WKB Approximation - An Example

An Example

Consider the second-order homogeneous linear differential equation

where . Substituting

results in the equation

To leading order (assuming, for the moment, the series will be asymptotically consistent) the above can be approximated as

In the limit, the dominant balance is given by

So δ is proportional to ε. Setting them equal and comparing powers renders

which can be recognized as the Eikonal equation, with solution

Looking at first-order powers of gives

This is the unidimensional transport equation, having the solution

where is an arbitrary constant. We now have a pair of approximations to the system (a pair because can take two signs); the first-order WKB-approximation will be a linear combination of the two:

Higher-order terms can be obtained by looking at equations for higher powers of ε. Explicitly

for . This example comes from Bender and Orszag's textbook (see references).