The equations satisfied by waves are
for water of depth
(1)
at
(2)
at
(3)
(1) can be solved by separation of variables technique. Assume
(there will also be an arbitrary factor of
which we deal with later). (1) becomes
since the left hand side is a function of
only and the right hand side is a function of
only, so both sides are equal to the same constant
If
the solution will be exponential, tending to
as
and the same problem occurs if
so
to give
where
This equation has solutions
The corresponding equation for
is
which has solutions
hence
at
so
and
In the same way we can solve (2) by the separation of variables method to find
Assume 
(ignoring the factor
)to give
as before, and as beforeelse
as
so put
to give
Now write
and
and since
we have
so by picking a suitable point on the wave surface and a suitable time we have