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expression is to match to (5.119); this requires that

i.e.

Finally, to be consistent with the development of this asymptotic expansion, we set

and then satisfies

where is given by (5.117). It is left as an exercise to show that the solution of
this equation, which satisfies the boundary and matching conditions, is

And, to complete our presentation, we comment that expanded for
produces

and the term here in contributes, apparently, to a change in the uniform flow at
infinity—which is impossible—and hence the need for a matched solution in the far
field. It was this observation that first alerted the earlier researchers to the difficulties
inherent in this problem; this complication is typical of flows in the limit

Another general area of study in fluid mechanics is gas dynamics, where the compress-
ibility of the fluid cannot be ignored. We have already seen some of these problems
(E3.2, E3.5 and Q3.9–3.11); we now look at another classical example.

E5.21 A piston problem
We consider the one-dimensional flow of a gas in a long, open-ended tube. The gas
is brought into motion by the action of a piston at one end, which moves forward at
a speed which is much less than the sound speed in the gas. (This is usually called the
acoustic problem.) The gas is modelled by the isentropic law for a perfect gas (pressure
and is described by the equations