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8 Waves
For long waves in shallow water
V =
gh
(8.15)
Capillary Waves
Surface waves are modified by surface tension S. If h is large compared with λ
V
2
=
2π s
ρλ
+
gλ
2π
(8.16)
The minimum value of λ is given by minimizing (8.16)
λ min = 2π
s
gρ
(8.17)
If λ is sufficiently large the second term dominates and the controlling factor being
mainly gravity. Thus, the velocity of the gravity waves is given by
V =
gλ/2π
(8.18)
If λ is very small, the first term in (8.16) dominates and the motion is mainly controlled by capillarity and
V =
2π s
ρλ
(8.19)
Acoustic Waves
∂ 2 ξ
∂t 2 = V
2 ∂ 2 ξ
∂ x 2
(plane wave equation for displacement)
(8.20)
∂ 2 P
∂t 2 = V
2 ∂ 2 P
∂ x 2
(plane wave equation for pressure)
(8.21)
where
V =
B/ρ 0
(8.22)
B being the bulk modulus of elasticity.
Sound Velocity in a Gas
V =
γ P
ρ
(Laplace formula)
(8.23)
8 Waves
For long waves in shallow water
V =
gh
(8.15)
Capillary Waves
Surface waves are modified by surface tension S. If h is large compared with λ
V
2
=
2π s
ρλ
+
gλ
2π
(8.16)
The minimum value of λ is given by minimizing (8.16)
λ min = 2π
s
gρ
(8.17)
If λ is sufficiently large the second term dominates and the controlling factor being
mainly gravity. Thus, the velocity of the gravity waves is given by
V =
gλ/2π
(8.18)
If λ is very small, the first term in (8.16) dominates and the motion is mainly controlled by capillarity and
V =
2π s
ρλ
(8.19)
Acoustic Waves
∂ 2 ξ
∂t 2 = V
2 ∂ 2 ξ
∂ x 2
(plane wave equation for displacement)
(8.20)
∂ 2 P
∂t 2 = V
2 ∂ 2 P
∂ x 2
(plane wave equation for pressure)
(8.21)
where
V =
B/ρ 0
(8.22)
B being the bulk modulus of elasticity.
Sound Velocity in a Gas
V =
γ P
ρ
(Laplace formula)
(8.23)