Chapter 5
The Exterior Helmholtz Problem

This chapter addresses the problem of computing the Helmholtz field, exterior to a closed surface or surfaces and subject to a particular boundary condition and to a specified incident Helmholtz field. Such problems include the determination of the sound pressure field that is produced by a radiating surface or the perturbation or scattering of an Helmholtz field that results from an obstruction. Let the domain of the Helmholtz field be the region E exterior to the closed boundary S, as illustrated in Figure 5.1.

Fig 5.1. The domain of the exterior Helmholtz problem

The problem is equivalent to the solution of the Helmholtz equation

2 f(p) + k2 f(p) = 0    (p E)
in the domain E. The exterior problem also requires a condition that the scattered and radiated waves are outgoing, this is known as the Sommerfeld radiation condition, stated earlier in Section 1.3. The boundary condition is assumed to take a general form
a(p) f(p) + b(p) v(p) = f(p)    (p S)
(5.1)
where a, b and f are complex-valued functions defined on the boundary.

Although the exterior Helmholtz equation can be solved by the finite element or finite difference methods ( see, for example, Harari and Hughes [30]), such methods are clearly awkward to apply and probably inefficient since the domain is infinite. The boundary element method has an important strategic advantage over the alternative methods in that it requires discretisation of the boundary only. The solution at points in the domain can then be obtained by a straightforward integration of the boundary functions.

The application of the BEM to Helmholtz radiation and scattering problems has been investigated by researchers over the past three decades or so. Early contributors (see references [18], [21]) applied what may be called elementary methods; methods derived in the standard way from the integral equations arising from Green's second theorem (generally known an the Helmholtz formula of the Helmholtz-Kirchoff equation) or a single- or double-layer representation. However, the resulting BEMs, the elementary methods, were subsequently found to give unreliable results for all but a relatively low range of wavenumber. They are generally reliable for wavenumbers such that kD < 4.0, where D represents the diameter of the body or the maximum distance between any two points on the boundary - much too restrictive a condition for most applications.

There has been a large number of contributors to the research and development of boundary element methods for the solution of the exterior Helmholtz problem. Alternative integral formulations of the exterior Helmholtz equation were introduced in references [12], [57], [66], [54] and [15] and these will be termed the improved formulations. A further non-standard BEM , based on the original integral equation formulations but being demonstrably more reliable, was introduced by Schenck [75] and this has since become very popular since it turns out to be rather easier to implement than the methods based on the alternative formulations. Early reviews of these and other methods are given in Burton [16], [17] and Kleinmann and Roach [51]. The monograph by Amini et al [3] contains a more recent review.

The main difficulty with the improved formulations is that they generally involve the Nk integral operator. Although methods can be developed for its discretisation, they are notoriously difficult to devise and program. Nevertheless, as far as this work is concerned, the discretisation of all the operators has been programmed for the simple elements covered in Chapter 2. Hence the presence of the Nk operator is no barrier to the implementation of these methods for the purposes of this work. In this Chapter the most important integral equation formulations and methods are considered. The subroutines HEBEM2, HEBEM3 and HEBEMA [33] for solving the two-, three- and axisymmetric three-dimensional problems are introduced.