1  Introduction

The underlying problem addressed in this paper is that of computing the acoustic resonant frequencies and mode shapes of an enclosed homogeneous isotropic fluid. Let the fluid occupy an arbitrary closed region D with boundary S, as illustrated in figure 1. In this application, and in the straightforward boundary-value problem, the finite element method is now a well established computational method [1-2]). The finite element method is outlined in this paper in order to consider its development and resulting eigenvalue problem with those of other methods. However, the primary concern in this paper is with modal analysis by methods derived from a boundary integral equation formulation of the Helmholtz (reduced wave) equation, boundary element methods.

The boundary element method has been developed over recent decades as an alternative to the finite element method and finite difference method for solving linear elliptic partial differential equations like the Helmholtz equation. Although the boundary element method is not as widely applicable as the other methods, to physical problems to which it can be applied it has received a lot of interest as it has the advantage that only the boundary requires discretization (rather than the full domain in the other methods). Early texts on the method include those of Jaswon and Symm [3] and Brebbia [4]. A more recent text that could also be a useful background reference for this paper is that of Chen and Zhou [5].
The development of boundary element methods for the solution of acoustic problems has received much attention over the last three decades or so. This is particularly true for the solution of exterior problems, since finite element methods are difficult to apply when the domain is large or infinite. For both interior and exterior problems, boundary elements have a further apparent advantage in that only the boundary requires discretization, as compared to the full domain when the finite element method is applied.
Although the advantage of the boundary element method over the finite element method for interior problems is less marked, the application of the boundary element method to interior acoustic problems has also received some attention. In Bernard et al [6], Kipp and Bernard [7] and Seybert and Cheng [8] the method is applied to the interior acoustic boundary-value problem. The development of the method for modal analysis has been more difficult since the eigenvalue problem that directly results is non-linear. However, such methods for acoustic modal analysis have been explored in [5,9-22,27,28].
The boundary element method has potential advantages over the finite element method both in ease of use and efficiency. In this paper we explore the practicalities of using methods based on the boundary element method to the interior acoustic modal analysis problem. In particular the so-called Dual Reciprocity Method [23] and the Multiple Reciprocity Method [24] are reviewed.
A new method, introduced in Kirkup and Amini [18], is outlined in section 6. The method is based on the interpolation of the matrix (that arises in the non-linear eigenvalue problem) with respect to the wavenumber. The interpolation method is equally applicable to general two-dimensional, general three-dimensional and axisymmetric three-dimensional problems. In this paper we concentrate on the solution of axisymmetric problems. A method is developed through describing the general axisymmetric surfaces as a sets of conical elements and approximating the boundary functions by a constant on each element. The method is verified through its application to the test problems of a cylinder and a sphere.
There are a number of applications for which a computational analysis of the acoustic modal properties of the enclosed fluid is useful. For example in Petyt et al [1] and Coyette anf Fyfe [16] methods are applied to the interior shape of a automobile, since the interior vehicle noise will be amplified at the acoustic resonance frequencies. In this paper, the method is applied to the internal enclosure of an axi-symmetric loudspeaker. The results from this are compared with results from the finite element method and experimental results.