Green's law

In fluid dynamics, Green's law, named for 19th-century British mathematician George Green, is a conservation law describing the evolution of non-breaking, surface gravity waves propagating in shallow water of gradually varying depth and width. In its simplest form, for wavefronts and depth contours parallel to each other (and the coast), it states:

${\displaystyle H_{1}\,\cdot \,{\sqrt[{4}]{h_{1}}}=H_{2}\,\cdot \,{\sqrt[{4}]{h_{2}}}}$   or   ${\displaystyle \left(H_{1}\right)^{4}\,\cdot \,h_{1}=\left(H_{2}\right)^{4}\,\cdot \,h_{2},}$

where ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the wave heights at two different locations – 1 and 2 respectively – where the wave passes, and ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ are the mean water depths at the same two locations.

Green's law is often used in coastal engineering for the modelling of long shoaling waves on a beach, with "long" meaning wavelengths in excess of about twenty times the mean water depth.^[1] Tsunamis shoal (change their height) in accordance with this law, as they propagate – governed by refraction and diffraction – through the ocean and up the continental shelf. Very close to (and running up) the coast, nonlinear effects become important and Green's law no longer applies.^[2][3]