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252 5. Some worked examples arising from physical problems

problem, these conditions (see e.g. Courant & Friedrichs, 1967) can be written as

and so

(Here, is the perturbation of the density.) This latter result confirms, from (5.126),
that behind the shock and, since ahead of the shock, we have
for Thus from (5.127) we obtain the implicit result

where F is an arbitrary function which, from (5.125), can be determined to give

(since, at the piston,
Thus the near-field solution is recovered, although this needs to be written
to accommodate the existence of the wave front there i.e.

where H is the Heaviside step function: We conclude
with the observation that the shock wave travels faster than the local sound speed
behind the shock; that is, from (5.128), as compared with
(and remember that ). Much more detail can be found in
any good text on gas dynamics.

As our final example, we use a similar technique to that employed in the previous
problem, but now in a quite different context: waves on the surface of water. (See
Q3.4 for a much simpler but related exercise.)

E5.22 A variable-depth Korteweg-de Vries equation for water waves
We consider the one-dimensional propagation of waves over water (incompressible),
which is modelled by an inviscid fluid without surface tension. The water is stationary
in the absence of waves, but the local depth varies on the same scale that is used to
measure the weak nonlinearity and dispersive effects in the governing equations. For
right-running waves, the appropriate far-field coordinates are and