## 4 Discussion  
The results obtained in this paper allow us to conclude that the modal analysis of the normal waves propagation, supplemented by an analytical calculation of Lamb wave attenuation, is an effective analytical method for calculating the propagation of AE signals along a waveguide. Using this method, signals that possess all the characteristic features of real signals in the shapes of their envelopes, their frequency spectrum and their spectrograms were obtained. To obtain a realistic form of model signals, it is necessary to consider all the components of the transmission coefficient: phase changes caused by the dispersion propagation of Lamb waves, the attenuation coefficient, and the characteristics of the AE sensor. The \(S_{0}\) wave is more susceptible to dispersive propagation in the considered frequency range from 30 to 500 kHz; as a result, the signal duration corresponding to the \(S_{0}\) mode is longer, and the peak amplitude, as a rule, is smaller than that of the \(A_{0}\) mode, even in the case of equal energy radiated in the form of these modes. The maximum mode transmission coefficient, \(A_{0}\), corresponds to the low-frequency region. The attenuation parameters of the \(A_{0}\) mode are determined mainly by the distance between the source and receiver of the AE waves and, to a small extent, depend on the thickness. The parameters of the passage of the \(S_{0}\) mode substantially depend on the thickness and, to a lesser extent, on the distance. The conditions for receiving the transverse component of the \(S_{0}\) mode oscillations are unfavorable due to the mismatch of the frequency ranges of the minimum attenuation coefficient and the maximum fraction of the transverse component of the \(S_{0}\) mode. Verification of the model was carried out by comparing the modeled signals with the signals obtained experimentally using a Hsu-Nielsen source. During this comparison, the waveform of the simulated and experimental signals, established to show a high degree of similarity between them, even though the experiment to simulate AE impulses was carried out under field conditions at industrial pipelines. The influence of factors such as the presence of welded joints, shutoff valves, as well as the presence of a light degree corrosion damage with a depth of up to 10% of the pipe wall's thickness did not significantly affect the similarity of the waveform between the real and modelled signals. The correlation coefficient, which can be interpreted as a numerical estimation of the waveform similarity, is about 0.6 due to the influence of multipath guided wave propagation, which was not taken into account due to differences between the transfer function of the reference sensor and the AE sensor that was used during the experiment. When applying the simulation results as reference signals, it should be considered that the signal from the defect does not fully correspond to the signal of the Hsu-Nielsen source. [PERSON] [4] established that the Hsu-Nielsen source corresponds to the monopole model, while stress-generated AE sources are almost universally composed of dipoles. In addition, it is necessary to consider that the simulation results presented in this paper correspond to a case when the source is located out-of-plane, while the depth of the stress-generated AE source is uncertain. [PERSON] [4] and [PERSON] [15] determined that the main difference between the in-plane and out-of-plane sources consists of the relationship between the A\({}_{0}\) and S\({}_{0}\) mode amplitudes. Since the relationships between the mode amplitudes are established in the papers [4; 15], this uncertainty could be considered due to the addition of a weighted coefficient for each mode, which is simulated separately in accordance with the scheme in Figure 3.  
Based on the result of the model verification, we can conclude that the analytically modeled signals could be used as reference signals for the AE signal filtration and as training sets for AE data processing algorithm verification.