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when we retain the O(1) term only. From (3.39), we find that

where we have retained the O(1) and terms (and we have assumed that H possesses
a suitable Taylor expansion). These two expansions, (3.41) and (3.42), match precisely
when we select and then

We conclude this important example by making a few observations.
First, the behaviour of the far-field solution, (3.43), as recovers the near-
field solution, and so (to this order, at least) the far-field solution is uniformly valid in
(although our original expansion, (3.25), exhibits a singular behaviour). The
solution (3.43), as Y increases, can be completely described by the characteristic lines,
in the form

along which These lines are therefore straight, but not par-
allel; they first intersect for some (which depends on the details of the function
and then the solution becomes multi-valued in —which is unacceptable,
unless we revert to an integral form of equation (3.38) and then admit a discontinuous
solution. This discontinuity, at a distance from the surface of the aerofoil, man-
ifests itself in the physical world as heralding the formation of a shock wave. It should be
no surprise that the characteristic variable plays a significant rôle in the solution of this
wave-type (hyperbolic) equation; indeed, the results described here can be obtained by
seeking asymptotic expansions for these, rather than for the functions themselves (see
Q3.9). A final point, which embodies an important idea, is to note that the near-field
solution takes essentially the correct form (to leading order) even for the far-field, in
the sense that the solution is replaced by where is the
appropriate approximation to the characteristic variable. One way to interpret this is
to regard as the correct solution, but that it is in the ‘wrong place’ i.e. it
is not constant along lines but rather, along lines

We have seen, in these two somewhat routine examples, and the similar problems in
the exercises, how the simplest type of singular perturbation problem can arise. The
other fundamentally different problem that we may encounter, just as for ordinary
differential equations, is where the small parameter multiplies the highest derivative: