## 3 Application to an Octagonal Viscoelastic Bar
### Waveguide Response
#### Forcing Function F\({}_{01}\)  
By using a sampling frequency _sf_ = 500 kHz and a total duration \(T\) = 2.5 ms, the calculated frequency response function (FRF) at distances source-receiver (\(z_{\text{R}}\)-\(z_{\text{S}}\)) equal to 1, 3 and 5 meters for the F\({}_{{}_{01}}\) force distribution, are represented in Figure 9(a) in terms of \(w\) displacement component of node 9. Damped FRFs (continuous lines) are represented along with the undamped FRFs (dotted lines) that have been calculated by considering a pure elastic steel for the octagonal bar assuming zero longitudinal and shear bulk wave attenuations, \(\kappa_{\text{L}}\) = 0.0 Np/\(\lambda\) and \(\kappa_{\text{T}}\) = 0.0 Np/\(\lambda\), respectively.  
A sampling frequency of 500 kHz, much higher than twice the maximum excited frequency in the system, was assumed in order to calculate an over sampled frequency response function determining a smoother time-transient representation after the inverse time Fourier transform is performed.  
Figure 9: Response for the \(z\)-dof of node 9 at three different distances source-receiver \(d\) = z\({}_{\text{R}}\)-z\({}_{\text{S}}\) for the damped waveguide (continuous line) and for the undamped waveguide (dotted line). (a) absolute value of the scaled frequency response function, (b) time-transient response.  
The frequency responses represented in Figure 9(a) have been scaled by the time spectrum of the applied force and their absolute value have been represented in a semi-logarithmic scale up to the Nyquist frequency (250 kHz).  
It can be seen that the frequency response has a smooth behaviour up to the point (at around 200 kHz) where the excited high order longitudinal modes, L(0,2) and L(0,3), have non zero \(w\) displacement at the central node. It must be remembered that a direct comparison of the modes behaviour in Figure 5 and the FRFs calculated in this section cannot be made since the responses have been calculated by using a 32 CST element mesh while the dispersion curves in Figure 5 were obtained for the mesh with 512 elements.  
Using a coarser mesh, in fact, simulate a stiffer waveguide for which the modes have higher frequency. In other words the curves in Figure 5 slightly shifts towards the higher frequencies when the 32 CST mesh is used. The smooth behaviour of the FRFs in Figure 9(a) was expected since the only excited L(0,1) mode exists from zero frequency and do not present cut-off. It is also evident how the response, presenting decreasing energy for increased distances, is affected by the considered material damping. Even if not shown here, the frequency response was also checked for the \(u\) and \(v\) displacements at node 9, resulting in magnitudes around 10 orders smaller of the \(w\) displacement, i.e. negligible. Additional numerical tests were carried out at the outer node 13, for which the response in the \(x\) and \(z\) directions was non null while in the \(y\)-direction was null, as expected for the symmetric conditions (load and waveguide geometry) of the problem.  
The frequency response functions of Figure 9(a) are transformed into the time-transient response by means of Equation (30) and represented in Figure 9(b). As it can be seen only the L(0,1) mode is present as expected. This mode propagates with very low dispersion retaining the pulse shape over long distances. The effect of material damping is well captured by the formulation as it can be easily seen by comparing the amplitudes of the damped L(0,1) pulse for increasing travelled distances in respect to the response calculated for the undamped case.  
In addition, the formulation captures properly the mode speed that is consistent with the value of the energy speed dispersion curve at 30 kHz represented in Figure 5(a).