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250 5. Some worked examples arising from physical problems

Figure 16. Sketch of a piston moving a gas (according to in an open tube.

where a is the (local) sound speed in the gas and u its speed along the tube. The initial
and boundary conditions are

and
with and V(0) = 0; see figure 16. (The problem of a closed tube,
described in a Lagrangian framework and using the method of multiple scales, is
discussed by Wang & Kassoy, 1990.)
We are already familiar with the result, in small-disturbance theories—which this
is for the simple, near-field wave-propagation problem is not uniformly
valid as t (or see E3.2. In particular, for disturbances propagating down
the tube (into which are described by e.g. u(x, t) ~ F(x – t), there will
be a breakdown where (or, equivalently, when with x – t =
O(1). Note that, in this problem, because the tube is open, there can be no disturbances
propagating back towards the piston. We introduce and (to make
the evaluation on as simple as possible) and then write and
to give

with
and

We seek an asymptotic solution, which is to be uniformly valid as in the form