274
P. Liu
When under the water, z 0 = H , and we can obtain
a H =
h
2
1
sinh k H
, b H = 0
The elliptical focal length is
a H =
h
2
1
sinh k H
, b H = 0
Obviously, the ellipse of the trajectory of the water particle is gradually flattened down from the water surface, but the focal length remains unchanged,
and the water mass particle at the bottom of the water vibrations horizontally
between the two focal particles.
Substituting the motion trajectory equation of the water mass particle
into the Lagrange equation of motion, and then simplifying it to obtain the
condition that satisfies the motion is
gb 0 −
σ 2 a 0
k
= 0
Substituting σ = 2π/T and k = 2π/λ into the above equation, the wave
speed and wave period can be obtained
a =
λ
T
=
gλ
2π
√
thk H, T =
2πλ
g
a 0
b 0
=
2πλ
g
√
cthk H
It can be seen that compared with deepwater propulsion waves, the speed
and period of shallow-water propulsion waves are not only related to the
wavelength, but also change with the depth of water. At the same wavelength, the period of shallow-water wave is larger than that of deepwater
wave, and the speed of shallow-water wave is smaller than that of deepwater
wave. Because when H /λ > 1/2, thkH ≈ 1, so substitute it into the above
formula and get the same result as the wave velocity and period of the deepwater wave. Therefore, in actual calculations, H /λ = 1/2 is often used as the
dividing line for deep and shallow water waves.
For waveform of shallow water waves, because t = 0, z 0 = 0, a = a 0 , b =
b 0 = h/2, and θ = σt − kx 0 = −kx 0 , the water surface wave equation can
be obtained by substituting front conditions into particle motion equation. θ
P. Liu
When under the water, z 0 = H , and we can obtain
a H =
h
2
1
sinh k H
, b H = 0
The elliptical focal length is
a H =
h
2
1
sinh k H
, b H = 0
Obviously, the ellipse of the trajectory of the water particle is gradually flattened down from the water surface, but the focal length remains unchanged,
and the water mass particle at the bottom of the water vibrations horizontally
between the two focal particles.
Substituting the motion trajectory equation of the water mass particle
into the Lagrange equation of motion, and then simplifying it to obtain the
condition that satisfies the motion is
gb 0 −
σ 2 a 0
k
= 0
Substituting σ = 2π/T and k = 2π/λ into the above equation, the wave
speed and wave period can be obtained
a =
λ
T
=
gλ
2π
√
thk H, T =
2πλ
g
a 0
b 0
=
2πλ
g
√
cthk H
It can be seen that compared with deepwater propulsion waves, the speed
and period of shallow-water propulsion waves are not only related to the
wavelength, but also change with the depth of water. At the same wavelength, the period of shallow-water wave is larger than that of deepwater
wave, and the speed of shallow-water wave is smaller than that of deepwater
wave. Because when H /λ > 1/2, thkH ≈ 1, so substitute it into the above
formula and get the same result as the wave velocity and period of the deepwater wave. Therefore, in actual calculations, H /λ = 1/2 is often used as the
dividing line for deep and shallow water waves.
For waveform of shallow water waves, because t = 0, z 0 = 0, a = a 0 , b =
b 0 = h/2, and θ = σt − kx 0 = −kx 0 , the water surface wave equation can
be obtained by substituting front conditions into particle motion equation. θ