The latter question can only be treated in connexion with the
dynamical theory (see below, S 11); but under all ordinary
circumstances the result is independent of the precise answer that may
be given. All that it is necessary to assume is that the effects of
the successive zones gradually diminish, whether from the increasing
obliquity of the secondary ray or because (on account of the
limitation of the region of integration) the zones become at last more
and more incomplete. The component vibrations at P due to the
successive zones are thus nearly equal in amplitude and opposite in
phase (the phase of each corresponding to that of the infinitesimal
circle midway between the boundaries), and the series which we have to
sum is one in which the terms are alternately opposite in sign and,
while at first nearly constant in numerical magnitude, gradually
diminish to zero. In such a series each term may be regarded as very
nearly indeed destroyed by the halves of its immediate neighbours, and
thus the sum of the whole series is represented by half the first
term, which stands over uncompensated. The question is thus reduced to
that of finding the effect of the first zone, or central circle, of
which the area is [pi][lambda]r.
We have seen that the problem before us is independent of the law of
the secondary wave as regards obliquity; but the result of the
integration necessarily involves the law of the intensity and phase of
a secondary wave as a function of r, the distance from the origin. And
we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
3, p. 46), determine the law of the secondary wave, by comparing the
result of the integration with that obtained by supposing the primary
wave to pass on to P without resolution.
Now as to the phase of the secondary wave, it might appear natural to
suppose that it starts from any point Q with the phase of the primary
wave, so that on arrival at P, it is retarded by the amount
corresponding to QP. But a little consideration will prove that in
that case the series of secondary waves could not reconstitute the
primary wave. For the aggregate effect of the secondary waves is the
half of that of the first Fresnel zone, and it is the central element
only of that zone for which the distance to be travelled is equal to
r. Let us conceive the zone in question to be divided into
infinitesimal rings of equal area. The effects due to each of these
rings are equal in amplitude and of phase ranging uniformly over half
a complete period. The phase of the resultant is midway between those
of the extreme elements, that is to say, a quarter of a period behind
that due to the element at the centre of the circle. It is accordingly
necessary to suppose that the secondary waves start with a phase
one-quarter of a period in advance of that of the primary wave at the
surface of resolution.
Further, it is evident that account must be taken of the variation of
phase in estimating the magnitude of the effect at P of the first
zone. The middle element alone contributes without deduction; the
effect of every other must be found by introduction of a resolving
factor, equal to cos [theta], if [theta] represent the difference of
phase between this element and the resultant. Accordingly, the
amplitude of the resultant will be less than if all its components had
the same phase, in the ratio
_ +1/2[pi]
/
| cos [theta]d[theta] : [pi],
_/-1/2[pi]
or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
reconcile the amplitude of the primary wave (taken as unity) with the
half effect of the first zone, the amplitude, at distance r, of the
secondary wave emitted from the element of area dS must be taken to be
dS/[lambda]r (1).
By this expression, in conjunction with the quarter-period
acceleration of phase, the law of the secondary wave is determined.
That the amplitude of the secondary wave should vary as r^-1 was to be
expected from considerations respecting energy; but the occurrence of
the factor [lambda]^-1, and the acceleration of phase, have sometimes
been regarded as mysterious. It may be well therefore to remember that
precisely these laws apply to a secondary wave of sound, which can be
investigated upon the strictest mechanical principles.
The recomposition of the secondary waves may also be treated
analytically. If the primary wave at O be cos kat, the effect of the
secondary wave proceeding from the element dS at Q is
dS dS
------------- cos k(at - [rho] + 1/4[lambda]) = ------------- sin k(at - [rho]).
[lambda][rho] [lambda][rho]