# The Boundary Element Method in Acoustics: A Survey
## 3 The Boundary Element Method and the Laplace Equation Stem
### Operator Notation and Further Integral Equations for the Laplace Problem  
In this Subsection, operator notation is introduced and further integral equations are introduced in order to illuminate some useful properties. Operator notation provides a shorthand and is an aid to communication. The Laplace integral operators are defined as follows:  
\[\left\{L\mu\right\}_{\Gamma}(\mathbf{p})=\int_{\Gamma}G(\mathbf{p},\mathbf{q})\;\mu(\mathbf{ q})dS_{q}\;, \tag{18}\]  
\[\left\{M\mu\right\}_{\Gamma}(\mathbf{p})=\int_{\Gamma}\frac{\partial G(\mathbf{p},\bm {q})}{\partial n_{q}}\;\mu(\mathbf{q})dS_{q}\;, \tag{19}\]  
\[\left\{M^{t}\mu\right\}_{\Gamma}\!\!\left(\mathbf{p};\mathbf{v}_{p}\right)=\frac{ \partial}{\partial v_{p}}\int_{\Gamma}G(\mathbf{p},\mathbf{q})\;\mu(\mathbf{q})dS_{q}\;, \tag{20}\]  
where \(\Gamma\) is a boundary (not necessarily closed), \(\mathbf{n}_{q}\) is the unique unit normal vector to \(\Gamma\) at \(\mathbf{q}\), \(v_{p}\) is a unit directional vector passing through \(\mathbf{p}\) and \(\mu(\mathbf{q})\) is a function defined for \(\mathbf{q}\in\Gamma\). With this notation, the Equations (14), which form the basis of the elementary direct method for the interior problem, can be written as  
\[\left\{M\varphi\right\}_{S}(\mathbf{p})-\left\{L\
u\right\}_{S}(\mathbf{p})=\left\{ \begin{array}{c}-\varphi(\mathbf{p})\text{ if }\mathbf{p}\in D\\ 0\text{ if }\mathbf{p}\in E\\ -\frac{1}{2}\varphi(\mathbf{p})\text{ if }\mathbf{p}\in S\end{array} \right. \tag{21}\]  
Similarly, the equations for the indirect method ((15) and (17)) in operator notation for the interior problem are as follows:  
\[\varphi(\mathbf{p})=\left\{L\sigma\right\}_{S}(\mathbf{p})\;\;\;(\mathbf{p}\in D\cup S), \tag{22}\]  
\[v(\mathbf{p})=\left\{M^{t}\sigma\right\}_{\Gamma}\!\!\left(\mathbf{p};\mathbf{n}_{p}\right) +\frac{1}{2}\sigma(\mathbf{p})\;\;\;(\mathbf{p}\in S). \tag{23}\]Further integral equations for the direct formulation can be obtained by differentiating (21) (as in the derivation of Equations (16), (17)) and for the indirect formulation by introducing a _double_ layer potential. Although they are generally unnecessary in solving Laplace problems, the equations resulting from differentiating the elementary integral equations or using double layer potentials are often used in the exterior acoustic problem, considered in the next Section. These equations are also useful in general and they will illuminate some important aspects. For example, differentiating Equation (21) with respect to the outward normal to the boundary,  
\[\frac{\partial}{\partial n_{p}}[M\varphi]_{S}(\mathbf{p})-\frac{\partial}{ \partial n_{p}}[L\
u]_{S}(\mathbf{p})=-\frac{1}{2}\frac{\partial\varphi(\mathbf{p})}{ \partial n_{p}}(\mathbf{p}\in S), \tag{24}\]  
which can be written in operator notation,  
\[\{N\varphi\}_{S}(\mathbf{p};\mathbf{n}_{p})-\{M^{t}\
u\}_{S}(\mathbf{p};\mathbf{n}_{p})=-\frac {1}{2}\
u(\mathbf{p})(\mathbf{p}\in S), \tag{25}\]  
in which a new operator, \(N\), has been introduced,  
\[\{N\mu\}_{\Gamma}(\mathbf{p};\mathbf{v}_{p})=\frac{\partial}{\partial\varphi_{p}} \int_{\Gamma}\frac{\partial G(\mathbf{p},\mathbf{q})}{\partial n_{q}}\;\mu(\mathbf{q})dS_{ q}. \tag{26}\]  
Further indirect integral equations may be obtained through presuming the field is the result of a double layer potential  
\[\varphi(\mathbf{p})=\{M\zeta\}_{S}(\mathbf{p})\;\;\;(\mathbf{p}\in D), \tag{27}\]  
\[\
u(\mathbf{p})=\{N\zeta\}_{S}(\mathbf{p};\mathbf{n}_{p})\;\;\;(\mathbf{p}\in S), \tag{28}\]  
and, taking into account the jump discontinuity,  
\[\varphi(\mathbf{p})=\{M\zeta\}_{S}(\mathbf{p})-\;\frac{1}{2}\zeta(\mathbf{p})\;\;\;(\mathbf{p }\in S). \tag{29}\]  
The integral equations for the exterior Laplace problem may be derived in the same way. In general, the equations are the same as for the interior problem, except for a change of sign. The equations that make up the direct formulation are  
\[\{M\varphi\}_{S}(\mathbf{p})-\{L\
u\}_{S}(\mathbf{p})=\left\{