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Since this shallow-water phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion. Using another normalization for the same frequency dispersion relation, the figure on the right shows that for a fixed wavelength λ the phase speed c p increases with increasing water depth. [1]
The equation says the matter wave frequency in vacuum varies with wavenumber (= /) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass ( ℏ ω 0 = m 0 c 2 {\displaystyle \hbar \omega _{0}=m_{0}c^{2}} ) and a quadratic part due to kinetic energy.
Tsunami generation and propagation, as computed with the shallow-water equations (red line; without frequency dispersion)), and with a Boussinesq-type model (blue line; with frequency dispersion). Observe that the Boussinesq-type model (blue line) forms a soliton with an oscillatory tail staying behind.
The KP equation can be used to model water waves of long wavelength with weakly non-linear restoring forces and frequency dispersion. If surface tension is weak compared to gravitational forces , λ = + 1 {\displaystyle \lambda =+1} is used; if surface tension is strong, then λ = − 1 {\displaystyle \lambda =-1} .
Frequency dispersion in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure.
In the mean horizontal-momentum equation, d(x) is the still water depth, that is, the bed underneath the fluid layer is located at z = −d. Note that the mean-flow velocity in the mass and momentum equations is the mass transport velocity Ũ , including the splash-zone effects of the waves on horizontal mass transport, and not the mean ...
Frequency dispersion in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure.
The KdV equation is a dispersive wave equation, including both frequency dispersion and amplitude dispersion effects. In its classical use, the KdV equation is applicable for wavelengths λ in excess of about five times the average water depth h , so for λ > 5 h ; and for the period τ greater than 7 h / g {\displaystyle \scriptstyle 7{\sqrt ...