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222 5. Some worked examples arising from physical problems

where C is an arbitrary constant. The solution in and is now found
by using the WKB method (see E4.5). (In this example, we will find an approximation
to the solution, in each of the three regions, by using only the appropriate local variable;
as we have seen in §4.3, all this could be expressed using formal multiple scales.)
For R > 0, is oscillatory and so in we seek a solution of equation (5.64)
in the form

where (as is a scaling to be determined, is a constant
and is to be written as a suitable asymptotic expansion when we know
Thus we obtain

where and the prime denotes the derivative with respect to r. Thus
we require and so we write

and hence we obtain

and so on. These two equations are readily solved, to give

where A is an arbitrary constant.
The corresponding solution for (the details of which are left as an exercise) is