Helmholtz integral formulation of the sonar equations

Abstract

The sonar equations provide a convenient way to calculate the detection range of actual and proposed sonar systems. They are simple, but approximate, relationships between parameters that describe the effects of source, target, and acoustic medium independently. The purpose of this paper is to derive an exact form of the active sonar equation by using the Helmholtz integral formulation to solve the boundary-value problem with source and target both present in the medium. The resulting equation involves combinations of linear integral operators; however, it is suitable for solution by numerical techniques already developed for radiation from objects of arbitrary shape. Furthermore, it is shown that these integral operators reduce to multiplicative factors which represent general definitions of the source level, transmission loss, and target strength when the source-to-target distance is large. Thus this work establishes a basis for the sonar equations as the limiting form of an exact boundary-value problem and presents formulas for calculating the sonar parameters.