## 2 Spectral Wave Equation for Uniform Cross-section Waveguides
### GW Dispersive Modal Properties  
From each \(\xi^{m}\), the phase velocity [m/s] and the attenuation [Np/m] of the \(m\)-_th_ guided wave can be computed, respectively, as:  
\[c^{m}_{ph}=\omega/\xi^{m}_{\text{Re}}\,,\ \ \ \ att^{m}=\xi^{m}_{\text{Im}} \tag{21}\]  
where an attenuation of \(\xi^{m}_{\text{Im}}\) Nepers per meter means that a wave of unit amplitude is reduced to an amplitude of \(e^{-\xi^{m}_{\text{Im}}}\) after traveling one meter (\(1\,\text{dB}=ln(10)\,/\,20\) Np ).  
From the half upper part of \(m\)-_th_ right wavestructure, \(\vec{\mathbf{U}}^{=}_{R}\), the complex modal spectral amplitude \(\vec{\mathbf{q}}\) for each finite element in the mesh can be extracted, and the modal element displacement \(\vec{\mathbf{u}}_{h}\), strain \(\vec{\mathbf{e}}_{h}\) and stress \(\vec{\mathbf{e}}^{=}_{h}\) amplitudes at a point reconstructed. All these mechanical variables are complex and behave harmonically in the time-wavenumber domain with frequency \(\omega\).  
The wave energy velocity [m/s] for the _m-th_ mode, \({V_{e}}^{m}\), i.e. the velocity at which a damped guided wave travels along the waveguide, can be obtained as [9]:  
\[{V_{e}}^{m}=\frac{\frac{1}{\Omega}\!\!\int\limits_{\Omega}\!\!\left(\mathbf{P}_{T }^{m}\cdot\hat{z}\right)d\Omega}{\frac{1}{\Omega}\!\!\int\limits_{\Omega}\! \left(\mathbf{S}_{T}^{m}+\mathbf{K}_{T}^{m}\right)d\Omega} \tag{22}\]  
where \(\hat{z}\) is a unit vector along the z-direction, \(\mathbf{P}_{T}^{m}\) is the power flow density vector [J/(s m\({}^{2}\))] (Poynting vector) at a point for the _m-th_ mode:  
\[\mathbf{P}_{T}^{m}=\!\!\left[P_{x}^{m}\;\;P_{y}^{m}\;\;P_{z}^{m}\right]^{\!\! \mathrm{T}}=-\frac{1}{2}\mathrm{Re}\!\left[\mathbf{s}_{h}^{m}\;\;\mathrm{ consj}\!\left(\overline{\mathbf{u}}_{h}^{m}\right)\right] \tag{23}\]  
where \(\overline{\mathbf{s}}_{h}^{m}\) is the classical 3\(\times\)3 stress tensor derived by simply reorganizing the 6\(\times\)1 stress vector \(\overline{\mathbf{\sigma}}_{h}^{m}\) in the opportune manner. In Equation (22) S\({}_{T}^{m}\) and K\({}_{T}^{m}\) are the time-averaged strain and kinetic energy densities [J/m\({}^{3}\)] (at a point) for the _m-th_ guided wave [11; 12]:  
\[\mathrm{S}_{T}^{m}=\frac{1}{2}\mathrm{Re}\!\left[\frac{1}{2}\!\!\left( \overline{\mathbf{\sigma}}_{h}^{m}\right)^{\!\!\mathrm{T}}\mathrm{consj}\!\left( \overline{\mathbf{\sigma}}_{h}^{m}\right)\right]\!,\qquad\mathrm{K}_{T}^{m}= \frac{1}{2}\!\!\left[\frac{\rho_{e}}{2}\!\!\left(\overline{\mathbf{u}}_{h}^{m} \right)^{\!\!\mathrm{T}}\mathrm{consj}\!\left(\overline{\mathbf{u}}_{h}^{m} \right)\right] \tag{24}\]  
For lossy media, the energy velocity coincides with the group velocity \(c_{g}^{m}\)[13], for which a simpler formula can be used [4]. Iterative routines are built in Matlab to formulate and solve Equation (18) and to calculate and collect the guided wave features (\(\xi\), \(c_{ph}\), \(att\), \(V_{e}\)), while varying the input frequency. The dispersive damped spectrum of the waveguide in terms of wavenumbers, phase velocity, attenuation and energy velocity can thus be represented for the frequency range of interest. The calculated eigendata at each frequency also serves to compute the damped time-transient response.