Oscillatory integral theory for uniform representation of wave functions


  • J. M. Arnold


This paper describes and evaluates a method of constructing asymptotic representations of wave functions, using only local geometrical properties of ray trajectories, which are uniform on and through caustics, shadow boundaries, or any other singularities which may appear in the conventional ray optical description of a wave field. This method is similar in philosophy to the geometrical methods of Maslov and Duistermäat, but can be carried out by means of elementary geometry on local ray trajectories, without recourse to the complicated topological structures employed by mathematicians. Thus it is possible to consider realistic applications of uniform methods to quite complicated problems, unobscured by the difficult language of differential topology favored by mathematicians as a framework for uniform geometrical asymptotics. The method is based on the fundamental concept of an angular spectrum of reference waves which may, but need not, be plane waves. The reference wave spectrum is used only as an intermediate device in the construction of the uniform representation and does not feature in the final form of the wave function. An application to cusped caustics in inhomogeneous media is considered.