Wave Shoaling - Mathematics

Mathematics

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,

where s is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed must be compensated by an increase in energy density E. This can be formulated as a shoaling coefficient relative to the wave height in deep water.

Let us follow Phillips (1977) and Mei (1989) and denote the phase of a wave ray as

.

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

,

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 .