3 Hydrodynamics
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during the propagation process, which will cause dispersion. For gravity
waves, the dispersion relationship is ω 2 = gk tanh k H and the wave speed
varies with wavelength. In nonuniform media, wave velocity is also related
to the direction of wave propagation. This formula can further show that for
deepwater waves, the dispersion relationship is ω =
√
gk=
g
2π
λ , and the
wave frequency is inversely proportional to the square root of the wavelength;
for shallow water waves, the dispersion relationship is ω =
√
ghk =
gh
2π
λ ,
where the wave frequency is inversely proportional to the wavelength and
proportional to the water depth. As shown in Fig. 3.82.
Another basic solution is
ϕ = B cosh k(z + H ) cos kx cos ωt
This is a typical standing wave velocity potential function. In this kind
of wave, the wave function does not propagate along the x-axis, but makes
periodic movement in place. As shown in Fig. 3.83, the shape of the standing
wave is fixed, and there are antinodes and nodes, and the distance between
adjacent belly (node) particles is λ/2, and the distance between adjacent belly
and nodes is λ/4.
Fig. 3.82 Gravity wave dispersion relationship