2[pi]
f(bt - x) = c sin -------- (bt - x) (5),
[lambda]
we shall have
cdS 2[pi]
[zeta]' = ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r) (6)."
2[lambda]r [lambda]
It is then verified that, after integration with respect to dS, (6)
gives the same disturbance as if the primary wave had been supposed to
pass on unbroken.
The occurrence of sin [phi] as a factor in (6) shows that the relative
intensities of the primary light and of that diffracted in the
direction [theta] depend upon the condition of the former as regards
polarization. If the direction of primary vibration be perpendicular
to the plane of diffraction (containing both primary and secondary
rays), sin [phi] = 1; but, if the primary vibration be in the plane of
diffraction, sin [phi] = cos [theta]. This result was employed by
Stokes as a criterion of the direction of vibration; and his
experiments, conducted with gratings, led him to the conclusion that
the vibrations of polarized light are executed in a direction
_perpendicular_ to the plane of polarization.
The factor (1 + cos [theta]) shows in what manner the secondary
disturbance depends upon the direction in which it is propagated with
respect to the front of the primary wave.
If, as suffices for all practical purposes, we limit the application
of the formulae to points in advance of the plane at which the wave is
supposed to be broken up, we may use simpler methods of resolution
than that above considered. It appears indeed that the purely
mathematical question has no definite answer. In illustration of this
the analogous problem for sound may be referred to. Imagine a flexible
lamina to be introduced so as to coincide with the plane at which
resolution is to be effected. The introduction of the lamina (supposed
to be devoid of inertia) will make no difference to the propagation of
plane parallel sonorous waves through the position which it occupies.
At every point the motion of the lamina will be the same as would have
occurred in its absence, the pressure of the waves impinging from
behind being just what is required to generate the waves in front. Now
it is evident that the aerial motion in front of the lamina is
determined by what happens at the lamina without regard to the cause
of the motion there existing. Whether the necessary forces are due to
aerial pressures acting on the rear, or to forces directly impressed
from without, is a matter of indifference. The conception of the
lamina leads immediately to two schemes, according to which a primary
wave may be supposed to be broken up. In the first of these the
element dS, the effect of which is to be estimated, is supposed to
execute its actual motion, while every other element of the plane
lamina is maintained at rest. The resulting aerial motion in front is
readily calculated (see Rayleigh, _Theory of Sound_, S 278); it is
symmetrical with respect to the origin, i.e. independent of [theta].
When the secondary disturbance thus obtained is integrated with
respect to dS over the entire plane of the lamina, the result is
necessarily the same as would have been obtained had the primary wave
been supposed to pass on without resolution, for this is precisely the
motion generated when every element of the lamina vibrates with a
common motion, equal to that attributed to dS. The only assumption
here involved is the evidently legitimate one that, when two systems
of variously distributed motion at the lamina are superposed, the
corresponding motions in front are superposed also.
The method of resolution just described is the simplest, but it is
only one of an indefinite number that might be proposed, and which are
all equally legitimate, so long as the question is regarded as a
merely mathematical one, without reference to the physical properties
of actual screens. If, instead of supposing the _motion_ at dS to be
that of the primary wave, and to be zero elsewhere, we suppose the
_force_ operative over the element dS of the lamina to be that
corresponding to the primary wave, and to vanish elsewhere, we obtain
a secondary wave following quite a different law. In this case the
motion in different directions varies as cos[theta], vanishing at
right angles to the direction of propagation of the primary wave. Here
again, on integration over the entire lamina, the aggregate effect of
the secondary waves is necessarily the same as that of the primary.
In order to apply these ideas to the investigation of the secondary
wave of light, we require the solution of a problem, first treated by
Stokes, viz. the determination of the motion in an infinitely extended
elastic solid due to a locally applied periodic force. If we suppose
that the force impressed upon the element of mass D dx dy dz is
DZ dx dy dz,
being everywhere parallel to the axis of Z, the only change required
in our equations (1), (2) is the addition of the term Z to the second
member of the third equation (2). In the forced vibration, now under
consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
expressing the resulting motion, are to be supposed proportional to
e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
periodic time. Under these circumstances the double differentiation
with respect to t of any quantity is equivalent to multiplication by
the factor -n^2, and thus our equations take the form
d[delta] \
(b^2[Delta]^2 + n^2)[xi] + (a^2 - b^2) -------- = 0 |
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[eta] + (a^2 - b^2) -------- = 0 > (7).
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[zeta] + (a^2 - b^2) -------- = -Z |
dx /
It will now be convenient to introduce the quantities.[=omega]1,
[=omega]2, [=omega]3 which express the _rotations_ of the elements of
the medium round axes parallel to those of co-ordinates, in accordance
with the equations
d[xi] d[eta] d[eta] d[zeta]
[=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
dy dx' dz dy
d[zeta] d[xi]
[=omega]2 = ------- - ----- (8).
dx dz
In terms of these we obtain from (7), by differentiation and
subtraction,
(b^2[Delta]^2 + n^2) [=omega]3 = 0 \
(b^2[Delta]^2 + n^2) [=omega]1 = dZ/dy > (9).
(b^2[Delta]^2 + n^2) [=omega]2 = -dZ/dx /