dH dG
-- = - [pi]vG, -- = [pi]vH - 1 (18).
dv dv
We now proceed to consider more particularly the distribution of light
upon a screen PBQ near the shadow of a straight edge A. At a point P
within the geometrical shadow of the obstacle, the half of the wave to
the right of C (fig. 18), the nearest point on the wave-front, is
wholly intercepted, and on the left the integration is to be taken
from s = CA to s = [oo]. If V be the value of v corresponding to CA,
viz.
/ / 2(a + b) \
V= / ( ---------- ).CA, (19),
\/ \ ab[lambda] /
we may write
_[oo] _[oo]
/ / \^2 / / \^2
I^2 = ( | cos 1/2[pi]v^2.dv ) + ( | sin 1/2[pi]v^2.dv ) (20),
\ _/v / \ _/v /
or, according to our previous notation,
I^2 = (1/2 - Cv)^2 + (1/2 - Sv)^2 = G^2 + H^2 (21).
Now in the integrals represented by G and H every element diminishes
as V increases from zero. Hence, as CA increases, viz. as the point P
is more and more deeply immersed in the shadow, the illumination
_continuously_ decreases, and that without limit. It has long been
known from observation that there are no bands on the interior side of
the shadow of the edge.
[Illustration: FIG. 18.]
The law of diminution when V is moderately large is easily expressed
with the aid of the series (16), (17) for G, H. We have ultimately G =
0, H = ([pi]V)^-1, so that
I^2 = 1/[pi]^2V^2,
or the illumination is inversely as the square of the distance from
the shadow of the edge.
For a point Q outside the shadow the integration extends over _more_
than half the primary wave. The intensity may be expressed by
I^2 = (1/2 + Cv)^2 + (1/2 + Sv)^2 (22);
and the maxima and minima occur when
dC dS
(1/2 + C_v) -- + (1/2 + S_v) -- = 0,
dV dV
whence
sin 1/2[pi]V^2 + cos 1/2[pi]V^2 = G (23).
When V = 0, viz. at the edge of the shadow, I^2 = 1/2; when V = [oo],
I^2 = 2, on the scale adopted. The latter is the intensity due to the
uninterrupted wave. The quadrupling of the intensity in passing
outwards from the edge of the shadow is, however, accompanied by
fluctuations giving rise to bright and dark bands. The position of
these bands determined by (23) may be very simply expressed when V is
large, for then sensibly G = 0, and
1/2[pi]V^2 = 3/4[pi] + n[pi] (24),
n being an integer. In terms of [delta], we have from (2)
[delta] = (3/8 + 1/2n)[lambda] (25).
The first maximum in fact occurs when [delta] = 3/8[lambda]
-.0046[lambda], and the first minimum when [delta] = 7/8[lambda]
-.0016[lambda], the corrections being readily obtainable from a table
of G by substitution of the approximate value of V.
The position of Q corresponding to a given value of V, that is, to a
band of given order, is by (19)
a + b / / b[lambda](a + b) \
BQ = ----- AD = V / ( ----------------- ) (26).
a \/ \ 2a /
By means of this expression we may trace the locus of a band of given
order as b varies. With sufficient approximation we may regard BQ and
b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB
is axis of y and a perpendicular through A the axis of x, and
rationalizing (26), we have
2ax^2 - V^2[lambda]y^2 - V^2a[lambda]y = 0,
which represents a hyperbola with vertices at O and A.
From (24), (26) we see that the width of the bands is of the order
[sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation
upon the width of the source of light, in order that the bands may be
properly formed. If [omega] be the apparent magnitude of the source
seen from A, [omega]b should be much smaller than the above quantity,
or
[omega] < [sqrt] {[lambda](a + b)/ab} (27).