According to this law, which is based on linearized shallow water equations, the spatial variations of the wave height  (twice the amplitude  for sine waves, equal to the amplitude for a solitary wave) for travelling waves in water of mean depth  and width  (in case of an open channel) satisfy
where  is the fourth root of  Consequently, when considering two cross sections of an open channel, labeled 1 and 2, the wave height in section 2 is:
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  a constant, say 1 metre or yard.
For refracting long waves in the ocean or near the coast, the width  can be interpreted as the distance between wave rays. The rays (and the changes in spacing between them) follow from the geometrical optics approximation to the linear wave propagation. In case of straight parallel depth contours this simplifies to the use of Snell's law.
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  is the group speed (equal to the phase speed in shallow water),  is the mean wave energy density integrated over depth and per unit of horizontal area,  is the gravitational acceleration and  is the water density.