## 2 Spectral Wave Equation for Uniform Cross-section Waveguides
### CST Damped Finite Element for Waveguide Problems  
The starting point of the SFE formulation consists in building a spectrally formulated finite element to mesh the waveguide cross-section. Considering a constant strain triangular (CST) element, with 3 dofs per node associated to the displacements \(u\), \(v\) and \(w\), the displacement vector \(\mathbf{u}\)  at a point within the generic _e-th_ element can be approximated as:  
\[\mathbf{u}_{h}(\mathbf{x},t)=\mathbf{N}(x,y)\mathbf{q}(z,t) \tag{8}\]where \(\mathbf{N}(x,y)\) is a 3\(\times\)9 matrix with the element shape functions and \(\mathbf{q}(z,t)\) is a 9\(\times\)1 column vector containing the _e-th_ element nodal displacements. The approximate strain vector can be expressed as:  
\[\mathbf{\varepsilon}_{h}(\mathbf{x},t)=\mathbf{B}_{xy}\mathbf{q}+\mathbf{B}_{z} \mathbf{q}_{z} \tag{9}\]  
where \(\mathbf{B}_{xy}=\mathbf{L}_{x}\mathbf{N}_{,x}+\mathbf{L}_{y}\mathbf{N}_{,y}\) and \(\mathbf{B}_{z}=\mathbf{L}_{z}\mathbf{N}\) are 6\(\times\)9 matrices in which \(\mathbf{N}_{xx}\) and \(\mathbf{N}_{,y}\) are the differentiations of the shape functions matrix with respect to the \(x\) and \(y\), respectively, and \(\mathbf{q}_{z}\) is the derivative of the nodal displacements with respect to \(z\). The discrete weak form of the governing equation can be obtained by applying the [PERSON]'s principle. If \(n_{el}\) is the total number of finite elements used to mesh the cross-section, the Lagrangian of the waveguide reads:  
\[L=\frac{1}{2}\bigcup_{e=1}^{n_{el}}\bigcup_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\left[\rho^{\epsilon}\dot{\mathbf{u}}_{k}^{\mathrm{T}}\dot{\mathbf{u} }_{k}-\mathbf{\varepsilon}_{h}^{\mathrm{T}}\tilde{\mathbf{C}}^{\epsilon} \mathbf{\varepsilon}_{k}+\mathbf{u}_{k}^{\mathrm{T}}\mathbf{t}^{\epsilon} \right]dxdydz \tag{10}\]  
where \(\Omega_{e}\) is the _e-th_ element surface domain and the overdot means derivative with respect to time. After the substitution of eqs. (8)-(9) into Equation (10), some algebra leads to:  
\[\bigcup_{e=1}^{n_{el}}\left[\mathbf{k}_{3}\mathbf{q}_{,z}-\mathbf{k}_{,2} \mathbf{q}_{,z}-\mathbf{k}_{,4}\mathbf{q}-\mathbf{m}\dot{\mathbf{q}}+\mathbf{ f}\right]=0 \tag{11}\]  
where:  
\[\mathbf{k}_{3}=\int_{\Omega_{e}}\left(\mathbf{B}_{z}^{\mathrm{T}}\tilde{ \mathbf{C}}^{\epsilon}\mathbf{B}_{z}\right)dxdy\]  
\[\mathbf{k}_{2}=\int_{\Omega_{e}}\left(\mathbf{B}_{xy}^{\mathrm{T}}\tilde{ \mathbf{C}}^{\epsilon}\mathbf{B}_{z}-\mathbf{B}_{z}^{\mathrm{T}}\tilde{ \mathbf{C}}^{\epsilon}\mathbf{B}_{xy}\right)dxdy\]  
\[\mathbf{k}_{1}=\int_{\Omega_{e}}\left(\mathbf{B}_{xy}^{\mathrm{T}}\tilde{ \mathbf{C}}^{\epsilon}\mathbf{B}_{xy}\right)dxdy\]  
\[\mathbf{m}=\int_{\Omega_{e}}\left(\rho^{\epsilon}\mathbf{N}^{\mathrm{T}} \mathbf{N}\right)dxdy\;,\qquad\qquad\mathbf{f}=\int_{\Omega_{e}}\left( \mathbf{N}^{\mathrm{T}}\mathbf{t}^{\epsilon}\right)dxdy\]  
In the above equations \(\rho^{\epsilon}\) is the element mass density, \(\tilde{\mathbf{C}}^{\epsilon}=\tilde{\mathbf{C}}^{\epsilon}(t)\) is the 6\(\times\)6 viscoelastic operator and \(\mathbf{t}^{\epsilon}\) is traction vector for the generic _e-th_ element. Those quantities are considered to be constant over \(\Omega_{e}\).