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251

which gives (from (5.121)

and so on. Further, we assume that the boundary condition at the piston can be
expressed as a Taylor expansion about t:

the validity of which certainly requires that remains finite as (and so
must be finite).
From (5.123) we find that

where is an arbitrary function, but and are otherwise undetermined at this
stage. From (5.124), we multiply the first by and then add to it (5.124b),
which eliminates and to produce

The terms in are now replaced by using (5.126) to give the equation
for

This is a nonlinear equation which, for given is readily solved. However, this
solution is incomplete without the weak acoustic shock wave that propagates ahead of
this solution; we must therefore write down the conditions for the insertion of a shock
(discontinuity).
First, from (5.122), this initial condition requires that To further deter-
mine we impose the Rankine-Hugoniot conditions that define the jump conditions
across the shock. The conditions ahead are undisturbed; let the conditions behind the
shock be denoted by the subscript ‘s’ and write the speed of the shock as In this