376
8 Waves
v g =
dω
dk
=
gh
(5)
∴ v g = v p
(b) kh >> 1, tanh(kh) = 1 and (1) becomes
ω
2
= gk +
S
ρ
k
3
ω 2
k 2 =
g
k
+
S
ρ
k
v p =
ω
k
=
g
k
+
Sk
ρ
(6)
v g = v p + k
dv p
dk
= v p +
k
2
S
ρ
−
g
k 2
g
k
+
Sk
ρ
−1/2
Using (6)
v g
v p
=
g
2k
+
3
2
ks
ρ
g
k
+
ks
ρ
(7)
For short wavelengths k is larger, the first term in both the numerator and
denominator will be smaller and v g =
3
2
v p , while for long wavelengths, k is
smaller and v g =
1
2
v p .
8.42 With reference to prob. (8.41) for small ripples, λ is small and k is large so
that the second term in (6) dominates over the first term in the radical.
v p =
Sk
ρ
=
2π S
λρ
=
2π × 0.075
0.01 × 1000
= 0.217 m/s
v g =
3
2
v p = 0.325 m/s
8 Waves
v g =
dω
dk
=
gh
(5)
∴ v g = v p
(b) kh >> 1, tanh(kh) = 1 and (1) becomes
ω
2
= gk +
S
ρ
k
3
ω 2
k 2 =
g
k
+
S
ρ
k
v p =
ω
k
=
g
k
+
Sk
ρ
(6)
v g = v p + k
dv p
dk
= v p +
k
2
S
ρ
−
g
k 2
g
k
+
Sk
ρ
−1/2
Using (6)
v g
v p
=
g
2k
+
3
2
ks
ρ
g
k
+
ks
ρ
(7)
For short wavelengths k is larger, the first term in both the numerator and
denominator will be smaller and v g =
3
2
v p , while for long wavelengths, k is
smaller and v g =
1
2
v p .
8.42 With reference to prob. (8.41) for small ripples, λ is small and k is large so
that the second term in (6) dominates over the first term in the radical.
v p =
Sk
ρ
=
2π S
λρ
=
2π × 0.075
0.01 × 1000
= 0.217 m/s
v g =
3
2
v p = 0.325 m/s