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170 4. The method of multiple scales

Thus a solution for which is periodic in T requires

which possess the general solution

where and are arbitrary—possibly complex which then requires
the complex conjugate to be included—constants. We see that

(a) if then and are oscillatory and so is oscillatory in
both T and
(b) if then there exists a solution which grows exponentially;
(c) if then (see (4.42)) one of or is constant and the other grows
linearly.

The method of multiple scales has enabled us, albeit in the limit to describe
all the essential features of solutions of Mathieu’s equation, and how these change as
the parameters select different positions and regions in The corresponding
problem for n = 2 is discussed in Q4.12, and related exercises are given in Q4.13 &
4.14.
We now turn to an important class of problems that are exemplified by the equation
(cf. Q2.24)

where is given. In the simplest problem of this type, takes one sign
throughout the given domain (D) i.e. a > 0 (oscillatory) or a < 0 (exponential). A
more involved situation arises if a changes sign in the domain: a turning-point problem
(see Q2.24). The intention here is to examine the solution of the equation in the case
so that, for the coefficient is slowly varying. We use the
method of multiple scales to analyse this problem and hence give a presentation of the
technique usually referred to as ‘WKB’. (This is after Wentzel, 1926; Kramers, 1926;
Brillouin, 1926, although the essential idea can be traced back to Liouville and Green.
Some authors extend the label to WKBJ, to include Jeffreys, 1924.) We will formulate
an oscillatory problem and use this example to describe the WKB approach.

E4.5 WKB method for a slowly-varying oscillation
We consider