Influid dynamics,wave shoaling is the effect by whichsurface waves, entering shallower water, increase inwave height. It is caused by the fact that thegroup velocity, 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 inenergy density in order to maintain a constant energy flux.^[2] Shoaling waves will also exhibit a reduction inwavelength while thefrequency 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.

Particularly in awaterbodyshallow enough for its surface to be affected by its bottom and wheredepth contours parallel the shore, awave packet that does dissipate its energy bybreaking will rise in height as it enters yet shallower water.^[3] This is plainly evident fortsunamis as they wax in height when approaching acoastline, often with devastating results.

Waves nearing the coast experience changes in wave height through different effects. Some of the important wave processes arerefraction,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 by changes in mean water depth – without alterations in wave propagation direction or energydissipation. Pure wave shoaling occurs forlong-crested waves propagatingperpendicular to the parallel depthcontour lines of a mildly sloping sea-bed. Then the wave height${\displaystyle H}$ at a certain location can be expressed as:^[4][5]

${\displaystyle H=K_S{H}_0,}$

with${\displaystyle K_S}$ the shoaling coefficient and${\displaystyle H_0}$ the wave height in deep water. The shoaling coefficient${\displaystyle K_S}$ depends on the local water depth${\displaystyle h}$ and the wavefrequency${\displaystyle f}$ (or equivalently on${\displaystyle h}$ and the wave period${\displaystyle T=1/f}$). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth${\displaystyle h}$ is larger than about half the deep-waterwavelength${\displaystyle L_0=g{T}^2/(2\pi ).}$

For non-breaking waves, theenergy flux associated with the wave motion, which is the product of thewave energy density with thegroup velocity, between twowave rays is aconserved quantity (i.e. a constant when following the energy of awave packet from one location to another). Under stationary conditions the total energy transport must be constant along the wave ray – as first shown byWilliam Burnside in 1915.^[6]For waves affected by refraction and shoaling (i.e. within thegeometric optics approximation), therate of change of the wave energy transport is:^[5]

${\displaystyle \frac{d}{ds}(b{c}_{g}E)=0,}$

where${\displaystyle s}$ is the co-ordinate along the wave ray and${\displaystyle b{c}_{g}E}$ is the energy flux per unit crest length. A decrease in group speed${\displaystyle c_g}$ and distance between the wave rays${\displaystyle b}$ must be compensated by an increase in energy density${\displaystyle E}$. This can be formulated as a shoaling coefficient relative to the wave height in deep water.^[5][4]

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

${\displaystyle H\sqrt[4]{h}=\text{constant},}$

with${\displaystyle h}$ the mean water depth,${\displaystyle H}$ the wave height and${\displaystyle \sqrt[4]{h}}$ thefourth root of${\displaystyle h.}$

FollowingPhillips (1977) andMei (1989),^[7][8] denote thephase of awave ray as

.

The localwave number vector is the gradient of the phase function,

${\displaystyle \mathbf{k}=\mathrm{\nabla }S}$,

and theangular frequency is proportional to its local rate of change,

${\displaystyle \omega =-\mathrm{\partial }S/\mathrm{\partial }t}$.

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;

${\displaystyle \frac{\mathrm{\partial }k}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\omega }{\mathrm{\partial }x}=0}$.

Assuming stationary conditions (${\displaystyle \mathrm{\partial }/\mathrm{\partial }t=0}$), this implies that wave crests are conserved and thefrequency must remain constant along a wave ray as${\displaystyle \mathrm{\partial }\omega /\mathrm{\partial }x=0}$.As waves enter shallower waters, the decrease ingroup velocity caused by the reduction in water depth leads to a reduction inwave length${\displaystyle \lambda =2\pi /k}$ because the nondispersiveshallow water limit of thedispersion relation for the wavephase speed,

${\displaystyle \omega /k\equiv c=\sqrt{gh}}$

dictates that

${\displaystyle k=\omega /\sqrt{gh}}$,

i.e., a steady increase ink (decrease in${\displaystyle \lambda }$) as thephase speed decreases under constant${\displaystyle \omega }$.

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