In fluid dynamics, wave shoaling is the effect by which surface waves entering shallower water change in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.
In other words, as the waves approach the shore and the water gets shallower, the waves get taller, slow down, and get closer together.
Waves nearing the coast change wave height through different effects. 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 changes in wave propagation direction and dissipation. Pure wave shoaling occurs for long-crested waves propagating perpendicular to the parallel depth contour lines of a mildly sloping sea-bed. Then the wave height at a certain location can be expressed as:
with the shoaling coefficient and the wave height in deep water. The shoaling coefficient depends on the local water depth and the wave frequency (or equivalently on and the wave period ). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth is larger than about half the deep-water wavelength
When waves enter shallow water they slow down. Under stationary conditions, the wave length is reduced. The energy flux must remain constant and the reduction in group (transport) speed is compensated by an increase in wave height (and thus wave energy density).
Convergence of wave rays (reduction of width ) at Mavericks, California, producing high surfing waves. The red lines are the wave rays; the blue lines are the wavefronts. The distances between neighboring wave rays vary towards the coast because of refraction by bathymetry (depth variations). The distance between wavefronts (i.e. the wavelength) reduces towards the coast because of the decreasing phase speed.
Shoaling coefficient as a function of relative water depth describing the effect of wave shoaling on the wave height – based on conservation of energy and results from Airy wave theory. The local wave height at a certain mean water depth is equal to with the wave height in deep water (i.e. when the water depth is greater than about half the wavelength). The shoaling coefficient depends on where is the wavelength in deep water: with the wave period and the gravity of Earth. The blue line is the shoaling coefficient according to Green's law for waves in shallow water, i.e. valid when the water depth is less than 1/20 times the local wavelength 
where is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed and distance between the wave rays must be compensated by an increase in energy density . This can be formulated as a shoaling coefficient relative to the wave height in deep water.
For shallow water, when the wavelength is much larger than the water depth – in case of a constant ray distance (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law:
with the mean water depth, the wave height and the fourth root of
Simplifying to one dimension and cross-differentiating it is now easily seen that the above definitions indicate simply that the rate of change of wavenumber is balanced by the convergence of the frequency along a ray;
Wave height – Difference between the elevations of a crest and a neighbouring trough
Ursell number – Dimensionless number indicating the nonlinearity of long surface gravity waves on a fluid layer.
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