Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/418

Propagation of Undulations in a Non-conducting Medium.

784.] In this case $C=0$, and the equations become

$\left.{\begin{aligned}K\mu {\frac {d^{2}F}{dt^{2}}}+\nabla ^{2}F&=0,\\K\mu {\frac {d^{2}G}{dt^{2}}}+\nabla ^{2}G&=0,\\K\mu {\frac {d^{2}H}{dt^{2}}}+\nabla ^{2}H&=0.\end{aligned}}\right\}$

(9)

The equations in this form are similar to those of the motion of an elastic solid, and when the initial conditions are given, the solution can be expressed in a form given by Poisson ^[1], and applied by Stokes to the Theory of Diffraction^[2].

Let us write

$V={\frac {1}{\sqrt {K\mu }}}.$

(10)

If the values of $F$, $G$, $H$, and of ${\frac {dF}{dt}}$, ${\frac {dG}{dt}}$, ${\frac {dH}{dt}}$ are given at every point of space at the epoch ($t=0$), then we can determine their values at any subsequent time, $t$, as follows.

Let $O$ be the point for which we wish to determine the value of $F$ at the time $t$. With $O$ as centre, and with radius $Vt$, describe a sphere. Find the initial value of $F$ at every point of the spherical surface, and take the mean, ${\overline {F}}$, of all these values. Find also the initial values of ${\frac {dF}{dt}}$ at every point of the spherical surface, and let the mean of these values be ${\frac {\overline {dF}}{dt}}$.

Then the value of $F$ at the point $O$, at the time $t$, is

$\left.{\begin{aligned}F&={\frac {d}{dt}}({\overline {F}}t)+t{\frac {\overline {dF}}{dt}},\\{\text{Similarly}}\;\;G&={\frac {d}{dt}}({\overline {G}}t)+t{\frac {\overline {dG}}{dt}},\\H&={\frac {d}{dt}}({\overline {H}}t)+t{\frac {\overline {dH}}{dt}}.\end{aligned}}\right\}$

(11)

785.] It appears, therefore, that the condition of things at the point $O$ at any instant depends on the condition of things at a distance $Vt$ and at an interval of time $t$ previously, so that any disturbance is propagated through the medium with the velocity $V$.

Let us suppose that when $t$ is zero the quantities ${\mathfrak {A}}$ and ${\dot {\mathfrak {A}}}$ are

1. Mem. de l' Acad., tom, iii, p. 130.
2. Cambridge Transactions, vol. ix, p. 10 (1850).