Wave shoaling
Waves grow taller as water shallows, conserving energy flux.
Wave shoaling is the process by which surface waves increase in height as they move into shallower water. This occurs because the group velocity, which carries wave energy, decreases with depth, forcing the energy density to rise to maintain a constant energy flux. Shoaling waves also slow down and shorten in wavelength while their frequency remains unchanged.
Quick Facts
- Conserved quantity
- energy flux (product of wave energy density and group velocity)
- First shown by
- William Burnside in 1915
- Shoaling coefficient relation
- Green's law for shallow water with constant ray distance
- Green's law formula
- wave height proportional to the fourth root of mean water depth
Facts from the source article.
Physics
For non-breaking waves, the energy flux, which is the product of wave energy density and group velocity, remains constant along a wave ray under stationary conditions. This conservation was first demonstrated by William Burnside in 1915. Within the geometric optics approximation, changes in group speed and the distance between wave rays must be balanced by a change in energy density. This relationship can be expressed as a shoaling coefficient relative to deep-water wave height. In shallow water where wavelength greatly exceeds depth and depth contours are parallel, wave shoaling follows Green's law: wave height is proportional to the fourth root of mean water depth.
Water wave refraction{{anchor|Refraction}}
The phase of a wave ray is denoted, with the local wave number vector as the gradient of the phase function and angular frequency proportional to its local rate of change. These definitions have direct analogues in Hamiltonian mechanics and the Hamilton-Jacobi theory. Under stationary conditions, wave crests are conserved and frequency remains constant along a wave ray. As waves enter shallower water, the decrease in group velocity due to reduced depth causes a reduction in wavelength. This follows from the nondispersive shallow-water dispersion relation, where phase speed decreases with depth, leading to a steady increase in wavenumber.
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