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Wikipedia / Wave shoaling

In fluid dynamics, wave shoaling is the effect by which surface waves, entering shallower water, increase in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, decreases 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.

Particularly in a waterbody shallow enough for its surface to be affected by its bottom and where depth contours parallel the shore, a wave packet that does dissipate its energy by breaking will rise in height as it enters yet shallower water. This is plainly evident for tsunamis as they wax in height when approaching a coastline, often with devastating results.

Overview

Waves nearing the coast experience changes in 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 by changes in mean water depth – without alterations in wave propagation direction or energy dissipation. Pure wave shoaling occurs for long-crested waves propagating perpendicular to the parallel depth contour lines at a certain location can be expressed as:

and the wave frequency is larger than about half the deep-water wavelength

Physics

For non-breaking waves, the energy flux associated with the wave motion, which is the product of the wave energy density with the group velocity, between two wave rays is a conserved quantity (i.e. a constant when following the energy of a wave packet from one location to another). Under stationary conditions the total energy transport must be constant along the wave ray – as first shown by William Burnside in 1915. For waves affected by refraction and shoaling (i.e. within the geometric optics approximation), the rate of change of the wave energy transport is:

. This can be formulated as a shoaling coefficient relative to the wave height in deep water.

For shallow water, when the wavelength (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law:

the fourth root

Water wave refractionRefraction

Following Phillips (1977) and Mei (1989), denote the phase of a wave ray as

  1. .

The local wave number vector is the gradient of the phase function,

  1. ,

and the angular frequency is proportional to its local rate of change,

  1. .

The equations above and the following equation have direct analogues in Hamiltonian mechanics and the Hamilton-Jacobi theory of classical mechanics. 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;

  1. .

), this implies that wave crests are conserved and the frequency. As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

dictates that

  1. ,

i.e., a steady increase in k) as the phase speed.

See also

Notes