## 3 Theoretical Framework
### Validation  
The magnitude of \(|K_{33}(\Lambda,f)|\) reaches the maximal values at the sets \((\Lambda,f_{n}(\Lambda))\) specified by the roots of characteristic equation (5); theoretically it is infinite with real \(\zeta_{n}\). Therefore, the ridges in the level-line plots depicting \(|K_{33}|\) as a function of \(\Lambda\) and \(f\) visually show the SAW/PSAW dispersion curves in the plane \((\Lambda,f)\). To confirm the ability of the developed [PERSON]'s matrix based model to correctly simulate SAWs generated in the samples, we had to make sure that the ridges pass through the experimentally obtained points \((\Lambda_{j},f_{j})\) indicating the peaks of the frequency spectrum for a sample with known material parameters. First of all, such a validating comparison has been performed for a simple two-layer specimen Ni/Si. The material properties of the isotropic nickel coating were taken from the handbooks (\(C_{11}=247\,\mathrm{GPa}\), \(C_{44}=122\,\mathrm{GPa}\), and \(\rho=8900\,\mathrm{kg/m^{3}}\)) while the silicon substrate is the same in all experiments (Table 1). In this sample, the thickness of the coating (\(h_{1}=15\,\mathrm{nm}\)) was very small as compared with the range of SAW wavelengths (2-15 \(\upmu\)m) in the measurements. Therefore, its influence was insignificant and the only Rayleigh-like SAW was excited, and, as expected, the only ridge of the calculated surface \(|K_{33}(\Lambda,f)|\) passed through the experimental points (Figure 6). This has confirmed both the measurement and simulation accuracy. Figure 7 presents the results of the theory-to-experiment comparisons with a more complex three-layer specimen Ni/HSQ400/Si. The core layer is a pure HSQ polymer, still without NWs, annealed at the temperature \(t=400\)\({}^{\circ}\)C. This material is isotropic, therefore, it was not too difficult to determine its material constants manually. It required just a few turns to fit the \(|K_{33}|\) ridges (Figure 7b) to the experimental points \((\Lambda_{j},f_{j})\) (Figure 7a). The fit is achieved with the HSQ parameters \(C_{11}=11.89\) GPa, \(C_{44}=2.97\) GPa, and \(\rho=1000\) kg/m\({}^{3}\). The real roots of Equation (5) calculated with these parameters allowed tracing the dispersion curves \(f=f_{n}(\Lambda)\) (Figure 7c). Obviously, the curves go along the \(|K_{33}|\) ridges but not the whole length. For example, the two upper-right experimental points get on the ridge in Figure 7b, but in Figure 7c this area is blank. The reason is that this part of the ridge is already associated with the complex pole \(\zeta_{n}\), i.e., the corresponding SAW actually becomes PSAW (leaky wave). The transformation of a traveling surface wave into a leaky wave occurs when its phase velocity \(c_{n}\) becomes greater than the velocity \(v_{s}\) of the bulk shear waves in the bottom half-space (i.e., in the silicon substrate in the case). Or equivalently, when its slowness \(s_{n}=1/c_{n}\) is less then the substrate's \(S\)-wave slowness \(S3=1/v_{s}\). Depicting dispersion curves in the frequency-slowness plane is more convenient, since, in contrast with the phase velocity curves that can come down from infinity, the slowness magnitudes vary in a limited range specified by the frequency independent bulk-wave slownesses. In Figure 7d and similar figures below, the latter are shown by horizontal dashed lines. They are indicated by the symbols \(P1\), \(P2\), \(P3\), and \(S1\), \(S2\), \(S3\), which relate to longitudinal and transversal (\(P\) and \(S\)) bulk waves propagating in the \(x\)-direction in each of the sublayers numbered from top to bottom. One can see that those two points, getting in the blank area in Figure 7c, lie below the \(S3\) line in Figure 7d. Figure 6: Validating example for the test sample Ni/Si. Calculated within the model developed dispersion curve of the only Rayleigh-like SAW indicated by the ridge in the \(|K_{33}|\) level-line image (yellow) passes through the TGM measured points (markers); low-frequency markers (\(f<250\) MHz) are just hardware interference.