## 2 Spectral Wave Equation for Uniform Cross-section Waveguides
### Harmonic Elastodynamic Response in the Frequency-wavenumber Domain  
Considering a generic cross-section waveguide, as the one represented in Figure 2, a standard finite element mesh assembling procedure on Equation (12) leads to a system of \(M\) equations:  
\[\mathbf{\bar{K}}_{3}\mathbf{\bar{U}}_{z-}-\mathbf{\bar{K}}_{2}\mathbf{\bar{U}}_{ z}-\mathbf{\bar{K}}_{1}\mathbf{\bar{U}}+\omega^{2}\mathbf{\bar{M}}\mathbf{\bar{U}}=- \mathbf{\bar{F}} \tag{14}\]  
where \(\mathbf{\bar{U}}(z)\) and \(\mathbf{\bar{F}}(z)\) are the frequency dependent column vectors of assembled nodal displacements \(\mathbf{\bar{q}}(z)\) and forces \(\mathbf{\bar{f}}(z)\), respectively. The \(z\)-dependence of Equation (14) is treated by means of the Fourier integral transform. Applying the spatial Fourier transform to Equation (14) leads to the spectral form of the governing equation as:  
\[\left[\,\xi^{2}\mathbf{\bar{K}}_{3}-i\xi\mathbf{\bar{K}}_{2}+\mathbf{\bar{K}}_ {1}-\omega^{2}\mathbf{\bar{M}}\,\right]\mathbf{\overline{U}}=\mathbf{\overline {F}} \tag{15}\]  
where:  
\[\mathbf{\overline{F}}=\int\limits_{-\infty}^{\infty}\mathbf{\bar{F}}(z)e^{-i \xi z}dz\qquad\qquad\mathbf{\overline{U}}=\int\limits_{-\infty}^{\infty} \mathbf{\bar{U}}(z)e^{-i\xi z}dz \tag{16}\]  
are the Fourier transform of \(\mathbf{\bar{F}}(z)\) and \(\mathbf{\bar{U}}(z)\), respectively, and where \(\xi\) is their spatial frequency (wavenumber) along the \(z\)-direction. The matrices \(\mathbf{\bar{K}}_{1}\) and \(\mathbf{\bar{K}}_{3}\) are complex symmetric matrices, while the matrix \(\mathbf{\bar{K}}_{2}\) is a complex skew symmetric matrix. The matrix \(\mathbf{M}\) is real and symmetric. In line with other works [4,10], an \(M\)\(\times\)\(M\) transformation diagonal matrix \(\mathbf{T}\) is introduced to eliminate the imaginary unit in Equation (15). The elements of \(\mathbf{T}\) corresponding to the \(u\) and \(v\) displacement components are equal to 1, while those corresponding to \(w\) are equal to the imaginary unit. The matrices in Equation (15) are pre-multiplied by \(\mathbf{T}\) and post-multiplied by \(\mathbf{T}\). This algebra manipulation does not alter the symmetric matrices and leads to a complex symmetric matrix \(\mathbf{\bar{K}}_{2}=\mathbf{T}\left(-i\mathbf{\bar{K}}_{2}\right)\mathbf{T}\). The final form of the discrete wave equation  
Figure 2: Schematic representation of the plane CST cross-section mesh. _Algorithms_**2009**, \(2\)  
\[\left[\xi^{2}\vec{\mathbf{K}}_{3}+\xi\vec{\mathbf{K}}_{2}+\vec{\mathbf{K}}_{1}- \omega^{2}\vec{\mathbf{M}}\right]_{M}\vec{\mathbf{U}}=\vec{\mathbf{F}} \tag{17}\]  
In order to calculate the frequency response for a given harmonic force, first the homogeneous part of Equation (17), which has the form of a two-parameter \(\left(\xi-\omega\right)\) eigensystem, must be solved. Considering the wave frequency \(\omega\) a real positive quantity, Equation (17) can be solved by assuming \(\xi\) as the eigenvalue parameter transforming the \(M\)-dimensional second order eigenproblem in the axial wavenumber into a \(2M\) first order form as:  
\[\left[\mathbf{A}-\xi\mathbf{B}\right]_{2M}\vec{\mathbf{V}}=\vec{\mathbf{P}} \tag{18}\]  
where:  
\[\mathbf{A}= \begin{bmatrix}0&\vec{\mathbf{K}}_{1}-\omega^{2}\mathbf{M}\\ \vec{\mathbf{K}}_{1}-\omega^{2}\mathbf{M}&\vec{\mathbf{K}}_{2}\end{bmatrix} _{2M }\mathbf{B}= \begin{bmatrix}\vec{\mathbf{K}}_{1}-\omega^{2}\mathbf{M}&0\\ 0&-\vec{\mathbf{K}}_{3}\end{bmatrix} _{2M}\vec{\mathbf{V}}= \begin{bmatrix} \vec{\mathbf{U}}\\ \vec{\mathbf{e}}\mathbf{U}\end{bmatrix} _{2M}\mathbf{P}= \begin{bmatrix}0\\ \vec{\mathbf{F}}\end{bmatrix} _{2M} \tag{19}\]  
Non trivial solutions of the homogeneous part of Equation (18) for a given input frequency \(\omega\), leads to \(m=1,2,3,...,2M\) complex wavenumbers \(\xi^{m}=\xi^{m}_{\text{Re}}+i\xi^{m}_{\text{Im}}\) and the corresponding right \(\vec{\mathbf{V}}^{=}_{R}\) and left \(\vec{\mathbf{V}}^{=}_{L}\) eigenvectors. Solving the problem at different frequencies leads the waveguide spectrum \(\xi-\omega\) to be represented.