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Any initial conditions that do not conform to this pattern would require a different
asymptotic, and multi-scale, structure. Other examples that correspond to E4.8 are set
as exercises Q4.23–4.26; see also Q4.27.
Before we describe one last general application of the method of multiple scales—
perhaps a rather surprising one—we note a particular limitation on the method.

4.5 A LIMITATION ON THE USE OF THE METHOD OF MULTIPLE SCALES
The foregoing examples that have shown how to generate asymptotic solutions of par-
tial differential equations appear reasonably routine and highly successful. However,
there is an underlying problem that is not immediately evident and which cannot be
ignored. In the context of wave propagation, which is the most common application
of this technique to partial differential equations, we encounter difficulties if the pre-
dominant solution is a non-dispersive wave i.e. waves with different wave number (k)
all travel at the same speed. To see how this difficulty can arise, we will examine an
example which is close to that introduced in E4.8.

E4.9 Dispersive/non-dispersive wave propagation
We consider the equation (cf. (4.73))

where is a given constant and we seek a solution in the form
where

The equation then becomes

and we look for a solution periodic in in the form of a harmonic wave:

where all this follows the procedure described in E4.8.
Here, we find that

but is yet to be determined. At the next order, the equation for can be written