# Wave shoaling: Wikis

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# Encyclopedia

--Oyvind Breivik 18:20, 18 March 2010 (UTC)

Surfing on shoaling and breaking waves.
Wave shoaling.

In fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height (which is about twice the amplitude). It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, changes with water depth.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height when following a wave packet as it moves from one location towards another location with less water depth[1]. This is particularly important with respect to the devastating effects of tsunamis when they reach the coast.

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.

Let us follow Mei (1989)[2] and denote the phase of a wave ray as $S = S(\mathbf{x},t), 0\leq S<2\pi$. The local wave number vector is the gradient of the phase function, $\mathbf{k} = \nabla S$, and the angular frequency is proportional to its local rate of change, $\omega = -\partial S/\partial t$. Simplifying to one dimension and cross-differentiating it is now easily seen that the above definitions indicate simply that the number of wave crests per unit length (i.e., points of constant phase S) is conserved along a wave ray; $\frac{\partial k}{\partial t} + \frac{\partial \omega}{\partial x} = 0$. Assuming stationary conditions ($\partial/\partial t = 0$), this implies that the frequency must remain constant along a wave ray as $\partial \omega / \partial x = 0$ .

Consequently, as waves enter shallower waters under stationary conditions, the decrease in group velocity caused by changes in water depth must be compensated by an increase in wave energy density (and thus also an increase in significant wave height).

However, the wave length λ = 2π / k must also decrease because the shallow water limit of phase and group velocity, $c = c_g = \sqrt{gh}$ dictates that $k = \omega/\sqrt{gh}$, i.e, a steady increase in k (decrease in λ) as the phase speed decreases.

## References

1. ^ WMO (1998). Guide to Wave Analysis and Forecasting. 702 (2nd edition ed.). World Meteorological Organization. ISBN 92-63-12702-6.
2. ^ Mei, Chiang C. (1989). The Applied Dynamics of Ocean Surface Waves. World Scientific. ISBN 9971-50-773-0.
• Dean, R.G.; Dalrymple, R.A. (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering. 2. Singapore: World Scientific. ISBN 978-9810204204.
• Goda, Y. (2000). Random Seas and Design of Maritime Structures. Advanced Series on Ocean Engineering. 15 (2nd edition ed.). Singapore: World Scientific. ISBN 978 981 02 3256 6.