# The Boundary Element Method in Acoustics: A Survey
## 3 The Boundary Element Method and the Laplace Equation Stem
### Elementary Integral Equation Formulation for the Interior Problem  
The boundary element method is not based on the direct solution of the PDE, but rather its reformulation as an integral equation. Historically, the integral equation reformulation of the PDE has followed two distinct routes, termed the direct method and the indirect method. The direct method is based on Green's second identity  
\[\int_{D}(\varphi(\mathbf{q})\
abla^{2}\psi(\mathbf{q})-\psi(\mathbf{q})\
abla^{2}\varphi( \mathbf{q}))\;dV_{q}=\int_{S}\varphi(\mathbf{q})\frac{\partial\psi(\mathbf{q})}{\partial n _{q}}-\psi(\mathbf{q})\frac{\partial\varphi(\mathbf{q})}{\partial n_{q}}\;dS_{q}, \tag{11}\]  
where \(\varphi\) and \(\psi\) are twice-differentiable scalar function in a domain \(D\) that is bounded by the closed surface \(S\). If \(\varphi\) is a solution of Laplace's equation, \(\
abla^{2}\varphi=0\), then  
\[\int_{D}\varphi(\mathbf{q})\
abla^{2}\psi(\mathbf{q})\;dV_{q}=\int_{S}\varphi(\mathbf{q} )\frac{\partial\psi(\mathbf{q})}{\partial n_{q}}-\psi(\mathbf{q})\frac{\partial \varphi(\mathbf{q})}{\partial n_{q}}\;dS_{q}. \tag{12}\]  
Green's third identity can be derived from the second identity by choosing \(\psi(\mathbf{q})=G(\mathbf{p},\mathbf{q})\), where \(G\) is a Green's function. A Green's function is a fundamental solution of the partial differential equation, in this case Laplace's equation, that is the effect or influence a unit source at the point \(\mathbf{p}\) has at the point \(\mathbf{q}\), and is defined by \(\
abla^{2}G(\mathbf{p},\mathbf{q})=-\delta(\mathbf{p}-\mathbf{q})\). For the two-dimensional Laplace equation \(G(\mathbf{p},\mathbf{q})=-\frac{1}{2\pi}\ln(r)\) and for three-dimensional problems \(G(\mathbf{p},\mathbf{q})=\frac{1}{4\pi r}\) where \(r=\left|\mathbf{p}-\mathbf{q}\right|\). The substitution of the Green's function into Equation (12) gives  
\[\int_{D}\varphi(\mathbf{q})\
abla^{2}G(\mathbf{p},\mathbf{q})\;dV_{q}=\int_{S}\varphi(\bm {q})\frac{\partial G(\mathbf{p},\mathbf{q})}{\partial n_{q}}-G(\mathbf{p},\mathbf{q})\frac{ \partial\varphi(\mathbf{q})}{\partial n_{q}}\;dS_{q}\]  
or  
\[-\int_{D}\varphi(\mathbf{q})\delta(\mathbf{p}-\mathbf{q})\;dV_{q}=\int_{S}\varphi(\mathbf{q} )\frac{\partial G(\mathbf{p},\mathbf{q})}{\partial n_{q}}-G(\mathbf{p},\mathbf{q})\frac{ \partial\varphi(\mathbf{q})}{\partial n_{q}}\;dS_{q}. \tag{13}\]  
Hence, as a result of the properties of the Dirac delta function,  
\[\int_{S}\varphi(\mathbf{q})\frac{\partial G(\mathbf{p},\mathbf{q})}{\partial n_{q}}-G(\bm {p},\mathbf{q})\frac{\partial\varphi(\mathbf{q})}{\partial n_{q}}\;dS_{q}=\left\{ \begin{array}{c}-\varphi(\mathbf{p})\text{ if }\mathbf{p}\in D\\ 0\text{ if }\mathbf{p}\in E\\ -c(\mathbf{p})\varphi(\mathbf{p})\text{ if }\mathbf{p}\in S\end{array} \right., \tag{14}\]  
where \(c(\mathbf{p})\) is the angle (in 2D, divided by \(2\pi\)) or solid angle (in 3D, divided by \(4\pi\)) subtended by the interior domain at the boundary point \(\mathbf{p}\). (Similarly, for exterior problems, \(c(\mathbf{p})\) similarly relates to the exterior angle). If the boundary is smooth at \(\mathbf{p}\) then \(c(\mathbf{p})=\frac{1}{2}\), and, for simplicity, this is the value that will be used for the remainder of this paper.