# The Boundary Element Method in Acoustics: A Survey
## 6 Areas of Discussion
### The Coupling Parameter
Marburg noted that many papers had inadvertently used \(\mu=-\frac{i}{k}\) as the coupling parameter. Given that its value was not thought to be critical, it might be thought that this would work as well as \(\mu=\frac{i}{k}\). However, Marburg demonstrated that \(\frac{i}{k}\) usually returns significantly more accurate results, and much faster convergence with the iterative methods for solving the systems of equations. The parameter \(\mu=-\frac{i}{k}\) was not used out of choice in the papers reviewed therein, but rather through the confusion of the signs that underlie the basic definitions in the mathematical model. These results have been confirmed by Galkowski [116]. If \(L_{k}\) is coupled with its derivative, as it is in the Burton and Miller equation, then the following equation is obtained, for one side of the equation  
\[\left\{(L_{k}+\mu(M_{k}^{t}+\frac{1}{2}I)v)\right\}_{S}\left( \boldsymbol{p};\boldsymbol{n}_{p}\right)=\int_{S}(G_{k}+\mu\frac{\partial G_{ k}}{\partial r}\frac{\partial r}{\partial n_{p}})v(\boldsymbol{q})dS+\frac{1}{2} \mu v(\boldsymbol{p})\] \[=\int_{S}\frac{e^{ikr}}{4\pi r^{2}}\bigg{(}r+\mu ikr\frac{ \partial r}{\partial n_{p}}-\mu\frac{\partial r}{\partial n_{p}}\bigg{)}v( \boldsymbol{q})\ dS\ +\frac{1}{2}\mu v(\boldsymbol{p}).\]  
With \(\mu=\frac{i}{k}\) there is significant cancellation in \(r+\mu ikr\frac{\partial r}{\partial n_{p}}\) when \(\frac{\partial r}{\partial n_{p}}\approx 1\), possibly with a diagonalising effect on the operator, and this is also presumed for the other combinations of operators. Another approach is to consider the condition of the matrices that arise in the BEM [402; 403; 404]. Earlier, it was discussed that the \(N_{k}\) operator was a pseudo-differential operator, rather than an integral operator. As a result of this, the \(\left\|N_{SS,k}\right\|\) increases with the number of elements \(n_{s}\), whereas for the other integral operators, the norm of the matrix is independent of the number of elements. This is also supported by Equations (45) and (46); the diagonal components of the \(N_{SS,k}\) matrix are \(O\Big{(}\frac{1}{n}\Big{)}=O(n_{S})\) for two- dimensional and axisymmetric problems and \(O\Big{(}\sqrt{n_{S}}\Big{)}\) for three-dimensional problems. It follows that \(\left\|N_{SS,k}\right\|=O(n_{S})\) or \(O\Big{(}\sqrt{n_{S}}\Big{)}\). The scene is therefore set for two conflicting interests in choosing the coupling parameter \(\mu=\frac{i}{k}\) is near-optimal in conditioning the combined operator, as a result of mathematical analysis whereas \(\mu=\frac{i}{n_{S}}\) or \(\mu=\frac{i}{\sqrt{n_{S}}}\) (put simply), balancing the matrix norms, as a result of numerical analysis. Without the latter correction, the \(N_{SS,k}\) matrix will apparently provide an increasing dominant potential for (quadrature-induced) error as the number of elements increases. However, in Section 6.1, the modern form of the combined method was outlined, involving pre-conditioning and iterative methods for solving the linear systems. For example, with the preconditioner in (96), the \(N_{k}\) operator is avoided and, in such cases, the analysis of the previous two paragraphs is not applicable.