Green's Law (Wave Height in Changing Depth)

A long wave in shallow water slows down as the depth falls. The energy flow of the wave stays constant, so a slower wave must grow taller. Green's law, from George Green in 1838, puts a number on that for long, non-breaking waves in water of slowly changing depth: the wave height times the fourth root of the depth stays constant, $Hh^{1/4}=\mathrm{constant}$. It follows from the linearized shallow-water equations, and it is why a tsunami grows as it comes in from deep water.

In $H_2=H_1(h_1/h_2)^{1/4}$, $H_1$ is the wave height where the water is $h_1$ deep and $H_2$ the height where it is $h_2$ deep, with $h_1$ the deeper place. The fourth root makes the growth gentle: when the depth has decreased by a factor of sixteen the waves are twice as high, so 1 m in 16 m of water becomes 2 m in 1 m. The page solves for any of the four quantities, so it can also find the depth at which a wave reaches a given height. The form here is the depth-only one, for a wave on an open coast where the width of the wave front does not change. Where the width $b$ of a channel also changes, the law is $H\sqrt{b}\,h^{1/4}=\mathrm{constant}$, and the wave height doubles when the width is reduced by a factor of four.

A tsunami of 1 m in water 4 000 m deep, arriving in 10 m of water, has a depth ratio of 400 and reaches $(400)^{1/4}$, or 4.4721 m, about 4.5 m. Working backwards gives the same pairs: 2 m from 1 m with a deep depth of 16 m puts the shallow depth at 1 m, and with a shallow depth of 1 m it puts the deep depth at 16 m. The height in deep water is an input to this page, and the run-up on land is a separate calculation.

The law is for waves long compared with the depth and for gradual slopes. On a steep shelf, or where the wave breaks, it can over-predict the height.