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100. (a) Since the speed of sound is lower in air than in water, the speed of sound in the airwater mixture
is lower than in pure water (see Table 181). Frequency is proportional to the speed of sound (see
Eq. 1839 and Eq. 1841), so the decrease in speed is “heard” due to the accompanying decrease in
frequency.
(b) This follows from Eq. 183 and Eq. 182 (with ∆’s replaced by derivatives). Thus,
1
v
2
=
ρ
B
=
ρ
V
¯
¯
¯
dp
dV
¯
¯
¯
=
ρ
V
¯
¯
¯
¯
dV
dp
¯
¯
¯
¯
.
(c) Returning to the ∆ notation, and letting the absolute values be “understood,” we write ∆
V
=
∆
V
w
+∆
V
a
as indicated in the problem. Subject to the approximations mentioned in the problem,
our equation becomes
1
v
2
=
ρ
w
V
w
µ
∆
V
w
∆
p
+
∆
V
a
∆
p
¶
=
ρ
w
V
w
∆
V
w
∆
p
+
ρ
w
ρ
a
V
a
V
w
µ
ρ
a
V
a
∆
V
a
∆
p
¶
.
In a pure water system or a pure air system, we would have
1
v
2
w
=
ρ
w
V
w
∆
V
w
∆
p
or
1
v
2
a
=
ρ
a
V
a
∆
V
a
∆
p
.
Substituting these into the above equation, and using the notation
r
=
V
a
/V
w
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.
 Fall '08
 SPRUNGER
 Physics

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