For non-deforming waves, in the coordinate system moving with the same speed as the wave celerity, c, the wave profile is standing still and the velocity field becomes steady.
where [nabla] = ([partial derivative]/[partial derivative]x, [partial derivative/[partial derivative]y) is the two-dimensional horizontal gradient operator, c the wave celerity and [c.
For very shallow water waves of small amplitude, the wave celerity is independent of the wave period, and hence all component waves propagate by the same wave celerity, [square root of (gh)], which means waves propagate by keeping the same profile.
where u (x, t) is the wave function and c(x) is an arbitrary continuous or discontinuous function having the meaning of the local wave celerity.
Obtained travelling wave solutions can be applied in oceanography to study the wave transformation above complicated bottom relief, which can be presented as superposition of small sections, for which the wave celerity changes as c(x) ~ [x.