Dispersion to Fresnel and Cauchy

Dispersion was the subject of several memoirs by the founders of the elastic-solid theory.

So early as 1830 Cauchy’s attention was directed[76] to the possibility of constructing a mathematical theory of this phenomenon on the basis of Fresnel’s “Hypothesis of Finite Impacts”[77]—i.e. the assumption that the radius of action of one particle of the luminiferous medium on its neighbours is so large as to be comparable with the wave-length of light.

Cauchy supposed the medium to be formed, as in Navier’s theory of elastic solids, of a system of point-centres of force: the force between two of these point-centres, m at (x, y, z), and μ at (x + Δx, y + Δy,z + Δz), may be denoted by mμf(r), where r denotes the distance between m and μ. When this medium is disturbed by light-waves propagated parallel to the z-axis, the displacement being parallel to the x-axis, the equation of motion of m is evidently

where ξ denotes the displacement of m, (ξ + Δξ) the displacement of μ, and (r + ρ) the new value of r. Substituting for ρ its value, and retaining only terms of the first degree in Δξ, this equation becomes

By Taylor’s theorem, since ξ depends only on z, wo have

Substituting, and remembering that summations which involve odd powers of Δz: must vanish when taken over all the point-centres within the sphere of influence of m, we obtain an equation of the form

where α, β, γ denote constants. Each successive term on the right-hand side of this equation involves an additional factor (Δz)/λ2 as compared with the preceding term, where λ denotes the wave-length of the light: so if the radii of influence of the point-centres were indefinitely small in comparison with the wave-length of the light, the equation would reduce to

which is the ordinary equation of wave-propagation in one dimension in non-dispersive media. But if the medium is so coarse-grained that λ is not large compared with the radii of influence, we must retain the higher derivates of ξ. Substituting

in the differential equation with these higher derivates retained, we have

which shows that c1, the velocity of the light in the medium, depends on the wave-length λ; as it should do in order to explain dispersion.

Dispersion is, then, according to the view of Fresnel and Cauchy, a consequence of the coarse-grainedness of the medium.

Since the luminiferous medium was dispersive only within material bodies, it seemed natural to suppose that in these bodies the aether is loaded by the molecules of matter, and that dispersion depends essentially on the ratio of the wave-length to the distance between adjacent material molecules.

This theory, in one modification or another, held its ground until forty years later it was overthrown by the facts of anomalous dispersion.

The distinction between aether and ponderable matter was more definitely drawn in memoirs which were published independently in 1841–2 by F. E. Neumann[78] and Matthew O’Brien.[79]

These authors supposed the ponderable particles to remain sensibly at rest while the aether surges round them, and is acted on by them with forces which are proportional to its displacement. Thus[80] the equation of motion of the aether becomes

where C denotes a constant on which the phenomena of dispersion depend. For polarized plane waves propagated parallel to the axis of x, this equation becomes

and substituting

where τ denotes the period and V the velocity of the light, we have

an equation which expresses the dependence of the velocity on the period.