Wave shoaling

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. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water. This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

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:

where ${\displaystyle s}$ is the co-ordinate along the wave ray and ${\displaystyle c_{g}E}$ is the energy flux per unit crest length. A decrease in group speed ${\displaystyle c_{g}}$ 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.

...
Wikipedia