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When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. [1] The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp ...
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.
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 waves propagate over an elliptic-shaped underwater shoal on a plane beach. This example combines several effects of waves and shallow water, including refraction, diffraction, shoaling and weak non-linearity. In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves.
Boussinesq approximation (water waves) – nonlinear theory for waves in shallow water. Capillary wave – surface waves under the action of surface tension; Cnoidal wave – nonlinear periodic waves in shallow water, solutions of the Korteweg–de Vries equation; Mild-slope equation – refraction and diffraction of surface waves over varying ...
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.
The first and third of these equations are solved at constant x by waves moving in either the positive or negative y direction at a speed =, the speed of so-called shallow-water gravity waves without the effect of Earth's rotation. [4]
shallow-water waves with weakly non-linear restoring forces, long internal waves in a density-stratified ocean, ion acoustic waves in a plasma, acoustic waves on a crystal lattice. The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.