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Using the binomial theorem, with
D
2
large and
a
2
+
x
2
small, we approximate this
expression:
()
2
2
/2 .
LDa
x
D
≈++
The distance traveled by the direct wave is
LD
a
x
1
2
2
=+
−
b
g
. Using the binomial theorem, we approximate this expression:
2
1
x
D
≈+−
Thus,
LLD
aa
x
x
D
D
x
x
D
ax
D
21
22
2
2
2
2
2
−≈+
++
−−
−+
=
.
Setting this equal to
m
+
1
2
b
g
λ
, where
m
is zero or a positive integer, we find
xm
D
a
1
2
2
b
gb
g
λ
.
87.
The wave that goes directly to the receiver travels a distance
L
1
and the reflected
wave travels a distance
L
2
. Since the index of refraction of water is greater than that of air
this last wave suffers a phase change on reflection of half a wavelength. To obtain
constructive interference at the receiver, the difference
L
2
–
L
1
must be an odd multiple of
a half wavelength. Consider the diagram below. The right triangle on the left, formed by
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 Spring '08
 Any
 Physics

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