Cauchy vs Green

The work of Green proved a stimulus not only to MacCullagh but to Cauchy, who now (1839) published yet a third theory of reflexion.[31]

This appears to have owed its origin to a remark of Green’s,[32] that the longitudinal wave might be avoided in either of two ways—namely, by supposing its velocity to be indefinitely great or indefinitely small.

Green curtly dismissed the latter alternative and adopted the former, on the ground that the equilibrium of the medium would be unstable if its compressibility were negative (as it must be if the velocity of longitudinal waves is to vanish).

Cauchy, without attempting to meet Green’s objection, took up the study of a medium whose elastic constants are connected by the equation

so that the longitudinal vibrations have zero velocity; and showed that if the aethereal vibrations are supposed to be executed at right angles to the plane of polarization, and if the rigidity of the aether is assumed to be the same in all media, a ray which is reflected will obey the sine-law and tangent-law of Fresnel. The boundary-conditions which he adopted in order to obtain this result were the continuity of the displacement e and of its derivate

, where the axis of x is taken at right angles to the interface.[33] These are not the true boundary-conditions for general elastic solids; but in the particular case now under discussion, where the rigidity is the same in the two media, they yield the same equations as the conditions correctly given by Green.

The aether of Cauchy’s third theory of reflexion is well worthy of some further study.

It is generally known as the contractile or labile[34] aether, the names being due to William Thomson (Lord Kelvin), who discussed it long afterwards.[35] It may be defined as an elastic medium of (negative) compressibility such as to make the velocity of the longitudinal wave zero: this implies that no work is required to be done in order to give the medium any small irrotational disturbance. An example is furnished by homogeneous foam free from air and held from collapse by adhesion to a containing vessel.

Cauchy, as we have seen, did not attempt to refute Green’s objection that such a medium would be unstable; but, as Thomson remarked, every possible infinitesimal motion of the medium is, in the elementary dynamics of the subject, proved to be resolvable into coexistent wave-motions.

If, then, the velocity of propagation for each of the two kinds of wave-motion is real, the equilibrium must be stable, provided the medium either extends through boundless space or has a fixed containing vessel as its boundary.

When the rigidity of the luminiferous medium is supposed to have the same value in all bodies, the conditions to be satisfied at an interface reduce to the continuity of the displacement e, of the tangential components of curl e, and of the scalar quantity (k + n) div e across the interface.

When a transverse wave is incident on an interface, it gives rise in general to reflected and refracted waves of both the transverse and the longitudinal species.

In the case of the contractile aether, for which the velocity of propagation of the longitudinal waves is very small, the ordinary construction for refracted waves shows that the directions of propagation of the reflected and refracted longitudinal waves will be almost normal to the interface.

The longitudinal waves will therefore contribute only to the component of displacement normal to the interface, not to the tangential components: in other words, the only tangential components of displacement at the interface are those due to the three transverse waves-the incident, reflected, and refracted.

Moreover, the longitudinal waves do not contribute at all to curl e; and, therefore, in the contractile aether, the conditions that the tangential components of e and of n curl e shall be continuous across an interface are satisfied by the distortional part of the disturbance taken alone.

The condition that the component of e normal to the interface is to be continuous is not satisfied by the distortional part of the disturbance taken alone, but is satisfied when the distortional and compressional parts are taken together.

The energy carried away by the longitudinal waves is infinitesimal, as might be expected, since no work is required in order to generate an irrotational displacement.

Hence, with this aether, the behaviour of the transverse waves at an interface may be specified without considering the irrotational part of the disturbance at all, by the conditions that the conservation of energy is to hold and that the tangential components of e and of n curl e are to be continuous.

But if we identify these transverse waves with light, assuming that the displacement e is at right angles to the plane of polarization of the light, and assuming moreover that the rigidity n is the same in all media[36] (the differences between media depending on differences in the inertia ρ), we have exactly the assumptions of Fresnel’s theory of light: whence it follows that transverse waves in the labile aether must obey in reflexion the sinc-law and tangent-law of Fresnel.

The great advantage of the labile aether is that it overcomes the difficulty about securing continuity of the normal component of displacement at an interface between two media: the light-waves taken alone do not satisfy this condition of continuity; but the total disturbance consisting of light-waves and irrotational disturbance taken together does satisfy it; and this is ensured without allowing the irrotational disturbance to carry off any of the energy.[37]

William Thomson (Lord Kelvin, b. 1824, d. 1908) devoted much attention to the labile aether. He was at one time led to doubt the validity of this explanation of light[38].

When investigating the radiation of energy from a vibrating rigid globe embedded in an infinite elastic-solid aether, he found that in some cases the irrotational waves would carry away a considerable part of the energy if the aether were of the labile type.

