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After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies. Cnoidal waves are exact periodic solutions to the Korteweg–de Vries equation in shallow water, that is, when the wavelength of the wave is much greater than the depth of the water.
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.
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 ...
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 depth; Ocean surface wave – real water waves as seen in the ocean and sea; Stokes wave – nonlinear periodic waves in non-shallow water; Wave power ...
(Note: Most of the wave speeds calculated from the wavelength divided by the period are proportional to the square root of the length. Thus, except for the shortest wavelength, the waves follow the deep water theory described in the next section. The 8.5 m long wave must be either in shallow water or between deep and shallow.)
A wave breaks when it runs into shallow water, or when two wave systems oppose and combine forces. When the slope, or steepness ratio, of a wave, is too great, breaking is inevitable. Individual waves in deep water break when the wave steepness—the ratio of the wave height H to the wavelength λ—exceeds about 0.17, so for H > 0.17 λ.
In this deep-water case, the phase velocity is twice the group velocity. The red square traverses the figure in the time it takes the green dot to traverse half. The dispersion relation for deep water waves is often written as =, where g is the acceleration due to gravity.
The speed of a wave in water depends on the depth, so the ripples slow down as they pass over the glass. This causes the wavelength to decrease. If the junction between the deep and shallow water is at an angle to the wavefront, the waves will refract. In the diagram above, the waves can be seen to bend towards the normal.