Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XIV

by Josiah Willard Gibbs

XIV.

A COMPARISON OF THE ELASTIC AND THE ELECTRICAL THEORIES OF LIGHT WITH RESPECT TO THE LAW OF DOUBLE REFRACTION AND THE DISPERSION OF COLORS.

[American Journal of Science, ser. 3, vol. xxxv, pp. 467–475, June, 1888.]

It is claimed for the electrical^[1] theory of light that it is free from serious difficulties, which beset the explanation of the phenomena of light by the dynamics of elastic solids. Just what these difficulties are, and why they do not occur in the explanation of the same phenomena by the dynamics of electricity, has not perhaps been shown with all the simplicity and generality which might be desired. Such a treatment of the subject is however the more necessary on account of the ever-increasing bulk of the literature on either side, and the confusing multiplicity of the elastic theories. It is the object of this paper to supply this want, so far as respects the propagation of plane waves in transparent and sensibly homogeneous media. The simplicity of this part of the subject renders it appropriate for the first test of any optical theory, while the precision of which the experimental determinations are capable, renders the test extremely rigorous.

It is moreover, as the writer believes, an appropriate time for the discussion proposed, since on one hand the experimental verification of Fresnel's Law has recently been carried to a degree of precision far exceeding anything which we have had before,^[2] and on the other, the discovery of a remarkable theorem relating to the vibrations of a strained solid^[3] has given a new impulse to the study of the elastic theory of light. Let us first consider the facts to which a correct theory must conform.

It is generally admitted that the phenomena of light consist in motions (of the type which we call wave-motions) of something which exists both in space void of ponderable matter, and in the spaces between the molecules of bodies, perhaps also in the molecules themselves. The kinematics of these motions is pretty well understood; the question at issue is whether it agrees with the dynamics of elastic solids or with the dynamics of electricity.

In the case of a simple harmonic wave-motion, which alone we need consider, the wave-velocity (${\text{V}}$) is the quotient of the wave-length ($l$) by the period of vibration ($p$). These quantities can be determined with extreme accuracy. In media which are sensibly homogeneous but not isotropic the wave-velocity ${\text{V}},$ for any constant value of the period, is a quadratic function of the direction cosines of a certain line, viz., the normal to the so-called "plane of polarization." The physical characteristics of this line have been a matter of dispute. Fresnel considered it to be the direction of displacement. Others have maintained that it is the common perpendicular to the wave-normal and the displacement. Others again would define it as that component of the displacement which is perpendicular to the wave-normal. This of course would differ from Fresnel's view only in case the displacements are not perpendicular to the wave-normal, and would in that case be a necessary modification of his view. Although this dispute has been one of the most celebrated in physics, it seems to be at length substantially settled, most directly by experiments upon the scattering of light by small particles, which seems to show decisively that in isotropic media at least the displacements are normal to the "plane of polarization," and also, with hardly less cogency, by the diflSculty of accounting for the intensities of reflected and refracted light on any other supposition.^[4] It should be added that all diversity of opinion on this subject has been confined to those whose theories are based on the dynamics of elastic bodies. Defenders of the electrical theory have always placed the electrical displacement at right angles to the "plane of polarization." It will, however, be better to assume this direction of the displacement as probable rather than as absolutely certain, not so much because many are likely to entertain serious doubts on the subject, as in order not to exclude views which have at least a historical interest.

The wave-velocity, then, for any constant period, is a quadratic function of the cosines of a certain direction, which is probably that of the displacement, but in any case determined by the displacement and the wave-normal. The coefficients of this quadratic function are functions of the period of vibration. It is important to notice that these coefficients vary separately, and often quite differently, with the period, and that the case does not at all resemble that of a quadratic function of the direction-cosines multiplied by a quantity depending on the period.

In discussing the dynamics of the subject we may gain something in simplicity by considering a system of stationary waves, such as results from two similar systems of progressive waves moving in opposite directions. In such a system the energy is alteniately entirely kinetic and entirely potential. Since the total energy is constant, we may set the average kinetic energy per unit of volume at the moment when there is no potential energy, equal to the average potential energy per unit of volume when there is no kinetic energy.^[5] We may call this the equation of energies. It will contain the quantities $l$ and $p,$ and thus furnish an expression for the velocity of either system of progressive waves. We have to see whether the elastic or the electric theory gives the expression most conformed to the facts.

