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: 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.

[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] According to this law, which is based on linearized shallow water equations, the spatial variations of the wave height

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

The rays (and the changes in spacing between them) follow from the geometrical optics approximation to the linear wave propagation.

[6] In case of straight parallel depth contours this simplifies to the use of Snell's law.

Green's law also corresponds to constancy of the mean horizontal wave energy flux for long waves:[4][5] where

is the mean wave energy density integrated over depth and per unit of horizontal area,

The oscillation period (and therefore also the frequency) of shoaling waves does not change, according to Green's linear theory.

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

[9] Starting point are the linearized one-dimensional Saint-Venant equations for an open channel with a rectangular cross section (vertical side walls).

These equations describe the evolution of a wave with free surface elevation

are denoting partial derivatives with respect to space and time.

along the channel axis is brought into account by denoting them as

The above two equations can be combined into one wave equation for the surface elevation: In the Liouville–Green method, the approach is to convert the above wave equation with non-homogeneous coefficients into a homogeneous one (neglecting some small remainders in terms of

The next step is to apply a coordinate transformation, introducing the travel time (or wave phase)

(1), become: Now the wave equation (1) transforms into: The next step is transform the equation in such a way that only deviations from homogeneity in the second order of approximation remain, i.e. proportional to

for travelling waves of permanent form propagating in either the negative or positive

For the inhomogeneous case, considering waves propagating in the positive

-direction, Green proposes an approximate solution: Then Now the left-hand side of Eq.

Since the theory is linear, solutions can be added because of the superposition principle.

Waves varying sinusoidal in time, with period

-direction follows directly from substituting the solution for the surface elevation

the flow velocities shoal to leading order as[8] This could have been anticipated since for a horizontal bed