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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.
x is the horizontal coordinate and the wave propagation direction (meters), z is the vertical coordinate, with the positive z direction pointing out of the fluid layer (meters), λ is the wave length (meters), T is the wave period . As derived below, the horizontal component ū S (z) of the Stokes drift velocity for deep-water waves is ...
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.
Several integral properties of Stokes waves on deep water as a function of wave steepness. [23] The wave steepness is defined as the ratio of wave height H to the wavelength λ. The wave properties are made dimensionless using the wavenumber k = 2π / λ, gravitational acceleration g and the fluid density ρ.
The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic, sound or water waves). The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation: =
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Some of the important wave processes are refraction, diffraction, reflection, wave breaking, wave–current interaction, friction, wave growth due to the wind, and wave shoaling. In the absence of the other effects, wave shoaling is the change of wave height that occurs solely due to changes in mean water depth – without alterations in wave ...
In shallow water, with the water depth small compared to the wavelength, the individual waves break when their wave height H is larger than 0.8 times the water depth h, that is H > 0.8 h. [25] Waves can also break if the wind grows strong enough to blow the crest off the base of the wave.