A note on uniform asymptotic wave diffraction by a wedge

Abstract

Summary New expressions for asymptotically uniform Green’s functions for high-frequency wave diffraction when a plane, cylindrical or point wave field is incident on an ideal wedge are derived. They are useful for deriving a uniform asymptotic expression for the exact solution in terms of the high-frequency diffracted and geometrical optics far field. The present method is simple and consists of differentiating out the singularities of the integral representations and using new representations for trigonometrical sums that arise when the wedge angle is a rational multiple of �. The new results make explicit the continuity of the fields across shadow and reflection boundaries. The geometrical theory of diffraction (GTD) was introduced by Keller (1) as an asymptotic method for the solution of diffraction problems at high frequencies. Keller’s GTD has proved to be very powerful in a wide variety of applications to physical problems. The method uses the saddle point method to derive asymptotic approximations from the exact solution of canonical wedge problems to derive so called ”diffraction coefficients”. The method gives a useful physical representation of the total far-field in terms of the geometrical optics and a diffracted field term. One drawback of Keller’s method is that it breaks down at shadow and reflection boundaries where the diffracted term predicts infinite fields. To overcome this defect, two uniform asymptotic techniques have been developed recently, namely the uniform asymptotic theory (UAT) of edge diffraction and the uniform geometrical theory of diffraction (UTD). A comparison of both these methods is given in the work of Boersma (2) where it is shown that the two different methods of asymptotically approximating integrals by pole subtraction or pole factorisation are equivalent if the complete asymptotic expansions are used. These asymptotic expressions are derived from integrals representing the exact solution. The starting point for our analysis is a particular periodic Green’s function for the solution of of the ideal wedge problem Rawlins(3) . This representation is related to the work of Sommerfeld, Carslaw, Macdonald and others at the beginning of the last century, see Rawlins(4) for a detailed history. The Sommerfeld approach was extended non-trivially to absorbing/impedance wedges by Mayuhzinets and Williams in the 1950’s, see Babich et al (5) for a comprehensive up to date history of general wedge problems solved by the so called Sommerfeld-Malyuzhinets technique. In order to derive useful physical results from these exact solutions the disposition of singularities and saddle points of integrals is crucial for getting accurate uniform results. Oberhettinger (6), (7) used asymptotic methods to derive results for diffraction by ideal