270
P. Liu
the particle trajectory equation into the Lagrange-type equation of motion,
and the pressure at the free surface after integration is
p a
ρ
=
σ 2
k
− g
r 0 cos(σ t − kx 0 ) +
1
2
σ
2 r
2
0 + C
where p a is atmospheric pressure; C is the free constant. Obviously, for the
existence of the above formula, it must be satisfied
σ 2
k
− g = 0
This shows that in the case of deepwater propulsion waves, the square value
of the angular velocity of the particle motion is equal to the product of the
curvature of the circle and the acceleration of gravity (Dispersion relation of
deepwater wave). Substituting σ = 2π/T and k = 2π/λ into the dispersion
equation, and we can obtain the wave velocity and the period
a =
λ
T
=
gλ
2π
, T =
2πλ
g
The wave velocity and wave period of a deepwater propulsion wave are
directly proportional to the wavelength. That is, the longer the wavelength,
the greater the wave speed and wave period.
For water surface wave,we have t = 0, z 0 = 0, r = h/2, as well as θ = σtkx 0 = -kx 0 . The water surface wave equation can be obtained by substituting
the front condition into the motion equation of the particle. θ is a changeable
parameter.
x = −
λ
2π
θ +
h
2
sin θ
z = −
h
2
cos θ
The curve obtained from this equation is the circle trochoid curve. The
circle trochoid curve is a curve drawn by a particle inside a circle as each circle
rolls along its tangent. If it is a curve drawn on a circle, it is called a cycloid,
which is plotted in Fig. 3.88. Similarly, we can obtain a curve equation of
wave surface at an arbitrary water depth
x = −
λ
2π
θ +
h
2
e
−kz 0 sin θ
P. Liu
the particle trajectory equation into the Lagrange-type equation of motion,
and the pressure at the free surface after integration is
p a
ρ
=
σ 2
k
− g
r 0 cos(σ t − kx 0 ) +
1
2
σ
2 r
2
0 + C
where p a is atmospheric pressure; C is the free constant. Obviously, for the
existence of the above formula, it must be satisfied
σ 2
k
− g = 0
This shows that in the case of deepwater propulsion waves, the square value
of the angular velocity of the particle motion is equal to the product of the
curvature of the circle and the acceleration of gravity (Dispersion relation of
deepwater wave). Substituting σ = 2π/T and k = 2π/λ into the dispersion
equation, and we can obtain the wave velocity and the period
a =
λ
T
=
gλ
2π
, T =
2πλ
g
The wave velocity and wave period of a deepwater propulsion wave are
directly proportional to the wavelength. That is, the longer the wavelength,
the greater the wave speed and wave period.
For water surface wave,we have t = 0, z 0 = 0, r = h/2, as well as θ = σtkx 0 = -kx 0 . The water surface wave equation can be obtained by substituting
the front condition into the motion equation of the particle. θ is a changeable
parameter.
x = −
λ
2π
θ +
h
2
sin θ
z = −
h
2
cos θ
The curve obtained from this equation is the circle trochoid curve. The
circle trochoid curve is a curve drawn by a particle inside a circle as each circle
rolls along its tangent. If it is a curve drawn on a circle, it is called a cycloid,
which is plotted in Fig. 3.88. Similarly, we can obtain a curve equation of
wave surface at an arbitrary water depth
x = −
λ
2π
θ +
h
2
e
−kz 0 sin θ