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172 4. The method of multiple scales

and a solution (with T = 0 at say), expressed in terms of is

This describes a fast oscillation (by virtue of the factor with a slow evolution of the
amplitude; these are the salient features of a WKB(J) solution. (Note that a property
of this solution is which is usually called ‘action’; typically, energy
and so is conserved—not the energy itself.)

The problem of finding higher-order terms in the WKB solution is addressed in Q4.15,
and the corresponding problem with is discussed in Q4.16, and
an interesting associated problem is discussed in Q4.17. We now consider the case of
a turning point.
It is apparent that the solution (4.47) is not valid if which is the case at a
turning point. In (4.43), we will write with throughout
the domain D, and analytic (to the extent that may be written as a uniformly
valid asymptotic expansion, as for This choice of has a
single (simple) turning point at x = 0; a turning point elsewhere can always be moved
to x = 0 by a suitable origin shift. The intention is to find a solution valid near the
turning point and then, away from this region, use the WKB method in x < 0 and
in x > 0. Thus the turning-point solution is to be inserted between the two WKB
solutions and, presumably, matched appropriately. We will present all these ideas, using
the method of multiple scales valid as in the following example.
E4.6 A turning-point problem
We consider

where both and are positive, O(1) constants, and analytic for
and The turning point is at x = 0, and the first issue is to decide
what scales to use in the neighbourhood of this point; this has already been addressed
in Let us write (and any scaling on y is redundant, in so far as the
governing equation is concerned, because the equation, (4.48), is linear) to give

and so a balance of terms is possible if i.e. fast scale. The
slow scale is simply However, a more convenient choice of the fast variable