3 Hydrodynamics
247
where F and G are waveform functions. On the surface of the water
u =
∂ξ
∂t
=
∂ξ
∂ x
∂ x
∂t
= a
∂ξ
∂ x
,
u
a
=
∂ξ
∂ x
Substituting
η = −H
∂ξ
∂ x
can get
u
a
= −
η
H
The above formula shows that the waveform remains unchanged in the
process of wave propagation.
Wave energy includes wave potential energy and wave kinetic energy. The
potential energy can be expressed as
E p = ρg
⎛
⎝
η
0
zdz
⎞
⎠ dx =
1
2
ρg
η
2 dx
Wave kinetic energy is
E k =
1
2
ρ H
u
2 dx =
1
2
ρ H
a
2 η 2
H 2 dx =
1
2
ρg
η
2 dx
It can be seen that E T = E p + E k = 2E k = 2E p . This conclusion was
first proposed by Rayleigh (as shown in Fig. 1.47) in 1876. If sine wave is
assumed, the wave function is
η = A cos(x − at)
where A is the amplitude. Substituting E P and E K , we get
E p = E k =
1
2
ρg
λ
0
η
2 dx =
1
2
ρg
λ
0
A
2 cos
2
(x − at)dx =
1
4
ρg A
2
λ