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

Rh Propagation of Undulations in a Non-conducting Medium.

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

{{numb form | $$ \left. \begin{align} K \mu \frac{d^2 F}{dt^2} + \nabla^2 F &= 0, \\ K \mu \frac{d^2 G}{dt^2} + \nabla^2 G &= 0, \\ K \mu \frac{d^2 H}{dt^2} + \nabla^2 H &= 0. \end{align} \right\} $$|(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, and applied by Stokes to the Theory of Diffraction.

Let us write

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

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