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Airy's Theory and Wave Breaking

Shallow waves become unstable when steepness grows high and depth gets smaller at the shore.

Side view of different types of breaking wave front

To understand the unsteep wave's speed as a function of its property in shallow water, a simplified Airy's dispersion relation is often used. By assuming that wavelength (λ) is much greater than (usually >20 times) the water depth (h), which is a reasonable assumption for shallow water near shore, the Airy's dispersion relation gives the following relation between celerity of the wave and water depth:

$C\approx {\sqrt {gh}}$

From the relation, the celerity decreases as the wave approaches the shore where water becomes shallow. This implies a compiling effect onto the slowing down frontal wave by the wave behind. As this compilation generates more wave steepness and the shore water gets shallower, the Airy's wave theory breaks down at very near shore. Here, the steepness metric of a wave can be measured by the ratio of the wave's maximum amlitude (η) to the wavelength (λ). Depending on the bed slope near shore and the wave's steepness, the wave could break into several general categories of forms near shore.