Short-wave diffraction on bodies with an arbitrary smooth surface

Abstract

The classical problem of the scattering of a high-frequency acoustic wave emitted by a point source is analyzed. The scattering occurs on an arbitrary smooth surface S of an obstacle. Below, we consider the time dependence of pressure to be monochromatic, i.e., and the boundary S of the obstacle to be acoustically solid: In the case of single reflection, the solution to this problem in the two-dimensional case was obtained by various asymptotic methods in [1‐3]. In [3], explicit asymptotic formulas were derived in the two-dimensional case for pressure in a reflected wave undergoing an arbitrary number of secondary reflections from the curvilinear boundary. In the three-dimensional case [4], the short-wave approximation was developed to determine pressure in the single-reflection case. In the present paper, we develop a method of investigating short-wave diffraction on obstacles with a complicated shape, which have an arbitrary smooth surface. This method is based on the estimate of Kirchhoff diffraction integrals, which uses the approach of many-dimensional stationary phase. The developed method makes it possible to determine for the first time and in closed form the amplitude of a multiply reflected high-frequency acoustic wave.