TZ x d / e^-ikr \
[=omega]2 = -------- . - . -- ( ------- ) (14).
4[pi]b^2 r dr \ r /
From (10), (13), (14) we see that, as might have been expected, the
rotation at any point is about an axis perpendicular both to the
direction of the force and to the line joining the point to the source
of disturbance. If the resultant rotation be [omega], we have
TZ [sqrt](x^2 + y^2) d /e^-ikr\
[=omega] = ------- . ----------------- . -- ( ------ ) =
4[pi]b^2 r dr \ r /
TZ sin[phi] d /e^-ikr\
= ----------- -- ( ------ ),
4[pi]b^2 dr \ r /
[phi] denoting the angle between r and z. In differentiating
e^(-ikr)/r with respect to r, we may neglect the term divided by r^2 as
altogether insensible, kr being an exceedingly great quantity at any
moderate distance from the origin of disturbance. Thus
-ik.TZ sin[phi] /e^-ikr\
[=omega] = --------------- . ( ------ ) (15),
4[pi]b^2 \ r /
which completely determines the rotation at any point. For a
disturbing force of given integral magnitude it is seen to be
everywhere about an axis perpendicular to r and the direction of the
force, and in magnitude dependent only upon the angle ([phi]) between
these two directions and upon the distance (r).
The intensity of light is, however, more usually expressed in terms of
the actual displacement in the plane of the wave. This displacement,
which we may denote by [zeta]', is in the plane containing z and r,
and perpendicular to the latter. Its connexion with [=omega]is
expressed by [=omega] = d[zeta]'/dr; so that
TZ sin [phi] /e^-ikr\
[zeta]' = ----------- . ( ------ ) (16),
4[pi]b^2 \ r /
where the factor e^int is restored.
Retaining only the real part of (16), we find, as the result of a
local application of force equal to
DTZ cos nt (17),
the disturbance expressed by
TZ sin [phi] /cos(nt - kr)\
[zeta]' = ------------ . ( ------------ ) (18).
4[pi]b^2 \ r /
The occurrence of sin [phi] shows that there is no disturbance
radiated in the direction of the force, a feature which might have
been anticipated from considerations of symmetry.
We will now apply (18) to the investigation of a law of secondary
disturbance, when a primary wave
[zeta] = sin(nt - kx) (19)
is supposed to be broken up in passing the plane x = 0. The first step
is to calculate the force which represents the reaction between the
parts of the medium separated by x = 0. The force operative upon the
positive half is parallel to OZ, and of amount per unit of area equal
to
-b^2D d[zeta]/dx = b^2kD cos nt;
and to this force acting over the whole of the plane the actual motion
on the positive side may be conceived to be due. The secondary
disturbance corresponding to the element dS of the plane may be
supposed to be that caused by a force of the above magnitude acting
over dS and vanishing elsewhere; and it only remains to examine what
the result of such a force would be.
Now it is evident that the force in question, supposed to act upon the
positive half only of the medium, produces just double of the effect
that would be caused by the same force if the medium were undivided,
and on the latter supposition (being also localized at a point) it
comes under the head already considered. According to (18), the effect
of the force acting at dS parallel to OZ, and of amount equal to
2b^2kD dS cos nt,
will be a disturbance
dS sin [phi]
[zeta]' = ------------ cos(nt - kr) (20),
[lambda]r
regard being had to (12). This therefore expresses the secondary
disturbance at a distance r and in a direction making an angle [phi]
with OZ (the direction of primary vibration) due to the element dS of
the wave-front.
The proportionality of the secondary disturbance to sin [phi] is
common to the present law and to that given by Stokes, but here there
is no dependence upon the angle [theta] between the primary and
secondary rays. The occurrence of the factor [lambda]r^-1, and the
necessity of supposing the phase of the secondary wave accelerated by
a quarter of an undulation, were first established by Archibald Smith,
as the result of a comparison between the primary wave, supposed to
pass on without resolution, and the integrated effect of all the
secondary waves (S 2). The occurrence of factors such as sin [phi], or
1/2(1 + cos [theta]), in the expression of the secondary wave has no
influence upon the result of the integration, the effects of all the
elements for which the factors differ appreciably from unity being
destroyed by mutual interference.
The choice between various methods of resolution, all mathematically
admissible, would be guided by physical considerations respecting the
mode of action of obstacles. Thus, to refer again to the acoustical
analogue in which plane waves are incident upon a perforated rigid
screen, the circumstances of the case are best represented by the
first method of resolution, leading to symmetrical secondary waves, in
which the normal motion is supposed to be zero over the unperforated
parts. Indeed, if the aperture is very small, this method gives the
correct result, save as to a constant factor. In like manner our
present law (20) would apply to the kind of obstruction that would be
caused by an actual physical division of the elastic medium, extending
over the whole of the area supposed to be occupied by the intercepting
screen, but of course not extending to the parts supposed to be
perforated.
On the electromagnetic theory, the problem of diffraction becomes
definite when the properties of the obstacle are laid down. The
simplest supposition is that the material composing the obstacle is
perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
mathematical skill, has solved the problem of the shadow thrown by a
semi-infinite plane screen. A simplified exposition has been given by
Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
that Fresnel's results, although based on an imperfect theory, require
only insignificant corrections. Problems not limited to two
dimensions, such for example as the shadow of a circular disk, present
great difficulties, and have not hitherto been treated by a rigorous
method; but there is no reason to suppose that Fresnel's results would
be departed from materially. (R.)