3 Hydrodynamics
281
Then definite solution problem for the dimensionless velocity potential
function is obtained.
β
2 ∂ 2 ϕ
∂ x 2 +
∂ 2 ϕ
∂z 2 = 0
∂ϕ
∂z
z=0
= 0
∂η
∂t + α
∂ϕ
∂ x
∂η
∂ x −
1
β 2
∂ϕ
∂z = 0, z
= 1 + αη
η
+
∂ϕ
∂t +
1
2
α
∂ϕ
∂ x
2
+
1
β
∂ϕ
∂z
2
= 0, z
= 1 + αη
Approximating the above equation set, keeping the first-order small terms
of α and β, then the approximate solution is
ϕ
≈ −
z 2
2
∂ V
∂ x β, V
=
∂ϕ
∂ x = η
−
1
4
αη
2
+
1
3
β
∂ 2 η
∂ x 2
The nonlinear differential equation obtained from the free surface condition is
∂η
∂t +
∂η
∂ x +
3
2
αη
∂η
∂ x +
1
6
β
∂ 3 η
∂ x 3 = 0
This equation is the dimensionless KdV equation. The third term in the
equation is a nonlinear effect and the fourth term is a dispersion effect.
Transform it into a dimensional form as
∂η
∂t
+
g H
1 +
3
2
η
H
∂η
∂ x
+
g H
H 2
6
∂ 3 η
∂ x 3 = 0
The solution of this equation has typical solitary wave properties, the
propagation direction is single, and the propagation speed is
a =
g H
1 +
1
2
A
H