## Appendix A Frequency-Dependent Constitutive Equation for the Dissipative Structure
### Construction of the Acoustic Impedance Boundary Operator for All Real Values of the Frequency  
This construction is based on the use of two boundary integral equations on \(\Gamma_{E}\). The first one is based on the use of a single- and double-layer potentials on \(\Gamma_{E}\). The second is obtained by the taking the normal derivative on \(\Gamma_{E}\) of the first one. We then obtained the following system relating \(\psi^{\mathrm{sol}}_{\Gamma_{E}}\) to \(v\), which then allows \(\mathbf{B}_{\Gamma_{E}}(\omega/c_{\mathrm{E}})\) to be defined using Equation (104),  
\[\begin{bmatrix}0\\ \mathbf{I}\,\psi^{\mathrm{sol}}_{\Gamma_{E}}\end{bmatrix}=\begin{bmatrix}- \mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})&\frac{1}{2}\,{}^{t}\mathbf{I} -{}^{t}\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E}})\\ \frac{1}{2}\,\mathbf{I}-\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E}})& \mathbf{S}_{\mathrm{S}}(\omega/c_{\mathrm{E}})\end{bmatrix}\begin{bmatrix} \psi_{\Gamma_{E}}\\ \
u\end{bmatrix}\,. \tag{106}\]  
The linear boundary integral operators \(\mathbf{S}_{\mathrm{S}}(\omega/c_{\mathrm{E}})\), \(\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E}})\), and \(\mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})\) are defined by  
\[\begin{split}<&\mathbf{S}_{\mathrm{S}}(\omega/c_{\mathrm{E}}) \,v\,,\delta v\,>\,=\,\int_{\Gamma_{E  
\[G(\mathbf{x}-\mathbf{y})=g(\|\mathbf{x}-\mathbf{y}\|)=-(4\pi)^{-1}\,e^{-ik\,r}/r\,, \tag{104}\]  
in which \(r=\|\mathbf{x}-\mathbf{y}\|\). In Equations (103)-(104), the brackets correspond to bilinear forms which allow the operators to be defined, and the functions \(\delta v\) and \(\delta\psi_{\Gamma_{E}}\) are associated with functions \(v\) and \(\psi_{\Gamma_{E}}\). Considering Equation (102), let \(\mathbf{H}(\omega/c_{\mathrm{E}})\) be the operator defined by  
\[\mathbf{H}(\omega/c_{\mathrm{E}})=\begin{bmatrix}-\mathbf{S}_{\mathrm{T}}( \omega/c_{\mathrm{E}})&\frac{1}{2}\,^{t}\mathbf{I}-{}^{t}\mathbf{S}_{\mathrm{ D}}(\omega/c_{\mathrm{E}})\\ \frac{1}{2}\,\mathbf{I}-\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E}})& \mathbf{S}_{\mathrm{S}}(\omega/c_{\mathrm{E}})\end{bmatrix}. \tag{105}\]  
Operator \(\mathbf{H}(\omega/c_{\mathrm{E}})\) has the symmetric property, \({}^{t}\mathbf{H}(\omega/c_{\mathrm{E}})=\mathbf{H}(\omega/c_{\mathrm{E}})\). In Equation (102), the first equation can be rewritten as \(\mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})\,\psi_{\Gamma_{E}}=\left( \frac{1}{2}\,^{t}\mathbf{I}-{}^{t}\mathbf{S}_{\mathrm{D}}(\omega/c_{\mathrm{E }})\right)v\). This classical boundary equation that allows the velocity potential to be calculated for a given normal velocity has a unique solution for all real \(\omega\), which does not belong to the set of frequencies for which \(\mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})\) has a null space that is not reduced to \(\{0\}\). This set of frequencies is called the set of spurious frequencies. Consequently, for a frequency that is a spurious frequency, \(\psi_{\Gamma_{E}}\) is the sum of solution \(\psi_{\Gamma_{E}}^{\mathrm{sol}}\) with an arbitrary element of the null space of \(\mathbf{S}_{\mathrm{T}}(\omega/c_{\mathrm{E}})\). The method then consists of using the second equation that is written as \(\psi_{\Gamma_{E}}^{\mathrm{sol}}=\left(\frac{1}{2}\,\mathbf{I}-\mathbf{S}_{ \mathrm{D}}(\omega/c_{\mathrm{E}})\right)\psi_{\Gamma_{E}}+\mathbf{S}_{ \mathrm{S}}(\omega/c_{\mathrm{E}})\,v\), and which yields solution \(\psi_{\Gamma_{E}}^{\mathrm{sol}}\) for all real \(\omega\). Concerning the practical construction of \(\psi_{\Gamma_{E}}^{\mathrm{sol}}\) for all real values of \(\omega\), using Equation (102), a particular elimination procedure is described in Section B.6.