Fundamental Results Concerning Integral Representations in Acoustic Radiation

Attention has of late been drawn to the use of integral equations for obtaining numerical solutions to problems of steady-state harmonic acoustic radiation. Essentially, three distinct methods have been reported: the Simple Source Method, the Surface Helmholtz Integral Equation, and the Interior Helmholtz Integral Relation. The present paper contains two major theoretical results. Firstly, nonexistence failure at certain frequencies of the simple-source method is clearly indicated. Secondly, it is shown that the surface Helmholtz integral equation fails by nonuniqueness at the same frequencies. In both cases, failure occurs at each of the discrete Dirichlet eigenfrequencies for the region enclosed by the vibrating surface. Physical interpretations and examples of the theoretical results are included.