## 1 Introduction  
A force with a high frequency content acting at a point on a slender solid with uniform cross section (waveguide) generates mechanical waves. These waves, generally termed as Guided Waves (GWs) due to the fact that the solid serves as a waveguide, propagate with defined wavelength, phase velocity, energy velocity, attenuation and wavestructure (wave shape over the waveguide cross-section). Due to the interaction between the wave's wavelength and the waveguide cross-section geometry, some or all of these wave features change when varying the frequency of propagation (dispersive behavior). Today, GWs are employed in several industrial applications for material characterization, impact and shock-induced wave propagation, acoustic focusing and advanced material design. Modeling these propagating waves in complex waveguides is not an easy task, especially at high frequency, where the wavelengths are small and the wavestructures have a complex shape. Among others, Spectral Finite Element (SFE) formulations are well established algorithms to simulate guided waves propagating in waveguides. The key point of SFE formulations consists in the hybrid displacement field adopted to describe the motion of a propagative guided wave. A transient wave is described, in fact, coupling a bi-dimensional finite element mesh over the waveguide cross-section and harmonic functions along the wave propagation direction. Next, a displacement-based variational scheme and the application of Fourier transform contractions leads to the governing wave equation in the frequency-wavenumber domain as a system of algebraic equations. Non trivial solutions of the homogenous part of the wave equation, easily calculated by using standard routines for eigenvalue problems, allow tracing the dispersive spectrum of the waveguide. Compared to other methods the main advantages in this calculation are: no missing roots; capability to handle waveguides of arbitrary cross-section; possibility to model high degree anisotropic waveguides with no extra effort; fast and reliable high frequency computation. A literature review on SFE formulations, acknowledged also as semi-analytical finite element (SAFE) formulations or waveguide finite element (WFE) methods, can be found in the book by [PERSON] and [PERSON] [1]. The SFE calculated waveguide spectrum, can also be exploited for the construction of the time-transient response due to the waves excited by a time-varying load. Time-transient response is essential to actuators design, quantitative non-destructive evaluation of cracks size and location as well as for structural health monitoring purposes. In brief, the time-transient response at a point can be obtained by means of the inverse Fourier transform, which turns into the time domain the waveguide frequency response calculated as a summation of the modal data contribution at that point weighted with the spectrum of the applied load. Applications of this procedure to calculate the time-transient response in undamped waveguides, such as plates [2] and pipes [3], have been proposed in the past. Lately, some work has been devoted to extend SFE procedures to damped waveguides. In particular [PERSON] _et al._, [4] generalized the SFE formulation to compute the modal spectrum in lossy waveguides by accounting for material damping, at the finite element level, by introducing complex constitutive viscoelastic relations. The scheme proposed in [4] has been next used to extract the damped dispersion curves in pipes [5] and helical waveguides [6], and subsequently exploited to extend SFE formulations to the calculation of the time-transient response in damped pipes [7] and rails [8]. In particular, in [7] where the scheme proposed in [4] was fully used, the SFE formulation is based on the analytical derivation of the complex stiffness matrices from the material acoustic properties. This also allows handling of frequency dependent linear viscoelastic rheological models, once the bulk properties are known.