Sir George Gabriel Stokes

Young and Fresnel proposed that:

• the vibrations of light* are performed at right angles to its direction of propagation
• this peculiarity is explained by a luminiferous medium that can resist attempts to distort its shape

This power distinguishes solid bodies from fluids which offer no resistance to distortion.

In other words, the aether behaves as an elastic solid.

The elastic-solid theory has one obvious difficulty at the outset.

If the aether were solid, how can the orbiting planets move through it at immense speeds without resistance?*

This was answered by Sir George Gabriel Stokes (b. 1819, d. 1903).

He remarked that such substances are like pitch and shoemaker’s wax. They are rigid as to be capable of elastic vibration, but are also plastic to permit other bodies to pass slowly through them.

The aether has this combination of qualities in an extreme degree, behaving like an elastic solid for vibrations so rapid as those of light, but yielding like a fluid to the much slower progressive motions of the planets.

Stokes’s explanation harmonizes with Fresnel’s hypothesis that the speed of longitudinal waves in the aether is indefinitely great compared with that of the transverse waves.

Experiment with actual substances shows that the ratio of the speed of longitudinal waves to that of transverse waves increases rapidly as the medium becomes softer and more plastic.

In attempting to set forth a parallel between light and the vibrations of an elastic substance, the investigator is compelled more than once to make a choice between alternatives.

Suppose:

• the aether vibrations are executed either parallel to the plane of polarization of the light or at right angles to it.
• the different refractive powers of different media are due either to differences in the inertia of the aether within the media, or to differences in its power of resisting distortion; or to both these causes combined.

There are several distinct methods for avoiding the difficulties caused by the presence of longitudinal vibrations.

A further source of diversity is to be found in that liability to error from which no man is free. It is therefore not surprising that the list of elastic-solid theories is a long one.

When the transversality of light was discovered, no general method had been developed for investigating mathematically the properties of elastic bodies.

But under the stimulus of Fresnel’s discoveries, some of the best intellects of the age were attracted to the subject.

The volume of Memoirs of the Academy which contains Fresnel’s theory of crystal-optics contains also a memoir by Claud Louis Mario Henri Navier (b. 1785, d. 1836), at that time Professor of Mechanics in Paris, in which the correct equations of vibratory motion for a particular type of elastic solid were for the first time given.

Navier supposed the medium to be ultimately constituted of an immense number of particles, which act on each other with forces directed along the lines joining them, and depending on their distances apart.

It showed that if e denote. the (vector) displacement of the particle whose undisturbed position is (x, y, z), and if p denote the density of the medium, the equation of motion is

where n denotes a constant which measures the rigidity, or power of resisting distortion, of the mediun, All such elastic properties of the body as the velocity of propagation of waves in it must evidently depend on the ratio n/ρ.

Among the referees of one of Navier’s papers was Augustine Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of the 19th century,[3].

He published in 1828[4] a discussion of it from an entirely different point of view.

Instead of assuming, as Navier had done, that the medium is an aggregate of point-centres of force, and thus involving himself in doubtful molecular hypotheses, he devised a method of directly studying the elastic properties of matter in bulk, and by its means showed that the vibrations of an isotropic solid are determined by the equation

here n denotes, as before, the constant of rigidity, and the constant k, which is called the modulus of compression,[5] denotes the ratio of a pressure to the cubical compression produced by it. Cauchy’s equation evidently differs from Navier’s in that two constants, k and n, appear instead of one.

The reason for this is that a body constituted from point-centres of force in Navier’s fashion has its moduli of rigidity and compression connected by the relation[6]

Actual bodies do not necessarily obey this condition; e.g. for india-rubber, k is much larger than … and there seems to be no reason why we should impose it on the aether.

In the same year Poisson[8] succeeded in solving the differential equation which had thus been shown to determine the wave-motions possible in an elastic solid. The solution, which is both simple and elegant, may be derived as follows:

Let the displacement vector e be resolved into two components, of which one c is circuital, or satisfies the condition

div c = 0,

while the other b is irrotational, or satisfies the condition

curl b = 0.

The equation takes the form

The terms which involve b and those which involve e must be separately zero, since they represent respectively the irrotational and the circuital parts of the equation. Thus, c satisfies the pair of equations

while b is to be determined from

A particular solution of the equations for c is easily seen to be

…

which represents a transverse plane wave propagated with velocity ✓(n/ρ). It can be shown that the general solution of the differential equations for c is formed of such waves as this, travelling in all directions, superposed on each other.[errata 1]

A particular solution of the equations for b is

…

which represents a longitudinal wave propagated with velocity

…

the general solution of the differential equation for b is formed by the superposition of such waves as this, travelling in all directions.

Poisson thus discovered that the waves in an elastic solid are of two kinds: those in c are transverse, and are propagated with velocity (n/ρ)

; while those in b are longitudinal, and are propagated with velocity {(k +

The latter are[9] waves of dilatation and condensation, like sound-waves; in the c-waves, on the other hand, the medium is not dilated or condensed, but only distorted in a manner consistent with the preservation of a constant density.[10]