Let us first apply the elastic theory to the case of the so-called vacuum. If we write $h$ for the amplitude measured in the middle between two nodal planes, the velocities of displacement will be as ${\frac {h}{p}},$ and the kinetic energy will be represented by ${\text{A}}{\frac {h^{2}}{p^{2}}},$ where ${\text{A}}$ is a constant depending on the density of the medium. The potential energy, which consists in distortion of the medium, may be represented by ${\text{B}}{\frac {h^{2}}{l^{2}}},$ where ${\text{B}}$ is a constant depending on the rigidity of the medium. The equation of energies, on the elastic theory, is therefore

${\text{A}}{\frac {h^{2}}{p^{2}}}={\text{B}}{\frac {h^{2}}{l^{2}}}$

(1)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {\text{B}}{\text{A}}}\cdot$

(2)

In the electrical theory, the kinetic energy is not determined by the simple formula of ordinary dynamics from the square of the velocity of each element, but is found by integrating the product of the velocities of each pair of elements divided by the distance between them. Very elementary considerations suffice to show that a quantity thus determined when estimated per unit of volume will vary as the square of the wave-length. We may therefore set ${\text{F}}l{\frac {h^{2}}{p^{2}}}$ for the kinetic energy, ${\text{F}}$ being a constant. The potential energy does not consist in distortion of the medium, but depends upon an elastic resistance to the separation of the electricities, which constitutes the electrical displacement, and is proportioned to the square of this displacement. The average value of the potential energy per unit of volume will therefore be represented in the electrical theory by ${\text{G}}h^{2}$ where ${\text{G}}$ is a constant, and the equation of energies will be

${\text{F}}l{\frac {h^{2}}{p^{2}}}={\text{G}}h^{2}$

(3)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {\text{G}}{\text{F}}}\cdot$

(4)

Both theories give a constant velocity, as is required. But it is instructive to notice the profound difference in the equations of energy from which this result is derived. In the elastic theory the square of the wave-length appears in the potential energy as a divisor; in the electrical theory it appears in the kinetic energy as a factor.

Let us now consider how these equations will be modified by the presence of ponderable matter, in the most general case of transparent and sensibly homogeneous bodies. This subject is rendered much more simple by the fact that the distances between the ponderable molecules are very small compared with a wave-length. Or, what amounts to the same thing, but may present a more distinct picture to the imagination, the wave-length may be regarded as enormously great in comparison with the distances between neighboring molecules. Whatever view we take of the motions which constitute light, we can hardly suppose them (disturbed as they are by the presence of the ponderable molecules) to be in strictness represented by the equations of wave-motion. Yet in a certain sense a wave-motion may and does exist. If, namely, instead of the actual displacement at any point, we consider the average displacement in a space large enough to contain an immense number of molecules, and yet small as measured by a wave-length, such average displacements may be represented by the equations of wave-motion; and it is only in this sense that any theory of wave-motion can apply to the phenomena of light in transparent bodies. When we speak of displacements, amplitudes, velocities (of displacement), etc., it must therefore be understood in this way.

The actual kinetic energy, on either theory, will evidently be greater than that due to the motion thus averaged or smoothed, and to a degree presumably depending on the direction of the displacement. But since displacement in any direction may be regarded as compounded of displacements in three fixed directions, the additional energy will be a quadratic function of the components of velocity of displacement, or, in other words, a quadratic function of the direction-cosines of the displacement multiplied by the square of the amplitude and divided by the square of the period.^[6] This additional energy may be understood as including any part of the kinetic energy of the wave-motion which may belong to the ponderable particles. The term to be added to the kinetic energy on the electric theory may therefore be written $f_{\text{D}}{\frac {h^{2}}{p^{2}}},$ where $f_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. The elastic theory requires a term of precisely the same character, but since the term to which it is to be added is of the same general form, the two may be incorporated in a smgle term of the form ${\text{A}}_{\text{D}}{\frac {h^{2}}{p^{2}}},$ where ${\text{A}}_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. We must, however, notice that both ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ are not entirely independent of the period. For the manner in which the flux of the luminiferous medium is distributed among the ponderable molecules will naturally depend somewhat upon the period. The same is true of the degree to which the molecules may be thrown into vibration. But ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ will be independent of the wave-length (except so far as this is connected with the period), because the wave-length is enormously great compared with the size of the molecules and the distances between them.

