## 3 Results  
This section contains the AE signals modeling results and verification of the simulation results and experimental data. Figure 5 shows the results of the modal analysis without the influence of signal absorption. Signals and their spectrograms simulating the propagation of Lamb wave zero modes for objects with different thicknesses and waveguide lengths are represented. Figure 5a,b shows that, with an increase in the path length, the signal duration increases due to a larger difference in the arrival times of the \(A_{0}\) and \(S_{0}\) modes frequency components. It should be noted that dispersive propagation also leads to a decrease in the peak of the AE signal due to the delocalization of various components of the signal. Figure 5b,c shows that a change in thickness affects the phase-frequency structure of the signal, while dispersion curves shift with increasing thickness toward lower frequencies. Figure 6a-d shows the model signals calculated considering all the influencing factors included in the modeling algorithm. The represented signals correspond to one distance between the AE source and the sensor, but different wall thickness values. Comparing the waveforms of the \(A_{0}\) and \(S_{0}\) modes, we can conclude that the waveform of the \(S_{0}\) mode is more dependent on the waveguide thickness than the waveform of the \(A_{0}\) mode. Frequency-dependent parameters, such as the transmission coefficient and the normal component of the extrema coordinates of the \(S_{0}\) mode, have an extremum depending on the \(f\)-\(h\) value. The complex correlation of the coordinates of the extrema for dependences \(K_{50}(f)\) and \(\delta_{50}(f)\), considered together with the resonant sensor transfer function, leads to the strong influence  
Figure 5: Model signals and wavelet spectrograms of the model signals at different waveguide lengths and wall thicknesses: (**a,d**) a thickness of 9 mm and a distance of 5 m; (**b,e**) a thickness of 9 mm and a distance of 10 m; (**c,f**) a thickness of 17 mm and a distance of 10 m.  
of waveguide thickness on the \(S_{0}\)-mode waveform, which is characterized by a lower transmission coefficient and a lower dispersion for small thicknesses of about 5-9 mm, and a larger transmission coefficient and greater dispersion for wall with thicknesses greater than 15 mm. To verify the model while considering the attenuation coefficients of the Lamb waves, we compared the simulation results with the experimental results described in Section 2.5. The simulated and measured signals corresponding to the same waveguide parameters are presented in Figures 7 and 8. Figure 6: Model signals calculated for a waveguide length of 5 m and various thicknesses (the blue graph is mode \(S_{0}\), and the red graph is mode \(A_{0}\)): (**a**) \(h\) = 5 mm; (**b**) \(h\) = 9 mm; (**c**) \(h\) = 17 mm; (**d**) \(h\) = 30 mm. Figure 7: Comparison of the modelled and experimental AE signals for a waveguide length of 6 m and a thickness of 9 mm: (**a**) the simulated signal, (**b**) the wavelet transform for the simulated signal and a superimposition of group velocities, (**c**) the measured signal, and (**d**) the wavelet transform for the measured signal and the superimposition of group velocities. The signals corresponding to the waveguide length equal to 6 m and thickness equal to 9 mm are represented in Figure 7a,c, and wavelet transforms are represented in Figure 7b,d. The experimental waveguide was the technological pipeline with a diameter of 500 mm. Comparing the simulated and experimental signals, we can conclude a high degree of correspondence; the ratio between the \(A_{0}\) and \(S_{0}\) mode amplitudes is equal to 3.7 for the simulated signal and 4.0 for the signals obtained experimentally. The time intervals between the \(A_{0}\) and \(S_{0}\) peaks are 421 us and 429 us for the simulated and measured data, respectively.