264
P. Liu
The water particle velocity is
u =
∂ϕ
∂ x
=
h
2
gk
ω
cosh k(z + H )
cosh k H
cos kx cos ωt
w =
∂ϕ
∂z
=
h
2
gk
ω
sinh k(z + H )
cosh k H
sin kx cos ωt
The trajectory of water particle is
x = x 0 +
h
2
cosh k(z 0 + H )
sinh k H
cos kx 0 sin ωt
z = z 0 +
h
2
sinh k(z 0 + H )
sinh k H
sin kx 0 sin ωt
3.8.5 Wave with Finite Amplitude
For waves with limited amplitude, because the nonlinear convection term
in the equation of motion cannot be ignored, it is more difficult to directly
obtain the solution of the governing equation, and some theoretical results
have certain limitations. Experiments have found that the shape of this type
of wave is no longer a sine (or cosine) curve, but a shape with a steeper wave
peak and a more flat wave surface. The shape of this wave is similar to the
shape of a trochoid curve. Therefore, in engineering practice, for the sake of
simplicity, we use the trochoid theory as an approximation. Compared with
the potential flow theory, this theory is different in that the motion characteristics of the water mass particle are given first, and then the correctness of
the hypothesis is verified according to the basic equations of water flow, so as
to establish the fluctuation law.
(1) Two-dimensional Lagrangian continuous equation and equation of
motion
When solving with the cycloid theory, it is required to track individual water
particles to study its motion law, so the continuous and motion equations
expressed by the Lagrangian method are needed. In a two-dimensional flow
space of incompressible fluid, we take a fixed rectangular coordinate system.
Then, the position of any water particle expressed in Lagrangian at time t is
x = x(a, b, t), z = z(a, b, t)
P. Liu
The water particle velocity is
u =
∂ϕ
∂ x
=
h
2
gk
ω
cosh k(z + H )
cosh k H
cos kx cos ωt
w =
∂ϕ
∂z
=
h
2
gk
ω
sinh k(z + H )
cosh k H
sin kx cos ωt
The trajectory of water particle is
x = x 0 +
h
2
cosh k(z 0 + H )
sinh k H
cos kx 0 sin ωt
z = z 0 +
h
2
sinh k(z 0 + H )
sinh k H
sin kx 0 sin ωt
3.8.5 Wave with Finite Amplitude
For waves with limited amplitude, because the nonlinear convection term
in the equation of motion cannot be ignored, it is more difficult to directly
obtain the solution of the governing equation, and some theoretical results
have certain limitations. Experiments have found that the shape of this type
of wave is no longer a sine (or cosine) curve, but a shape with a steeper wave
peak and a more flat wave surface. The shape of this wave is similar to the
shape of a trochoid curve. Therefore, in engineering practice, for the sake of
simplicity, we use the trochoid theory as an approximation. Compared with
the potential flow theory, this theory is different in that the motion characteristics of the water mass particle are given first, and then the correctness of
the hypothesis is verified according to the basic equations of water flow, so as
to establish the fluctuation law.
(1) Two-dimensional Lagrangian continuous equation and equation of
motion
When solving with the cycloid theory, it is required to track individual water
particles to study its motion law, so the continuous and motion equations
expressed by the Lagrangian method are needed. In a two-dimensional flow
space of incompressible fluid, we take a fixed rectangular coordinate system.
Then, the position of any water particle expressed in Lagrangian at time t is
x = x(a, b, t), z = z(a, b, t)