Acoustic radiation from vibrating surfaces at characteristic frequencies

Abstract

The application of boundary integral methods to the problem of acoustic radiation suffers from nonexistence or nonuniqueness problems at the characteristic frequencies of the associated interior Dirichlet problem. To overcome this difficulty, an iterative overdetermination scheme is developed. First, the surface integral Helmholtz equation is discretized, and at noncharacteristic frequencies, the resulting system of linear algebraic equations can be solved directly, yielding the values of the acoustic pressure at the boundary nodal points. At characteristic frequencies, the system matrix becomes ill-conditioned and the direct solution is not possible. An approximate surface impedance function is then assumed to serve as a basis for the computation of approximate farfield pressures. These are then utilized to form an overdetermined system, which is solved with the least-squares procedure. This solution can be used to compute the next-order approximation, and the process is continued. The method is applied to the cylindrical transducer problem and proves to yield good results at characteristic frequencies. It is also used to solve a structural radiation problem, for which no characteristic frequencies are found. Nonexistence of such frequencies for problems of this type is conjectured, although no general proof is given.