,, - 2 ) ar +-^- 2 1 ^(,, - 4 ) x + . . .

The expectation of intensity is therefore n, and this whether n be great or small.

The same conclusion holds good when the phases are unrestricted. From (4), § 2, if A = 1,

P 2 = 7i∙+22 cos (<x 2 — α 1 ) (2), where under the sign of summation are to be included the cosines of the ½n(n - 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P 2 is n.

The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, the mean intensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order n 2 , we can infer nothing as to the value of the sum of the series in comparison with n.

Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved that the probability of a resultant intermediate in amplitude between r and r + dr is - e~ r 'l n rdr ....... (3). n

The probability of an amplitude less than r is thus - ∕' r e- r2 l n rdr≈l-e~ r2 ' n (4).

n/o or, which is the same thing, the probability of an amplitude greater than r is e - r2/n . . . . . . . . (5).

The accompanying table gives the probabilities of intensities less than the fractious of n named in the first column. For example, the probability of intensity less than n is ∙6321.

It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations. [table]

The mean intensity, expressed by - Γ*e~ r2 l n .r i .rdr, udo is, as we have already seen, equal to τι.

It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two candles are twice as bright as one.

§ 5. Propagation of Waves in General.

It has been shown under Optics that a system of rays, however many reflexions or refractions they may have undergone, are always normal to a certain surface, or rather system of surfaces. From our present point of view these surfaces are to be regarded as wave-surfaces, that is, surfaces of constant phase. It is evident that, so long as the radius of curvature is very large in comparison with λ, each small part of a wave-surface propagates itself just as an infinite plane wave coincident with the tangent plane would do. If we start at time t with a given surface, the corresponding wavesurface at time t+dt is to be found by prolonging every normal by the length V dt, where V denotes the velocity of propagation at the place in question. If the medium be uniform, so that V is constant, the new surface is parallel to the old one, and this property is retained however many short intervals of time be considered in succession. A wave-surface thus propagates itself normally, and the corresponding parts of successive surfaces are those whieh lie upon the same normal. In this sense the normal may be regarded as a ray, but the idea must not be pushed to streams of light limited to pass through small apertures. The manner in which the phase is determined by the length of the ray, and the conditions under which energy may be regarded as travelling along a ray, will be better treated under the head of Huygens’s principle, and the theory of shadows (§ 10).

From the law of propagation, according to which the wavesurfaces are always as far advanced as possible, it follows that the course of a ray is that for which the time, represented by ∫ V -1 ds, is a minimum. This is Fermat’s principle of least time. Since the refractive index (μ) varies as V -1 ,...