# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 4 Results and Conclusions  
The energy scattering coefficients affiliated with the main waveguide Section 1 (see Figure 2) were calculated using both techniques, i.e., PET model and hybrid method, for the following percentages of the reflected energy in the sub-waveguide 3: 0, 2.3, 4.7, 10, 20, 30, 40, 50, 60, 70, 80, 92, and 99. For each computational process within the predefined percentage of the reflected energy value in the given sub-waveguide, we followed the procedure described in Section 3. The optimal total number of wavenumbers for a given configuration in the case of the PET model was identified using the minimalization of the self-proposed error function:  
\[error_{N_{i}}=0.2\times\left(\left|\frac{k_{A\omega\omega}^{FEM}-k_{A\omega \omega}^{PET}}{k_{A\omega}^{FEM}}\right|+\left|\frac{k_{S\omega\omega}^{FEM}-k_{S \omega}^{PET}}{k_{S\omega}^{FEM}}\right|+\left|\frac{k_{A\omega\omega}^{FEM}-k_ {A\omega}^{PET}}{k_{A\omega}^{FEM}}\right|+\left|\frac{k_{S\omega\omega}^{FEM}-k _{S\omega}^{PET}}{k_{S\omega}^{FEM}}\right|\right)+0.8\times\left|\frac{100- EB_{PET}}{100}\right|+\left(0.1\frac{N_{i}}{N_{initial}}\right)^{\frac{1}{3}}, \tag{36}\]  
where \(error_{N_{i}}\) refers to the error value for \(N_{i}\) considered wavenumbers, \(k_{ref}^{FEM}/k_{tr}^{FEM}/k_{ref}^{PET}\) denote the energy reflection coefficients of reflected/transmitted modes obtained using the FEM/PET approach, \(N_{initial}\) refers to the initial total number of wavenumbers, and \(N_{i}\) denotes the total number of wavenumbers for the \(i\)-th iteration. It should be noted that the hybrid approach (denoted by the \"FEM\" superscript) provides a reference to the PET model, which is obvious from Equation (36), where the individual error contributions of the PET model energy scattering coefficients are related to their FEM model counterparts. Moreover, the error formulae also use weighted functions, which have to emphasize the  
Figure 5: Energy scattering coefficients versus multiplier of [PERSON]’s modulus \(n_{\text{E}}\) for sub-waveguide 3.  
result from the energy balance criterion. The last term in Equation (36) is used to benefit the results with a lower considered number of wavenumbers and, at the same time, with an acceptable energy balance condition. Figure 6 displays the relationship between the energy scattering coefficients of A\({}_{0}\)/S\({}_{0}\) modes affiliated to the main waveguide Section 1 (see Figure 2), which were obtained by both presented methods. It is obvious that the PET approach shows very good, if not excellent, agreement compared to its FEM counterpart, where the maximum difference between the reference values of the scattering coefficients was not greater than 4% across the entire range of the percentage of reflected energy in sub-waveguide 3. The convergence characteristic of the semi-analytical method incorporating the PET approach was primarily influenced by the utilized value of the percentage of reflected energy in sub-waveguide 3. From Figure 7, it can be seen that the higher the percentage of reflected energy in the given sub-waveguide, the larger the number of real and pure imaginary wavenumbers in the given section. This behavior has a direct effect on the modified value of [PERSON]'s modulus, which needs to be accordingly reduced by the multiplier \(\mathrm{n_{E}}\). By comparison, the optimal value of the total number of wavenumbers, which ensures the best energy balance, has only a slightly increasing trend and tends to occur between 20 and 50. Additionally, the convergence effectiveness across the entire interval of percentages of reflected energy is clearly demonstrated by the graphs presented in Figures 8-10.