# The Boundary Element Method in Acoustics: A Survey
## 6 Areas of Discussion
### The Coupling Parameter  
The direct and indirect integral formulations of the exterior problem that that were free of the characteristic frequencies or non-uniqueness, were introduced in Section 4.3.4. The equations were defined with a coupling parameter \(\mu\), which weighs the contribution of derivative equation with the original equation in the direct method, or between the double layer potential and the single layer potential equations in the indirect method. Mathematically, the coupling parameter has a non-zero imaginary part that ensures that the equations are free of characteristic wavenumbers, and, in general, \(\mu\) is presumed to be an imaginary number. In accepting that \(\mu\) is imaginary, the starting point is \(-\infty<Im(\mu)<\infty\). If \(|\mu|\) is very small or very large then one or the other of the original equations would dominate, and the issues with the dominant equation would be evident with the hybrid equation, and this narrows its range; \(0\ll|\mu|\ll\infty\). The argument quickly shifted from introducing \(\mu\) in order to avoid the potential catastrophic errors in the original equations of Section 4.3.1 to considering an optimal value. For several decades, based originally on the works of Kress [395; 396; 397; 398] and the further works of Amini [399], there has been a strong recommendation, with supporting research, that \(\mu=\frac{i}{k}\) is reasonably close to the 'optimal' choice, as least for the simple boundaries, like spheres. Obviously \(\frac{i}{k}\) is unsuitable for low wavenumbers, as the parameter would be large, violating the condition \(|\mu|\ll\infty\), and the second equation would dominate and a cap on its value is recommended, for example,  
\[\mu=\left\{ \begin{array}{ll}1&\text{if }k\leq 1\\ \frac{i}{k}&\text{if }k>1\end{array} \right..\]  
The term 'optimal' in these paragraphs, refers to the condition of the combined operators over which the equation is being solved. The stronger the condition of the operator, the less able it is to magnify the solution, or magnify its error. The original equations, on their own, have'spikes' of ill-conditioning around their respective eigenvalues. The rationale is that in combining the equations, these spikes are significantly reduced, and the condition of the combined operator steers an even keel through the frequencies. It has been shown, for example for a sphere that \(\frac{i}{k}\) provides a generally well-conditioned combined operator. A recent paper by Zheng [400] showed that each eigenvalue of the combined operator follows a loop-shaped locus as \(\mu\) varies, joining the real axis when \(|\mu|=0,\infty\), and supporting \(\mu=\frac{i}{k}\), as this reasonably approximated the point on the locus that was furthest from the real axis. Zheng terms the inclusion of the derivative equation as 'adding damping', relating the analogy with, for example, Equation (91), wherein the damping term similarly shifts the eigenvalue off the real axis; the combined integral equations are not free from irregular frequencies, rather they are simply moved from the real axis into the complex plane. There is a simple case for the parameter with an inverse proportionality with the wavenumber. Considering the three-dimensional case, the Green's function (52) is the kernel of the \(L_{k}\) operator, and its magnitude does not change with \(k\left(\left|e^{ikr}\right|=1\right)\). However its derivative,  
\[\frac{\partial G_{k}}{\partial r}=\frac{e^{ikr}}{4\pi r^{2}}(ikr-1), \tag{97}\]  
a factor in the kernel of the \(M_{k}\) and \(M_{k}^{t}\) operators, is \(O(k)\). Hence, combining the operators of \(L_{k}\) with \(M_{k}\) or \(M_{k^{\prime}}^{t}\), as in Section 4.3.4, a parameter that is inversely proportional to \(k\) equalises the contribution from the two operators. Similarly, differentiating again, the kernel of the \(N_{k}\) is \(O(k^{2})\), and the same parameter has a similar function, when \(M_{k}\) or \(M_{k}^{t}\) are combined with it. A recent paper by Marburg [401] considered principally the sign of \(\mu\).