The potential energy on the elastic theory must be increased by a term of the form $b_{\text{D}}h^{2},$ where $b_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. For the ponderable particles must oppose a certain elastic resistance to the displacement of the ether, which in æolotropic bodies will presumably be different in different directions. The potential energy on the electric theory will be represented by a single term of the same form, say ${\text{G}}_{\text{D}}h^{2},$ where a quadratic function of the direction-cosines of the displacement, ${\text{G}}_{\text{D}},$ takes the place of the constant ${\text{G}},$ which was sufficient when the ponderable particles were absent. Both ${\text{G}}_{\text{D}}$ and $b_{\text{D}}$ will vary to some extent with the period, like ${\text{A}}_{\text{D}}$ and $f_{\text{D}},$ and for the same reason.

In regard to that potential energy, which on the elastic theory is independent of the direct action of the ponderable molecules, it has been supposed that in sdolotropic bodies the effect of the molecules is such as to produce an asolotropic state in the ether, so that the energy of a distortion varies with its orientation. This part of the potential energy will then be represented by ${\text{B}}_{\text{ND}}{\frac {h^{2}}{l^{2}}},$ where ${\text{B}}_{\text{ND}}$ is a function of the directions of the wave-normal and the displacement. It may easily be shown that it is a quadratic function both of the direction-cosines of the wave-normal and of those of the displacement Also, that if the ether in the body when undisturbed is not in a state of stress due to forces at the surface of the body, or if its stress is uniform in all directions, like a hydrostatic pressure, the function ${\text{B}}_{\text{ND}}$ must be symmetrical with respect to the two sets of direction-cosines.

The equation of energies for the elastic theory is therefore

${\text{A}}_{\text{D}}{\frac {h^{2}}{l^{2}}}={\text{B}}_{\text{ND}}{\frac {h^{2}}{l^{2}}}+b_{\text{D}}h^{2},$

(5)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {{\text{B}}_{\text{ND}}}{{\text{A}}_{\text{D}}-b_{\text{D}}p^{2}}}\cdot$

(6)

The equation of energies for the electrical theory is

${\text{F}}l^{2}{\frac {h^{2}}{p^{2}}}+f_{\text{D}}{\frac {h^{2}}{p^{2}}}={\text{G}}_{\text{D}}h^{2},$

(7)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {{\text{G}}_{\text{D}}}{\text{F}}}-{\frac {f_{\text{D}}}{{\text{F}}p^{2}}}\cdot$

(8)

It is evident at once that the electrical theory gives exactly the form that we want. For any constant period the square of the wave-velocity is a quadratic function of the direction-cosines of the displacement. When the period varies, this function varies, the different coefficients in the function varying separately, because ${\text{G}}_{\text{D}}$ and $f_{\text{D}}$ will not in general be similar functions.^[7] If we consider a constant direction of displacement while the period varies, ${\text{G}}_{\text{D}}$ and $f_{\text{D}}$ will only vary so far as the type of the motion varies, i.e., so far as the manner in which the flux distributes itself among the ponderable molecules and intermolecular spaces, and the extent to which the molecules take part in the motion are changed. There are cases in which these vary rapidly with the period, viz., cases of selective absorption and abnormal dispersion. But we may fairly expect that there will be many cases in which the character of the motion in these respects will not vary much with the period. ${\frac {{\text{G}}_{\text{D}}}{\text{F}}}$ and ${\frac {f_{\text{D}}}{\text{F}}}$ will then be sensibly constant and we have an approximate expression for the general law of dispersion, which agrees remarkably well with experiment.^[8]

If we now return to the equation of energies obtained from the elastic theory, we see at once that it does not suggest any such relation as experiment has indicated, either between the wave-velocity and the direction of displacement, or between the wave-velocity and the period. It remains to be seen whether it can be brought to agree with experiment by any hypothesis not too violent.

