Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/161

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

$\rho {\frac {\partial ^{2}\mathbf {c} }{\partial t^{2}}}=n\nabla ^{2}\mathbf {c}$, $\mathrm {div} \ \mathbf {c} =0$;

while b is to be determined from

$\rho {\frac {\partial ^{2}\mathbf {b} }{\partial t^{2}}}=\left(k+{\frac {4}{3}}n\right)\nabla ^{2}\mathbf {b}$, $\mathrm {curl} \ \mathbf {b} =0$.

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

$c_{x}=A\sin \lambda \left(z-t{\sqrt {\frac {n}{\rho }}}\right)$, $c_{y}=B\sin \lambda \left(z-t{\sqrt {\frac {n}{\rho }}}\right)$, $c_{z}=0$,

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

$b_{x}=0$, $b_{y}=0$, $b_{z}=C\sin \lambda \left(z-t{\sqrt {\frac {k+{\tfrac {4}{3}}n}{\rho }}}\right)$,

which represents a longitudinal wave propagated with velocity

${\sqrt {(k+{\tfrac {4}{3}}n)/\rho }}$;

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)^1/2; while those in b are longitudinal, and are propagated with velocity {(k + 4/3n)/ρ}^1/2 The latter are^[1] waves of dilatation and condensation, like sound-waves; in the c-waves, on the other hand, the medium is not dilated or condensed, but

1. Cf. Stokes, "On the Dynamical Problem of Diffraction," Camb. Phil. Trans., ix (1849).

Erratum

1. Correction: other should be amended to other.