3 Hydrodynamics
263
At a certain time, the free surface is a sine curve on the xoz plane, and the
height of the free surface at each particle is determined by the coordinate of
particle X. The intersection of free surface and ox is
x 0 =
nπ
k
, k = 0, ±1, ...,
The positions of these intersections do not change with time, so they are
called nodes. The horizontal distance between two adjacent peaks is called
the wavelength. Wave period is
T =
2π
ω
=
κλ
ω
=
2πλ
g
ω
2
= kg
Vibration speed of water particle
u =
∂ϕ
∂ x
=
h
2
gk
ω
e
kz cos kx cos ωt
w =
∂ϕ
∂z
=
h
2
gk
ω
e
kz sin kx cos ωt
The approximate trajectory of a water particle is
(z − z 0 ) = (x − x 0 )tgkx 0
For shallow standing wave standing waves, a constant water depth is H.
Let z = −H == constant, then
ω
2
= kg tanh k H
The velocity potential function is
ϕ =
h
2
g
ω
cosh k(z + H )
cosh k H
sin kx cos ωt
The shape of the free surface is
η =
h
2
sin kx sin ωt