246
P. Liu
From the horizontal equation of motion (Euler equation), we get
∂u
∂t
+ u
∂u
∂ x
= −
1
ρ
∂ p
∂ x
The z-direction velocity and acceleration are ignored. If the nonlinear term
is not included, there are
∂u
∂t
= −g
∂η
∂ x
In a certain period of time, the horizontal displacement of water particle
movement is
ξ =
udt
From the principle of conservation of mass, we can get
∂
∂ x
ξ H +
ηdx
= 0, η = −H
∂ξ
∂ x
∂ 2 ξ
∂t 2 = g H
∂ 2 ξ
∂ x 2 ,
Eliminate ξ and get
∂ 2 η
∂t 2 = g H
∂ 2 η
∂ x 2 ,
∂ 2 η
∂t 2 = a
2 ∂ 2 η
∂ x 2
This is the famous wave equation, which was deduced by the French scientist D’Alembert (as shown in Fig. 1.14) in 1747. Among them, water surface
wave velocity a =
√
g H, H is water depth.
If we set x 1 = x −at, x 2 = x +at and substitute it into the wave equation
about wave height, we can get
∂ 2 η
∂ x 1 ∂ x 2
= 0
Its solution is
η = F(x − at) + G(x + at)