## Appendix A Frequency-Dependent Constitutive Equation for the Dissipative Structure
### Symmetric Boundary Element Method without Spurious Frequencies  
The finite element method is used for discretizing the boundary integral operators \(\mathbf{S}_{\mathrm{S}}(\omega/c_{\mathrm{E}})\), \(\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E}})\), and \(\mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})\) (called a boundary element method). Let us consider a finite element mesh of boundary \(\Gamma_{E}\). Let \(\mathbf{V}=(V_{1},\ldots,V_{n_{E}})\) and \(\mathbf{\Psi}_{\Gamma_{E}}=(\Psi_{\Gamma_{E}},\ldots,\Psi_{\Gamma_{E},n_{E}})\) be the complex vectors of the degrees-of-freedom consisting of the values of \(v\) and \(\psi_{\Gamma_{E}}\) at the nodes of the mesh. Let \([S_{\mathrm{S}}(\omega/c_{\mathrm{E}})]\), \([S_{\mathrm{D}}(\omega/c_{\mathrm{E}})]\), and \([S_{\mathrm{T}}(\omega/c_{\mathrm{E}})]\) be the full complex matrices corresponding to the discretization of the operators defined in Equations (A34)-(A36). The complex matrices \([S_{\mathrm{S}}(\omega/c_{\mathrm{E}})]\) and \([S_{\mathrm{T}}(\omega/c_{\mathrm{E}})]\) are symmetric. The finite element discretization of Equation (A33) yields  
\[ \begin{bmatrix}\mathbf{0}\\ \left[\,E\,\right]\mathbf{\Psi}_{\Gamma_{E}}^{\mathrm{sol}}\end{bmatrix} =[H( \omega/c_{\mathrm{E}})]\, \begin{bmatrix}\mathbf{\Psi}_{\Gamma_{E}}\\ \mathbf{V}\end{bmatrix} ,\] (A45)  
in which the symmetric complex matrix \([H(\omega/c_{\mathrm{E}})]\) is written as  
\[ \begin{bmatrix}-[S_{\mathrm{T}}(\omega/c_{\mathrm{E}})]&\frac{1}{2}\left[\,E \,\right]^{T}-[S_{\mathrm{D}}(\omega/c_{\mathrm{E}})]^{T}\\ \frac{1}{2}\left[\,E\,\right]-[S_{\mathrm{D}}(\omega/c_{\mathrm{E}})]&[S_{ \mathrm{S}}(\omega/c_{\mathrm{E}})]\end{bmatrix} .\] (A46)  
In Equation (A45), \(\mathbf{\Psi}_{\Gamma_{E}}^{\mathrm{sol}}\) is the complex vector of the nodal unknowns corresponding to the finite element discretization of \(\psi_{\Gamma_{E}}^{\mathrm{sol}}\). The matrix \([\,E\,]\) is the non-diagonal \((n_{E}\times n_{E})\) real matrix corresponding to the discretization of identity operator \(\mathbf{I}\). The elimination of \(\mathbf{\Psi}_{\Gamma_{E}}\) in Equation (A45) yields a linear equation between \([\,E\,]\,\mathbf{\Psi}_{\Gamma_{E}}^{\mathrm{sol}}\) and \(\mathbf{V}\), which defines the symmetric \((n_{E}\times n_{E})\) complex matrix \([B_{\Gamma_{E}}(\omega/c_{\mathrm{E}})]\), which corresponds to the finite element discretization of boundary integral operator \(\mathbf{B}_{\Gamma_{E}}(\omega/c_{\mathrm{E}})\),  
\[[\,E\,]\,\mathbf{\Psi}_{\Gamma_{E}}^{\mathrm{sol}}=\left[B_{\Gamma_{E}}( \omega/c_{\mathrm{E}})\right]\mathbf{V}\,.\] (A47)  
Vector \(\mathbf{\Psi}_{\Gamma_{E}}\) is eliminated using a Gauss elimination with a partial row pivoting algorithm [138]. If \(\omega\) does not belong to the set of the spurious frequencies, then \([S_{\mathrm{T}}(\omega/c_{\mathrm{E}})]\) is invertible and the elimination in Equation (A45) is performed up to row number \(n_{E}\). If \(\omega\) coincides with a spurious frequency \(\omega_{\mathrm{a}}\) (i.e., \(\omega=\omega_{\mathrm{a}}\)), then \([S_{\mathrm{T}}(\omega_{\mathrm{a}}/c_{\mathrm{E}})]\) is not invertible, and its null space is a real subspace of \(\mathbb{C}^{n_{E}}\) of dimension \(n_{\mathrm{a}}<n_{E}\). In this case, the elimination in Equation (A45) is performed up to row number \(n_{E}-n_{\mathrm{a}}\). In practice, \(n_{\mathrm{a}}\) is unknown. During the Gauss elimination with a partial row pivoting algorithm, the elimination process is stopped when a \"zero\" pivot is encountered. It should be noted that when the elimination is stopped, the equations corresponding to row numbers \(n_{E}-n_{\mathrm{a}}+1,\ldots,n_{E}\) are automatically satisfied.