In order that ${\text{V}}^{2}$ may be a quadratic function of any set of direction-cosines, it is necessary that ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ shall be independent of the direction of the displacement, in other words, in the case of a crystal like Iceland spar, that the direct action of the ponderable molecules upon the ether, shall affect both the kinetic and the potential energy in the same way, whether the displacement take place in the direction of the optic axis or at right angles to it. This is contrary to everything which we should expect. If, nevertheless, we make this supposition, it remains to consider ${\text{B}}_{\text{ND}}.$ This must be a quadratic function of a certain direction, which is almost certainly that of the displacement If the medium is free from external stress (other than hydrostatic), ${\text{B}}_{\text{ND}},$ as we have seen, is symmetrical with respect to the wave-normal and the direction of displacement, and a quadratic function of the direction-cosines of each. The only single direction of which it can be a function is the common perpendicular to these two directions. If the wave-normal and the displacement are perpendicular, the direction-cosines of the common perpendicular to both will be linear fimctions of the direction-cosines of each, and a quadratic function of the direction-cosines of the common perpendicular will be a quadratic function of the direction-cosines of each. We may thus reconcile the theory with the law of double refraction, in a certain sense, by supposing that ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ are independent of the direction of displacement, and that ${\text{B}}_{\text{ND}}$ and therefore ${\text{V}}^{2}$ is a quadratic function of the direction-cosines of the common perpendicular to the wave-normal and the displacement. But this supposition, besides its intrinsic improbability so far as ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ are concerned, involves a direction of the displacement which is certainly or almost certainly wrong.

We are thus driven to suppose that the undisturbed medium is in a state of stress, which, moreover, is not a simple hydraulic stress. In this case, by attributing certain definite physical properties to the medium, we may make the function ${\text{B}}_{\text{ND}}$ become independent of the direction of the wave-normal, and reduce to a quadratic function of the direction-cosines of the displacement.^[9] This entirely satisfies Fresnel's Law, including the direction of displacement, if we can suppose ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ independent of the direction of displacement. But this supposition, in any case difficult for aeolotropic bodies, seems quite irreconcilable with that of a permanent (not hydrostatic) stress. For this stress can only be kept up by the action of the ponderable molecules, and by a sort of action which hinders the passage of the ether past the molecules. Now the phenomena of reflection and refraction would be very different from what they are, if the optical homogeneity of a crystal did not extend up very close to the surface. This implies that the stress is produced by the ponderable particles in a very thin lamina at the surface of the crystal, much less in thickness, it would seem probable, than a wave-length of yellow light. And this again implies that the power of the ponderable particles to pin down the ether, as it were, to a particular position is very great, and that the term in the energy relating to the motion of the ether relative to the ponderable particles is very important. This is the term containing the factor $b_{\text{D}},$ which it is difficult to suppose independent of the direction of displacement because the dimensions and arrangement of the particles are different in different directions. But our present hypothesis has brought in a new reason for supposing $b_{\text{D}}$ depend on the direction of displacement, viz., on account of the stress of the medium. A general displacement of the medium midway between two nodal planes, when it is restrained at innumerable points by the ponderable particles, will produce special distortions due to these particles. The nature of these distortions is wholly determined by the direction of displacement, and it is hard to conceive of any reason why the energy of these distortions should not vary with the direction of displacement, like the energy of the general distortion of the wave-motion, which is partly determined by the displacement and partly by the wave-normal.^[10]

But the difficulties of the elastic theory do not end with the law of double refraction, although they are there more conspicuous on account of the definite and simple law by which they can be judged. It does not easily appear how the equation of energies can be made to give anything like the proper law of the dispersion of colors. Since for given directions of the wave-normal and displacement, or in an isotropic body, ${\text{B}}_{\text{ND}}$ is constant, and also ${\text{A}}_{\text{D}}$ and $b_{\text{D}},$ except so far as the type of the vibration varies, the formula requires that the square of the index of refraction (which is inversely as ${\text{V}}^{2}$) should be equal to a constant diminished by a term proportional to the square of the period, except so far as this law is modified by a variation of the type of vibration. But experiment shows nothing like this law. Now the variation in the type of vibration is sometimes very important,—it plays the leading rôle in the phenomena of selective absorption and abnormal dispersion,—but this is certainly not always the case. It seems hardly possible to suppose that the type of vibration is always so variable as entirely to mask the law which is indicated by the formula when ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ (with ${\text{B}}_{\text{ND}}$) are regarded as constant. This is especially evident when we consider that the effect on the wave-velocity of a small variation in the type of vibration will be a small quantity of the second order.^[11]

The phenomena of dispersion, therefore, corroborate the conclusion which seemed to follow inevitably from the law of double refraction alone.

