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Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by shallow-water phase velocity √ gh as a function of relative depth h / λ. Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √ gh valid in shallow water.
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).
Boussinesq approximation (water waves) – Approximation valid for weakly non-linear and fairly long waves; Mild-slope equation – Physics phenomenon and formula; Shallow water equations – Set of partial differential equations that describe the flow below a pressure surface in a fluid; Stokes drift – Average velocity of a fluid parcel in a ...
Shallow-water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain (e.g. surface waves in a bath). In order for shallow-water equations to be valid, the wavelength of the phenomenon they are supposed to model has to be much larger than the depth of the ...
The standard example of a longitudinal wave is a sound wave or "pressure wave" in gases, liquids, or solids, whose oscillations cause compression and expansion of the material through which the wave is propagating. Pressure waves are called "primary waves", or "P-waves" in geophysics. Water waves involve both longitudinal and transverse motions ...
Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the dispersion relation.
A requirement for this in river currents is a flowing water with shallow depth in which the inertia of the water overcomes its gravity due to the supercritical flow speed (Froude number: 1.7 – 4.5, surpassing 4.5 results in direct standing wave [7]) and is therefore neither significantly slowed down by the obstacle nor pushed to the side.
The solution to these equations yields the following phase speed: =; this result is the same speed as for shallow-water gravity waves without the effect of Earth's rotation. [1] Therefore, these waves are non-dispersive (because the phase speed is not a function of the zonal wavenumber).