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Wave shoaling

Surfing on shoaling and breaking waves.
The phase velocity cp (blue) and group velocity cg (red) as a function of water depth h for surface gravity waves of constant frequency, according to Airy wave theory.
Quantities have been made dimensionless using the gravitational acceleration g and period T, with the deep-water wavelength given by L0 = gT2/(2π) and the deep-water phase speed c0 = L0/T. The grey line corresponds with the shallow-water limit cp =cg = √(gh).
The phase speed – and thus also the wavelength L = cpT – decreases monotonically with decreasing depth. However, the group velocity first increases by 20% with respect to its deep-water value (of cg = ⁠1/2⁠c0 = gT/(4π)) before decreasing in shallower depths.[1]

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 other words, as the waves approach the shore and the water gets shallower, the waves get taller, slow down, and get closer together.

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.

  1. ^ Wiegel, R.L. (2013). Oceanographical Engineering. Dover Publications. p. 17, Figure 2.4. ISBN 978-0-486-16019-1.
  2. ^ 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. Bibcode:1964DSRA...11..529L. doi:10.1016/0011-7471(64)90001-4. Archived from the original (PDF) on 2010-06-12. Retrieved 2010-03-25.
  3. ^ WMO (1998). Guide to Wave Analysis and Forecasting (PDF). Vol. 702 (2 ed.). World Meteorological Organization. ISBN 92-63-12702-6.

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