On geometry of fronts in wave propagations

Abstract

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front. 0. Introduction. In this note, we intend to develop a preparatory theory toward asymptotic analysis around singular loci that appear in wave propagation process with initial data whose C∞ singularities are located on certain smooth algebraic variety. For this purpose, we study the geometry of the (wave) front, i.e. the singular support S.S. of the solutions to the Cauchy problem where it loses C∞ smoothness. We give a concrete expression of defining function of the wave front in terms of the initial data (Theorem 10). We impose conditions on the initial data so that they are defined by a quasihomogeneous polynomial with isolated singularities. The key trick is to understand the (wave) front as a discriminantal set for a deformation of complete intersection singularity. This point of view has been exposed in [20]. As is well known, the fundamental solutions of hyperbolic operators are expressed by means of certain kinds of Gel′fand-Leray integrals. Thus the geometry of singular support (i.e. front) of the solutions is reduced to the study of singular support of these types of integrals. Homological and cohomological approach to the analysis of sharp-diffuse type of these integrals has been initiated by [3]. V. A. Vassiliev [21] uses F. Pham’s approach [15] analyzing homology classes to describe the ramification character of certain kinds of Gel′fand-Leray integrals, consequently to clarify the sharp-diffuse type (see Section 4 for 1991 Mathematics Subject Classification: Primary 35L25, 58G17, 33C75; Secondary 32S40, 78A05, 33C20. The paper is in final form and no version of it will be published elsewhere.