378
8 Waves
8.46 ω =
γ RT
M
k
v p =
ω
k
=
γ RT
M
v g =
dω
dk
=
γ RT
M
∴ v g = v p
8.47 v
2
p =
g
k
+
Sk
ρ
(1)
Maximize (1)
d(v 2
p )
dk
= −
g
k 2 +
S
ρ
= 0
∴ k =
2π
λ
=
gρ
S
∴ λ min = 2π
S
gρ
8.3.4 Sound Waves
8.48 Let the displacement be represented by
y = A cos(kx − ωt)
(1)
∂ y
∂ x
= −k A sin(kx − ωt)
but P = −B
∂ y
∂ x
= Bk A sin(kx − ωt)
where B is the bulk modulus:
B = v
2
ρ 0
P = [kρ 0 v
2 A] sin(kx − ωt)
(2)
P represents the change from standard pressure P 0 . The term in square bracket
represents maximum change in pressure and is called the pressure amplitude
P max . Then
8 Waves
8.46 ω =
γ RT
M
k
v p =
ω
k
=
γ RT
M
v g =
dω
dk
=
γ RT
M
∴ v g = v p
8.47 v
2
p =
g
k
+
Sk
ρ
(1)
Maximize (1)
d(v 2
p )
dk
= −
g
k 2 +
S
ρ
= 0
∴ k =
2π
λ
=
gρ
S
∴ λ min = 2π
S
gρ
8.3.4 Sound Waves
8.48 Let the displacement be represented by
y = A cos(kx − ωt)
(1)
∂ y
∂ x
= −k A sin(kx − ωt)
but P = −B
∂ y
∂ x
= Bk A sin(kx − ωt)
where B is the bulk modulus:
B = v
2
ρ 0
P = [kρ 0 v
2 A] sin(kx − ωt)
(2)
P represents the change from standard pressure P 0 . The term in square bracket
represents maximum change in pressure and is called the pressure amplitude
P max . Then