Numerical implementation of the lumped parameter model for the acoustic power output of a vibrating structure

In a previous paper, a lumped parameter model for the acoustic radiation from a vibrating structure was defined by dividing the surface of the structure into elements, expanding the acoustic field from each of the elements in a multipole expansion, and truncating all but the lowest-order terms in the expansion. Here, the lumped parameter model is implemented numerically by requiring the boundary condition for the normal surface velocity to be satisfied in a lumped parameter sense. This alleviates the difficulties typically encountered in enforcing the boundary condition, leading to a relatively simple numerical solution with well-defined convergence properties. The basis functions for the numerical analysis are taken as the acoustic fields of discrete simple, dipole, and tripole sources located at the geometrical centers of the surface elements. The different source types are used to represent the radiation from different kinds of surface elements: simple sources for elements in the plane of an infinite baffle, dipole sources for very thin structures which deform only in bending, and tripole sources for elements associated with parts of a structure enclosing a finite volume. The convergence of the numerical solution for the power output as a function of both frequency and element size is demonstrated through several example problems.