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In fluid dynamics, Green's law describes the evolution of non-breaking surface gravity waves propagating in shallow water of gradually varying depth and width. The law is named after George Green. In its simplest form, for wavefronts and depth contours parallel to each other (and the coast), it states:
Contents
where
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. 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.
Description
According to this law, which is based on linearized shallow water equations, the spatial variations of the wave height
where
with the subscripts 1 and 2 denoting quantities in the associated cross section. So, when the depth has decreased by a factor sixteen, the waves become twice as high. And the wave height doubles after the channel width has gradually been reduced by a factor four. For wave propagation perpendicular towards a straight coast with depth contours parallel to the coastline, take
For refracting long waves in the ocean or near the coast, the width
Green published his results in 1838, based on a method – the Liouville–Green method – which would evolve into what is now known as the WKB approximation. Green's law also corresponds to constancy of the mean horizontal wave energy flux for long waves:
where
Wavelength and period
Further, from Green's analysis, the wavelength
along a wave ray. The oscillation period (and therefor also the frequency) of shoaling waves does not change, according to Green's linear theory.
Derivation
Green derived his shoaling law for water waves by use of what is now known as the Liouville–Green method, applicable to gradual variations in depth