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 to constructing a mathematical theory of this phenomenon.

• Its basis was Fresnel’s “Hypothesis of Finite Impacts”—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)

• 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, we 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 was natural to suppose that:

• in these bodies the aether is loaded by the molecules of matter, and
• 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 40 years later when it was overthrown by anomalous dispersion.

The distinction between aether and ponderable matter was more definitely drawn in memoirs published independently in 1841–2 by:

• F. E. Neumann
• Matthew O’Brien

These authors supposed the ponderable particles to remain sensibly at rest while the aether surges around them, and is acted on by them with forces which are proportional to its displacement.

Thus, 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.