8.3 Solutions
375
8.39
ω 2
k 2 =
F
μ
+ αk
2
(1)
The phase velocity v p =
ω
k
=
F
μ
+ αk 2 =
F
μ
1 +
αμk 2
F
1/2
=
F
μ
1 + α
k 2 μ
2F
+ · · ·
(for small α)
(2)
Since k = 2π/λ, v p increases as λ decreases.
The group velocity is given by
v g = v p +
k dv p
dk
= v p + αk
2
μ
F
8.40 (a) ω =
S
ρ
k
3/2
v p =
ω
k
=
S
ρ
√
k
(b) v g = v p +
k dv p
dk
=
S
ρ
√
k +
kS
2ρ
√
k
=
3
2
S
ρ
√
k
(c) From (a) and (b) v g > v p
8.41 (a) ω
2
=
gk +
S
ρ
k
3
tanh(kh)
(1)
If kh << 1, then tanh (kh) = kh and (1) becomes
ω
2
=
gk +
S
ρ
k
3
kh
(2)
If the second term in the brackets is smaller than the first one
S
ρ
k 2 << g
ω
2
= ghk
2
∴ ω = k
gh
(3)
v p =
ω
k
=
gh
(4)
375
8.39
ω 2
k 2 =
F
μ
+ αk
2
(1)
The phase velocity v p =
ω
k
=
F
μ
+ αk 2 =
F
μ
1 +
αμk 2
F
1/2
=
F
μ
1 + α
k 2 μ
2F
+ · · ·
(for small α)
(2)
Since k = 2π/λ, v p increases as λ decreases.
The group velocity is given by
v g = v p +
k dv p
dk
= v p + αk
2
μ
F
8.40 (a) ω =
S
ρ
k
3/2
v p =
ω
k
=
S
ρ
√
k
(b) v g = v p +
k dv p
dk
=
S
ρ
√
k +
kS
2ρ
√
k
=
3
2
S
ρ
√
k
(c) From (a) and (b) v g > v p
8.41 (a) ω
2
=
gk +
S
ρ
k
3
tanh(kh)
(1)
If kh << 1, then tanh (kh) = kh and (1) becomes
ω
2
=
gk +
S
ρ
k
3
kh
(2)
If the second term in the brackets is smaller than the first one
S
ρ
k 2 << g
ω
2
= ghk
2
∴ ω = k
gh
(3)
v p =
ω
k
=
gh
(4)