Wave shoaling

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. For waves affected by refraction and shoaling (i.e. within the geometric optics approximation), the rate of change of the wave energy transport is:

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

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.

For shallow water, when the wavelength 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 satisfies Green's law:

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

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

Water wave refraction

Following Phillips (1977) and Mei (1989), denote the phase of a wave ray as

${\displaystyle S=S(\mathbf{x},t),0\le S<2\pi }$.

The local wave number vector is the gradient of the phase function,

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

and the angular 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 the frequency must remain constant along a wave ray as ${\displaystyle \mathrm{\partial }\omega /\mathrm{\partial }x=0}$. As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length ${\displaystyle \lambda =2\pi /k}$ because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

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

dictates that

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

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