386
ELECTROMAGNETIC THEORY OF LIGHT.
[784.
Propagation of Undulations in a Non-conducting Medium.
784.] In this case
, and the equations become
|
![{\displaystyle \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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5496c578167978cd5fe4ac93e88dd5895dbc6c) | (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
|
![{\displaystyle V={\frac {1}{\sqrt {K\mu }}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00ef0a260b20f425806fdfb38cd812ed5983225a) | (10) |
If the values of
,
,
, and of
,
,
are given at every point of space at the epoch (
), then we can determine their values at any subsequent time,
, as follows.
Let
be the point for which we wish to determine the value of
at the time
. With
as centre, and with radius
, describe a sphere. Find the initial value of
at every point of the spherical surface, and take the mean,
, of all these values. Find also the initial values of
at every point of the spherical surface, and let the mean of these values be
.
Then the value of
at the point
, at the time
, is
|
![{\displaystyle \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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c63737a280cae3e756d5c34dfc4aee062d331e9e) | (11) |
785.] It appears, therefore, that the condition of things at the point
at any instant depends on the condition of things at a distance
and at an interval of time
previously, so that any disturbance is propagated through the medium with the velocity
.
Let us suppose that when
is zero the quantities
and
are
- ↑ Mem. de l' Acad., tom, iii, p. 130.
- ↑ Cambridge Transactions, vol. ix, p. 10 (1850).