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In , wave shoaling is the effect by which surface waves, entering shallower water, change in . It is caused by the fact that the , 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 in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in while the 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 , non-breaking waves will increase in wave height as the enters shallower water.

(1998). 9789263127020, World Meteorological Organization. .
This is particularly evident for as they wax in height when approaching a , with devastating results.


Overview
Waves nearing the coast experience changes in wave height through different effects. Some of the important wave processes are , , reflection, , 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 alterations in wave propagation direction or energy . Pure wave shoaling occurs for waves propagating to the parallel depth of a mildly sloping sea-bed. Then the wave height H at a certain location can be expressed as:
H = K_S\; H_0,
with K_S the shoaling coefficient and H_0 the wave height in deep water. The shoaling coefficient K_S depends on the local water depth h and the wave f (or equivalently on h and the wave period T=1/f). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth h is larger than about half the deep-water L_0=gT^2/(2\pi).


Physics
For non-, the associated with the wave motion, which is the product of the density with the , between two wave rays is a conserved quantity (i.e. a constant when following the energy of a from one location to another). Under stationary conditions the total energy transport must be constant along the wave ray – as first shown by in 1915. For waves affected by refraction and shoaling (i.e. within the approximation), the rate of change of the wave energy transport is:
\frac{d}{ds}(b c_g E) = 0,
where s is the co-ordinate along the wave ray and b c_g E is the energy flux per unit crest length. A decrease in group speed c_g and distance between the wave rays b must be compensated by an increase in energy density E. This can be formulated as a shoaling coefficient relative to the wave height in deep water.
(1991). 9789810204204, World Scientific. .
(2025). 9789814282390, World Scientific. .

For shallow water, when the is much larger than the water depth – in case of a constant ray distance b (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law:

H\, \sqrt4{h} = \text{constant},
with h the mean water depth, H the wave height and \sqrt4{h} the of h.


Water wave refraction
Following Phillips (1977) and Mei (1989),
(1977). 9780521298018, Cambridge University Press. .
(1989). 9789971507732, World Scientific. .
denote the phase of a wave ray as
S = S(\mathbf{x},t), \qquad 0\leq S<2\pi.
The local 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 rate of change of wavenumber is balanced by the convergence of the frequency along a ray;
\frac{\partial k}{\partial t} + \frac{\partial \omega}{\partial x} = 0.
Assuming stationary conditions (\partial/\partial t = 0), this implies that wave crests are conserved and the must remain constant along a wave ray as \partial \omega / \partial x = 0. As waves enter shallower waters, the decrease in caused by the reduction in water depth leads to a reduction in \lambda = 2\pi/k because the nondispersive shallow water limit of the dispersion relation for the wave ,
\omega/k \equiv c = \sqrt{gh}
dictates that
k = \omega/\sqrt{gh},
i.e., a steady increase in k (decrease in \lambda) as the decreases under constant \omega.


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