## 2 Mathematical Model
### Simple and Double Layer Potentials  
The particular solutions of the Laplace and Helmholtz equations in the domain are the potentials of a simple and double layer on the boundary of the domain. The simple layer potential for the Helmholtz equation placed on the surface \(\Sigma\) is the function  
\[W_{\lambda}[\Sigma,h](x)=\int_{\Sigma}h(y)F_{\lambda}(x-y)d\sigma_{y},\quad F (x-y)=\frac{\exp\{-\lambda|x-y|\}}{4\pi|x-y|}, \tag{8}\]  
with \(x\in\mathbb{R}^{3}\setminus\Sigma\), and \(h\) is the density of a simple layer potential, which is a function given on the surface \(\Sigma\).  
If \(h\in C(\Sigma)\), then the simple layer potential \(W[\Sigma,h](x)\) is defined by Formula (8) even for \(x\in\Sigma\) (see [25]).  
\[W_{\lambda}[\Sigma,h]^{\pm}(x)=W_{\lambda}[\Sigma,h](x),\quad x\in\Sigma. \tag{9}\]  
Moreover, under certain smoothness conditions of the density \(h(x)\), the following formula is valid for the boundary values of the gradient of the simple layer potential [25]:  
\[\
abla W_{\lambda}[\Sigma,h]^{\pm}(x)=\
abla W_{\lambda}[\Sigma,h](x)\mp\frac {1}{2}h(x)\mathbf{n}(x)\qquad x\in\Sigma, \tag{10}\]  
where \(\
abla W_{\lambda}[\Sigma,h](x)\) is the so-called direct value of the gradient of a simple layer potential, determined by the formula  
\[\
abla W_{\lambda}[\Sigma,h](x)=\int_{\Sigma}h(y)\
abla_{x}F_{\lambda}(x-y)d \sigma_{y},\ \ x\in\Sigma,\]  
where the integral is considered in the sense of a principal value.  
The double layer potential for the Helmholtz equation \(U_{\lambda}[\Sigma,g]\) with density \(g\) set on the surface \(\Sigma\) is the following integral operator:  
\[U_{\lambda}[\Sigma,g](x)=\int_{\Sigma}g(y)\frac{\partial F_{\lambda}(x-y)}{ \partial n_{y}}d\sigma_{y},\qquad x\in\mathbb{R}^{3}\setminus\Sigma. \tag{11}\]  
The double layer potential with density \(g\in C(\Sigma)\), defined by Expression (11) at the points of the space \(\mathbb{R}^{3}\setminus\Sigma\), is a harmonic function on this set. The double layer potential can be extended on the surface \(\Sigma\) from the external and internal domains \(\Omega_{1}\) and \(\Omega_{2}\) (see [25]). The boundary values of the function \(U_{\lambda}[\Sigma,g]\) on the surface \(\Sigma\) are given by the expression:  
\[U_{\lambda}[\Sigma,g](x)^{\pm}=U_{\lambda}[\Sigma,g](x)\pm\frac{g(x)}{2} \qquad x\in\Sigma, \tag{12}\]with \(U_{\lambda}[\Sigma,g](x)\) being the direct value of the double layer potential obtained directly from Expression (11) at point \(x\in\Sigma\).  
If the double layer potential density \(g(x)\) satisfies a certain smoothness condition, then the vector field \(\
abla U_{\lambda}[\Sigma,h]\) can be continued onto the surface \(\Sigma\) from each of the domains \(\Omega_{1}\) and \(\Omega_{2}\). For the boundary values of the gradient of the function \(U_{\lambda}[\Sigma,h]\), the formula is valid [25]:  
\[\
abla U_{\lambda}[\Sigma,g]^{\pm}(x)=\
abla U_{\lambda}[\Sigma,g](x)\pm\frac{ 1}{2}\text{Grad}\,g(x),\quad x\in\Sigma. \tag{13}\]  
The direct value of the double layer potential's gradient at the boundary of a closed surface is the following expression:  
\[\
abla U_{\lambda}[\Sigma,g](x)=\int_{\Sigma}[\text{Grad}\,g(y)\times n(y)] \times\
abla_{x}F(x-y)d\sigma_{y}. \tag{14}\]  
The simple or double layer potential for the Laplace equation is a particular case of the Helmholtz potential with \(\lambda=0\).