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We now seek a solution of the set of equations (3.65)–(3.68), for and fixed
as bywriting

where q (and correspondingly represents each of p, T and u. The first terms
in each of these asymptotic expansions satisfy the equations

which follow from (3.65)–(3.68), respectively. These equations then give

and then, selecting the right-going wave (for simplicity), we have

for some f(x) at t = 0. For a solution-set which recovers the undisturbed state for
we also have

However, our experience with hyperbolic problems (see §3.2)is that asymptotic ex-
pansions like (3.69) are not uniformly valid as t (or x) for x – t = O(1) i.e.
in the far-field. Let us investigate the result of the non-uniformity directly, without
examining the details of the breakdown (which is left as an exercise).
The variables that we choose to use in the far-field are

for each q, so we have the identities

equations (3.65)–(3.68) become (when we retain only those terms relevant to the
determination of the dominant contributions to each Q):