1. The term electrical seems the most simple and appropriate to describe that theory of light which makes it consist in electrical motions. The cases in which any distinctively magnetic action is involved in the phenomena of light are so exceptional, that it is difficult to see any sufficient reason why the general theory should be called electromagnetic unless we are to call all phenomena electromagnetic which depend on the motions of electricity.
2. In the recent experiments of Professor Hastings relating to the index of refraction of the extraordinary ray in Iceland spar for the spectral line D₂ and a wave-normal inclined at about 31° to the optic axis, the difference between the observed and the calculated values was only two or three units in the sixth decimal place (in the seventh significant figure), which was about the probable error of the determinations. See Am. Jour. Sci. ser. 3, vol. xxxv, p. 60.
3. Sir Wm. Thomson has shown that if an elastic incompressible solid in which the potential energy of any homogeneous strain is proportional to the sum of the squares of the reciprocals of the principal elongations minus three is subjected to any homogeaeons strain by forces applied to its surface, the transmission of plane waves of distortion, superposed on this homogeneous strain, will follow exactly Fresnel's law (including the direction of displacement), the three principal velocities being proportional to the reciprocals of the principal elongations. It must be a surprise to mathematicians and physicists to learn that a theorem of such simplicity and beauty has been waiting to be discovered in a field which has been so carefully gleaned. See page 116 of the current volume (xxv) of the Philosophical Magazine.
4. "At the same time, if the above reasoning be valid, the question as to the direction of the vibrations in polarized light is decided in accordanoe with the view of Fresnel. . . . I confess I cannot see any room for doubt as to the result it leads to. . . . I only mean that if light, as is generally supposed, consists of transversal vibrations similar to those which take place in an elastic solid, the vibration must be normal to the plane of polarization." Lord Rayleigh "On the light from the Sky, its Polarization and Color;" Phil. Mag. (4), xli (1871), p. 109.
   "Green's dynamics of polarization by reflexion, and Stokes' dynamics of the diffraction of polarized light, and Stokes' and Rayleigh's dynamics of the blue sky, all agree in, as it seems to me, irrefragably, demonstrating Fresnel's original conclusion, that in plane polarized light the line of vibration is perpendicular to the plane of polarization." Sir Wm. Thomson, loc. citat.
5. The terms kinetic energy and potential energy will be used in this paper to denote these average values.
6. For proof in extenso of this proposition, when the motions are supposed electrical, the reader is referred to page 187 of this volume.
7. But ${\text{G}}_{\text{D}},f_{\text{D}},$ and ${\text{V}}^{2},$ considered as funotions of the direction of displacement, are all subject to any law of symmetry which may belong to the structure of the body considered. The resulting optical characteristics of the different crystallographic systems are given on pages 192–194.
8. This will appear most distinctly if we consider that ${\text{V}}$ divided by the velocity of light in vacuo gives the reciprocal of the index of refraction, and $p$ multiplied by the same quantity gives the wave-length in vacuo.
9. See note on page 224.
10. The reader may perhaps ask how the above reasoning is to be reconciled with the fact that the law of double refraction has been so often deduced from the elastic theory. The troublesome terms are $b_{\text{D}}$ and the variable part of ${\text{A}}_{\text{D}},$ which express the direct action of the ponderable molecules on the ether. So far as the (quite limited) reading and recollection of the present writer extend, those who have sought to derive the law of double refraction from the theory of elastic solids have generally either neglected this direct action—a neglect to which Professor Stokes calls attention more than once in his celebrated "Report on Double Refraction" (Brit. Assoc., 1862, pp. 264, 268)—or taking account of this action they have made shipwreck upon a law different from Fresnel's and contradicted by experiment.
11. See pages 190, 191 of this volume, or Lord Rayleigh's Theory of Sound, vol. i, p. 84.