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184 4. The method of multiple scales

and so we require

It is immediately evident that we have a non-uniformity as From (4.82) we see
that corresponds to a non-dispersive wave i.e. c = ±1 for all wave numbers, k.
When we set in (4.83), and use no solution exists, although for any
no complications arise and we may proceed.

This failure is not to be regarded as fatal: for the assumed initial data (the harmonic
wave), or any other reasonable initial conditions, an asymptotic solution (with
can be found by the method of strained coordinates. For a wave problem, as we have
seen (e.g. Q3.25), this simply requires a suitable representation of the characteristic
variables. Nevertheless, in the most extreme cases, when the method of multiple scales
is still deemed to be the best approach, the resulting asymptotic solution may not be
uniformly valid as T (or in our discussion of ordinary differential equations)
Typically, the multiple-scale solution will be valid for T (or no larger than O(1),
but this is usually an improvement on the validity of a straightforward asymptotic
expansion.

4.6 BOUNDARY-LAYER PROBLEMS
The examples that we have discussed so far involve, usually with a rather straightforward
physical interpretation, the slow evolution or development of an underlying solution.
Boundary-layer problems (see §2.6–2.8), on the other hand, might appear not to possess
this structure. Such problems have different—but matched—solutions away from, and
near to, a boundary. However, the solution of such problems expressed as a composite
expansion (see §1.10) exhibits precisely the multiple-scale structure: a fast scale which
describes the solution in the boundary layer, and a slow scale describing the solution
elsewhere. We will demonstrate the details of this procedure by considering again our
standard boundary-layer-type problem given in equation (2.63) (and see also (1.16)).

E4.10 A boundary-layer problem
We consider

with where and are constants independent of
and The boundary-layer variable for this problem (see (2.66)) is
and so we write which leads to equation (4.84)