# Wave shoaling

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.^{[2]} Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water.^{[3]} This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

## Mathematics

### Wave shoaling

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:^{[4]}

where is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed 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.^{[5]}^{[6]}

For shallow water, when the wavelength is much larger than the water depth, wave shoaling satisfies Green's law:

with the mean water depth, the wave height and the fourth root of

### Refraction effects

Following Phillips (1977) and Mei (1989),^{[7]}^{[8]} denote the phase of a wave ray as

- .

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

- ,

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

- .

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;

- .

Assuming stationary conditions (), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as . 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

- ,

i.e., a steady increase in *k* (decrease in ) as the phase speed decreases under constant .

## See also

## Notes

- ↑ Wiegel, R.L. (2013).
*Oceanographical Engineering*. Dover Publications. p. 17, Figure 2.4. ISBN 0-486-16019-X. - ↑ Longuet-Higgins, M.S.; Stewart, R.W. (1964). "Radiation stresses in water waves; a physical discussion, with applications" (PDF).
*Deep-Sea Research and Oceanographic Abstracts*.**11**(4): 529–562. - ↑ WMO (1998).
*Guide to Wave Analysis and Forecasting*(PDF).**702**(2 ed.). World Meteorological Organization. ISBN 92-63-12702-6. - ↑ Burnside, W. (1915). "On the modification of a train of waves as it advances into shallow water".
*Proceedings of the London Mathematical Society*. Series 2.**14**: 131–133. doi:10.1112/plms/s2_14.1.131. - ↑ Dean, R.G.; Dalrymple, R.A. (1991).
*Water wave mechanics for engineers and scientists*. Advanced Series on Ocean Engineering.**2**. Singapore: World Scientific. ISBN 978-981-02-0420-4. - ↑ Goda, Y. (2000).
*Random Seas and Design of Maritime Structures*. Advanced Series on Ocean Engineering.**15**(2 ed.). Singapore: World Scientific. ISBN 978-981-02-3256-6. - ↑ Phillips, Owen M. (1977).
*The dynamics of the upper ocean (2nd ed.)*. Cambridge University Press. ISBN 0-521-29801-6. - ↑ Mei, Chiang C. (1989).
*The Applied Dynamics of Ocean Surface Waves*. Singapore: World Scientific. ISBN 9971-5-0773-0.

## External links

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