260
P. Liu
Fig. 3.83 Standing wave
(3) Plane progressive wave
Plane progressive wave is a simple wave phenomenon. Its waveform profile
is a sine curve, and the waveform advances at a certain speed on the water
surface, and the waveform remains unchanged during the advance, but no
quality migration occurs.
For deepwater wave (ω 2 = gk), in order to ensure that z → −∞ has a
limit at u =
∂φ
∂ x , we get
ϕ(x, z, t) = Ae
kz sin(kx − ωt)
The free surface equation is
η =
h
2
cos(kx − ωt)
where h is the free surface wave height, H = 2Aω/g. At a certain time,
the contour of the free surface is a cosine wave curve. The velocity of water
particle is very small, which is divided into
dx
dt
= u =
∂ϕ
∂ x
=
h
2
ωe
kz cos(kx − ωt)
dz
dt
= w =
∂ϕ
∂z
=
h
2
ωe
kz sin(kx − ωt)
The x and z in the right term of the above formula are replaced by the x 0
and z 0 of the initial position of the particle.
(x − x 0 )
2
+ (z − z 0 )
2
=
h
2
2
e
2kz 0
P. Liu
Fig. 3.83 Standing wave
(3) Plane progressive wave
Plane progressive wave is a simple wave phenomenon. Its waveform profile
is a sine curve, and the waveform advances at a certain speed on the water
surface, and the waveform remains unchanged during the advance, but no
quality migration occurs.
For deepwater wave (ω 2 = gk), in order to ensure that z → −∞ has a
limit at u =
∂φ
∂ x , we get
ϕ(x, z, t) = Ae
kz sin(kx − ωt)
The free surface equation is
η =
h
2
cos(kx − ωt)
where h is the free surface wave height, H = 2Aω/g. At a certain time,
the contour of the free surface is a cosine wave curve. The velocity of water
particle is very small, which is divided into
dx
dt
= u =
∂ϕ
∂ x
=
h
2
ωe
kz cos(kx − ωt)
dz
dt
= w =
∂ϕ
∂z
=
h
2
ωe
kz sin(kx − ωt)
The x and z in the right term of the above formula are replaced by the x 0
and z 0 of the initial position of the particle.
(x − x 0 )
2
+ (z − z 0 )
2
=
h
2
2
e
2kz 0