# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 3 Numerical Verification of PET Model
Once the desired through-thickness variables for the incident, reflected, and scattered mode are identified, it is possible to calculate the energy scattering coefficients, which due to the power-normalization of stress/displacement components, can be expressed using complex amplitudes [16,43]:  
\[k_{R}^{n}=\frac{P_{AV~{}R}^{n}}{P_{AV~{}IN}}=\frac{\left|\
icefrac{{\alpha_{R} }}{{\alpha_{IN}}}\right|^{2}}{\left|\
icefrac{{\alpha_{I}}}{{\alpha_{IN}}} \right|^{2}}=\frac{\int_{0}^{t_{0}}\left(\widetilde{\sigma}_{x~{}R}^{n}\mu_{R }^{FEM}+\sigma_{x~{}R}^{FEM}\widetilde{u}_{R}^{n}-\widetilde{\tau}_{xx~{}R}^{ n}\
u_{R}^{FEM}-\tau_{xx~{}R}^{FEM}\widetilde{v}_{R}^{n}\right)dz\times \left\{2\int_{0}^{t_{0}}\left[\widetilde{\sigma}_{x~{}IN}\widetilde{u}_{IN}- \widetilde{\tau}_{xz~{}IN}\widetilde{v}_{IN}\right]dz\right\}^{-1}}{\int_{0}^ {t_{0}}\left(\widetilde{\sigma}_{x~{}IN}u_{IN}^{FEM}+\sigma_{x~{}IN}^{FEM} \widetilde{u}_{IN}-\widetilde{\tau}_{xx~{}IN}~{}v_{IN}^{FEM}\right)dz\times \left\{2\int_{0}^{t_{0}}\left[\widetilde{\sigma}_{x~{}IN}\widetilde{u}_{IN}- \widetilde{\tau}_{xz~{}IN}\widetilde{v}_{IN}\right]dz\right\}^{-1}}, \tag{34}\]  
\[k_{T}^{n}=\frac{P_{AV~{}T}^{n}}{P_{AV~{}IN}}=\frac{\left|\
icefrac{{\alpha_{T} }}{{\alpha_{IN}}}\right|^{2}}{\left|\
icefrac{{\alpha_{T}}}{{\alpha_{IN}}} \right|^{2}}=\frac{\int_{0}^{t_{0}}\left(\widetilde{\sigma}_{x~{}T}^{n}\mu_{T }^{FEM}+\sigma_{x~{}T}^{FEM}\widetilde{u}_{T}^{n}-\widetilde{\tau}_{xz~{}T}^ {n}\
u_{T}^{FEM}-\tau_{xz~{}T}^{FEM}\widetilde{v}_{T}^{n}\right)dz\times \left\{2\int_{0}^{t_{0}}\left[\widetilde{\sigma}_{x~{}IN}\widetilde{u}_{IN}^{FEM}- \widetilde{\tau}_{xz~{}IN}\widetilde{v}_{IN}\right]dz\right\}^{-1}}{\int_{0}^ {t_{0}}\left(\widetilde{\sigma}_{x~{}IN}u_{IN}^{FEM}+\sigma_{x~{}IN}^{FEM} \widetilde{u}_{IN}-\tau_{xz~{}IN}^{FEM}\widetilde{v}_{IN}\right)dz\times \left\{2\int_{0}^{t_{0}}\left[\widetilde{\sigma}_{x~{}IN}\widetilde{u}_{IN}- \widetilde{\tau}_{xz~{}IN}\widetilde{v}_{IN}\right]dz\right\}^{-1}}, \tag{35}\]  
where \(P_{AV~{}IN}/P_{AV~{}R}^{n}/P_{AV~{}T}^{m}\) express time-averaged acoustic power flow of incident/reflected/transmitted modes, \(k_{R}/k_{T}\) are energy scattering coefficients of reflected/transmitted modes, \(\
icefrac{{\alpha_{IN}}}{{\alpha_{R}}}/\
icefrac{{\alpha_{T}}}{{\alpha_{T}}}\) are complex amplitudes of incident/reflected/transmitted modes, the superscript \(FEM\) denotes the total stress/displacement field obtained by FEM computation, the subscript/superscript \(n\) denotes the \(n\)-th considered Lamb wave mode and the subscripts \(IN/R/T\) in the case of displacement/stress components represent affiliation to incident/reflected/transmitted modes. As stated above, the PET approach was applied to sub-waveguide 3 (interfaces 3,6--see Figure 2), where, due to the fact that the incident mode is antisymmetric (A\({}_{0}\)), the amount of transmitted/reflected energy in the concerned sub-waveguide is related to the power of the A\({}_{0}\) mode in this waveguide. The identification of the desired value of Young's modulus, which induces the given ratio between transmitted and reflected energy for A\({}_{0}\) mode in sub-waveguide 3, was performed using the hybrid approach presented above. Figure 5 displays the dependence of energy scattering coefficients for the reflected and transmitted A\({}_{0}\)/S\({}_{0}\) waves versus the multiplier of Young's modulus \(n_{E}\) for sub-waveguide 3. It should be noted that the given values were also used by the semi-analytical model incorporating the PET approach. In the  
Figure 4: Discretization of the scattering region including its close vicinity; the region where the PET approach was adopted is marked by blue color. next step, both methods, i.e., semi-analytical and hybrid approaches, were then used to evaluate the energy scattering coefficients for selected fractions of reflected or transmitted energy of A\({}_{0}\) mode in sub-waveguide 3.