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Figure 17. Wave propagation in stationary water over variable depth.

where is to be determined. The non-dimensional equations are

with

and

Here (u, w) are the velocity components of the flow, p the pressure in the fluid relative
to the hydrostatic pressure (with pressure constant at the surface) and is
the surface of the water. The bottom is represented by the function see
figure 17.
We seek a solution in the familiar form:

where q represents each of u, w, p and The leading-order problem gives, from
(5.129),

with

This set is easily solved; the relevant solution (in which is not a function of z) is