A numerical solution for the general radiation problem based on the combined methods of superposition and singular-value decomposition

Abstract

In the method of wave superposition, the acoustic field, due to a complex radiator, is expressed in terms of a Fredholm integral equation of the first kind called the ‘‘superposition integral equation.’’ In general, Fredholm integral equations of the first kind are ill-posed and therefore difficult to solve numerically. In this paper, it will be shown that a simple collocation procedure, when combined with the singular-value decomposition, can yield accurate results for the numerical solution of the superposition integral. As an example of the application of the method, the acoustic radiation from a circular cylinder will be analyzed using this numerical procedure and compared to the exact solution. It is shown that, for this problem, the accuracy of the numerical solution can be judged by evaluating how well the superposition solution approximates the specified boundary condition on the surface of the radiator. An example is also given of a problem which has no exact solution. In this situation, it is suggested, without proof, that the accuracy of the numerical solution can be judged in a similar manner by evaluating the error in the superposition solution’s satisfaction of the boundary condition.