Effect by which surface waves entering shallower water change in wave height
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.
Overview
Waves nearing the coast change wave height through different effects. Some of the important wave processes are
refraction,
diffraction,
reflection,
wave breaking,
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 changes in wave propagation direction and
dissipation. Pure wave shoaling occurs for
long-crested waves propagating
perpendicular to the parallel depth
contour lines of a mildly sloping sea-bed. Then the wave height at a certain location can be expressed as:[4][5]
with the shoaling coefficient and the wave height in deep water. The shoaling coefficient depends on the local water depth and the wave
frequency (or equivalently on and the wave period ). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth is larger than about half the deep-water
wavelength
Physics
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 – as first shown by
William Burnside in 1915.[6]
For waves affected by refraction and shoaling (i.e. within the
geometric optics approximation), the
rate of change of the wave energy transport is:[5]
where is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed and distance between the wave rays must be compensated by an increase in energy density . This can be formulated as a shoaling coefficient relative to the wave height in deep water.[5][4]
For shallow water, when the
wavelength is much larger than the water depth – in case of a constant ray distance (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies
Green's law:
with the mean water depth, the wave height and the
fourth root of
and the
angular frequency is proportional to its local rate of change,
.
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;
.
Assuming stationary conditions (), this implies that wave crests are conserved and the
frequency must remain constant along a wave ray as .
As waves enter shallower waters, the decrease in
group velocity caused by the reduction in water depth leads to a reduction in
wave length because the nondispersive
shallow water limit of the
dispersion relation for the wave
phase speed,
dictates that
,
i.e., a steady increase in k (decrease in ) as the
phase speed decreases under constant .
See also
Airy wave theory – Fluid dynamics theory on the propagation of gravity waves
Breaking wave – Wave that becomes unstable as a consequence of excessive steepness