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The shallow-water equations in unidirectional form are also called (de) Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below). The equations are derived [ 2 ] from depth-integrating the Navier–Stokes equations , in the case where the horizontal length scale is much greater than the vertical ...
The Camassa–Holm equation can be written as the system of equations: [2] + + =, = + + (), with p the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water layer on a horizontal bed.
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 gravity wave; Undertow (water waves) – Return flow below nearshore water waves.
When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations. Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation η {\displaystyle \eta } :
In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: {{math|c g = 1 / 2 c p. [7] The group velocity also turns out to be the energy transport velocity.
Adhémar Jean Claude Barré de Saint-Venant (French pronunciation: [ademaʁ ʒɑ̃ klod baʁe də sɛ̃ vənɑ̃]; 23 August 1797 – 6 January 1886) [1] was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as the Saint-Venant equations that are a fundamental set of equations used in ...
The frictional term in the shallow water equations, is nonlinear in both the velocity and water depth. In order to understand the latter, one can infer from the / (+) term that the friction is strongest for lower water levels. Therefore, the crest "catches up" with the trough because it experiences less friction to slow it down.
Shallow water equations — are also nonlinear and do have amplitude dispersion, but no frequency dispersion; they are valid for very long waves, λ > 20 h. Boussinesq equations — have the same range of validity as the KdV equation (in their classical form), but allow for wave propagation in arbitrary directions, so not only forward ...