This difficulty, however, was removed by the observation[39] that it is sufficient for the fulfilment of Fresnel’s laws if the velocity of the irrotational waves in one of the two media is very small, without regard to the other medium.

Following up this idea, Thomson assumed that in space void of ponderable matter the aether is practically incompressible by the forces concerned in light-waves, but that in the space occupied by liquids and solids it has a negative compressibility, so as to give zero velocity for longitudinal aether-waves in these bodies.

This assumption was based on the conception that material atoms move through space without displacing the aether: a conception which, as Thomson remarked, contradicts the old scholastic axiom that two different portions of matter cannot simultaneously occupy the same space.[40]

He supposed the aether to be attracted and repelled by the atoms, and thereby to be condensed or rarefied.[41]

The year 1839, which saw the publication of MacCullagh’s dynamical theory of light and Cauchy’s theory of the labile aether, was memorable also for the appearance of a memoir by Green on crystal-optics.[42]

This really contains two distinct theories, which respectively resemble Cauchy’s First and Second Theories: in one of them, the stresses in the undisturbed state of the aether are supposed to vanish, and the vibrations of the aether are supposed to be executed parallel to the plane of polarization of the light.

In the other theory, the initial stresses are not supposed to vanish, and the aether-vibrations are at right angles to the plane of polarization. The two investigations are generally known as Green’s First and Second ‘Theories of crystal-optics.

The foundations of both theories are, however, the same. Green first of all determined the potential energy of a strained crystalline solid; this in the most general case involves 27 constants, or 21 if there is no initial stress.[43] If, however, as is here assumed, the medium possesses the planes of symmetry at right angles to each other, the number of constants reduces to 12, or to 9 if there is no initial stress; if e denote the displacement, the potential energy per unit volume may be written The usual variational equation

then yields the differential equations of motion, namely:

and two similar equations.

These differ from Cauchy’s fundamental equations in having greater generality: for Cauchy’s medium was supposed to be built up of point-centres of force attracting each other according to some function of the distance; and, as we have seen, there are limitations in this method of construction, which render it incompetent to represent the most general type of elastic solid.

Cauchy’s equations for crystalline media are, in fact, exactly analogous to the equations originally found by Navier for isotropic media, which contain only one elastic constant instead of two.

The number of constants in the above equations still exceeds the three which are required to specify the properties of a biaxal crystal: and Green now proceeds to consider how the number may be reduced. The condition which he imposes for this purpose is that for 2 of the 3 waves whose front is parallel to a given plane, the vibration of the aethereal molecules shall be accurately in the plane of the wave: in other words, that two of the three waves shall be purely distortional, the remaining one being consequently a normal vibration. This condition gives five relations,[44] which may be written:—

where μ denotes a new constant.[45] Thus the potential energy per unit volume may be written

This expression contains the correct number of constants, namely, 4: 3 of them represent the optical constants of a biaxal crystal. 1 (namely, μ) represents the square of the velocity of propagation of longitudinal waves.

The two sheets of the wave-surface which correspond to the two distortional waves form a Fresnel’s wave-surface.

The third sheet, which corresponds to the longitudinal wave, being an ellipsoid.

The directions of polarization and the wave-velocities of the distortional waves are identical with those assigned by Fresnel, provided it is assumed that the direction of vibration of the aether-particles is parallel to the plane of polarization; but this last assumption is of course inconsistent with Green’s theory of reflexion and refraction.

In his Second Theory, Green, like Cauchy, used the condition that for the waves whose fronts are parallel to the coordinate planes, the wave-velocity depends only on the plane of polarization, and not on the direction of propagation. He thus obtained the equations already found by Cauchy—

G - f = H - g = I - h.

The wave-surface in this case also is Fresnel’s, provided it is assumed that the vibrations of the aether are executed at right angles to the plane of polarization.

The principle which underlies the Second Theories of Green and Cauchy is that the aether in a crystal resembles an elastic solid which is unequally pressed or pulled in different directions by the unmoved ponderable matter.

This idea appealed strongly to W. Thomson (Kelvin), who long afterwards developed it further,[46] arriving at the following interesting result:—Let an incompressible solid, isotropic when unstrained, be such that its potential energy per unit volume is

where q denotes its modulus of rigidity when unstrained, and α

2 , denote the proportions in which lines parallel to the axes of strain are altered; then if the solid be initially strained in a way defined by given values of α, β, γ, by forces applied to its surface, and if waves of distortion be superposed on this initial strain, the transmission of these waves will follow exactly the laws of Fresnel’s theory of crystal-optics, the wave-surface being