Uniform asymptotic solutions of the reduced wave equation

Abstract

x = (x1 , 3c2 , 3 , x ) and v is a formal series in inverse powers of k, has been employed with great success. The geometrical theory of diffraction developed by Keller [2] to extend geometrical optics into regions where it predicts an incomplete asymptotic solution, has increased the range of usefulness of expansions of the above form. However, in transition regions near caustics, shadow boundaries or source points, where the amplitude terms v become unbounded, modified forms of asymptotic expansions are needed. The boundary layer theory which involves the stretching of coordinates to produce a boundary layer expansion, which must then be matched with outer expansions of the above form, is one approach towards the problem of transition regions [3], [4]. We will be concerned here with a second method, that of uniform asymptotic expansions. It is similar to that developed by Langer, Olver (see [5], [6], respectively), and others, to treat turning point problems for ordinary differential equations. It has been used by Ludwig [7] and by Lewis [8], et al., to obtain asymptotic expansions near caustics and shadow boundaries, respectively. While specific problems have been solved by the uniform expansion method, a general theory, of the type that exists for turning point problems, has not been