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254 5. Some worked examples arising from physical problems

where the surface boundary condition finally gives

and is arbitrary at this stage. For rightward propagation, we select

so that the characteristic variable becomes

which, with D = 1 (constant depth), recovers the standard result: Note
that, in this calculation, we have taken the evaluation at the surface to be on z = 1;
see below.
At the next order, equations (5.129) give

with
and

The boundary conditions at the surface, have been written by invoking
Taylor expansions, and so become the corresponding boundary conditions evaluated
on z = 1, valid as (and for sufficiently smooth surface waves). Again, this set
is fairly easily solved; the details are left as an exercise, but some of the intermediate
results are

and