Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/448

Rh 422 W AYE T H E II Y ance 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 priori 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 parti cular 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. Elastic Of such special forms of the wave theory the most solid famous is that which assimilates light to the transverse theory, 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 casss 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 pro pagation 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 perpen dicular 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 con clusions 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 recon ciled with the view that the rigidity is the same in different isotropic media. And if we pass over this diffi culty, and inquire what kind of double refraction a crystalline solid would admit of, we find no such corre spondence 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. 1 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 Electro- favour in which light is regarded as an electromagnetic magnetic phenomenon. The dielectric medium is conceived to be tneor y. subject to a rapidly periodic &quot;electric displacement,&quot; the variations of which have the magnetic properties of an electric current. On the basis of purely electrical observa tions Maxwell calculated the velocity of propagation of such disturbances, and obtained a value not certainly dis tinguishable from the velocity of light. Such an agree ment 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 founda tions 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 mean time, 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 vibra tion, the analytical expression for an infinite train of plane waves is Acos j (Vt-x) + a j ..... (1), ( A ) in which A represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the ampli tude, and its nature depends upon the medium and must there fore here be left an open question. The phase of the wave at a given time and place is represented by o. 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, ,-,* = A/ In experimenting upon sound we are able to determine inde pendently T,, and V ; but on account of itssmallness the periodic time of luminous vibrations eludes altogether our means of observa tion, and is only known indirectly from A and V by means of (2). There is nothing arbitrary in the use of a circular function to re present 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 wave length, 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 wave lengths in harmonical progression. A well-known characteristic of waves of type (1) is that any Compo- number of trains of various amplitudes and phases, but of the same s ition oi ivave-length, are equivalent to a single train of the same type. Thus waves oJ 2 A cos &amp;lt; ( Vt - a-) + a like period. = 2Acosa.cos (Y( -x) -2Asin a. sin (Vt-x) A A = Pcos where .... (3), .... (4), 2(Asina) ,r% tan 0= -f- T (5). 2(Acosa) An important particular case is that of two component trains only. T WO . . I 2ir ,, T . ), , I 2 *Vv, i &amp;gt; I trains o: Acos I (Vt-x) + a I +A cos | (Vt-x)** *&amp;gt; wayeg _ = PcosJ ~(Vt-x) + ( A where ?- = A 2 + A - + 2AA cos(a-o ) .... (6). 1 See Stokes, &quot;Report on Double Refraction,&quot; Brit. Assoc. Report, 1862, p. 253.