## 3 Theoretical Framework  
### Premises  
The theoretical model is based on the analysis of dispersion curves of guided surface acoustic waves. Resonance peaks in the frequency spectra of the acquired signals (e.g., Figure 4a) are dots on the wavelength-frequency plane \((\Lambda,f)\) (e.g., Figure 4b). They indicate that at the corresponding wavelength-frequency pairs \((\Lambda_{j},f_{j})\), the laser beam excites SAWs. To use this information for the probing (i.e., quantitative evaluation) of the effective material constants of HSQ-NW composite medium, one needs, first, a mathematical and computer model allowing adequate prediction of the SAW characteristics for a given set of input sample's parameters (geometry, elastic moduli, and density). Second, it is necessary to specify an objective function which determines the deviation of the calculated SAW dispersion characteristics from the experimentally obtained ones. The minimization of this function in the space of input parameters of the specimen should result in approaching of the calculated characteristics to the experimental data. Additionally, the set of variable parameters providing the global minimum can be treated as the effective sample's parameters within the framework of the model used. The analysis of the scanning electronic microscope pictures of cleaved specimens (e.g., Figure 5a) tells us that, in general, the samples can be considered as three-layer elastic waveguides (Figure 5b). Counting top-down, the HSQ-NW composite is the core (second) layer covered by an electrical-contact coating (first layer) and underlain by a thick silicon substrate which may be simulated by an elastic half-space (third domain). It is expected that the total thickness of the first two layers deposited on the substrate is about 1 \(\upmu\)m while the coating takes about one-tenth. It is obvious that, due to a complex microstructure of the nanowired-based material, its elastic properties can be simulated neither by the properties of separate constituents (HSQ polymer or NW material) nor by their jointly averaged characteristics. Vertical elongation of the nano-inclusions results in a strong anisotropy of elastic properties that manifests itself in a large difference between vertical and horizontal deformations. Consequently, within the mathematical model, its stress-strain relations should be described by a transversely isotropic elastic medium with the horizontal plane of isotropy \((x,y)\) (Figure 5b). Such materials are specified by five independent elastic moduli C\({}_{ijkl}\) and density \(\rho\)  
Figure 4: Typical frequency spectrum of the laser-generated signal shown in the insert TGM acquired for the SAW (surface acoustic wave) wavelength \(\Lambda=5.85\upmu\)m (**a**). Typical dispersion diagram obtained in experiment (**b**). The material parameters of the CrPt alloy in the coating and Si[111] in the substrate are generally known and may be taken from literature. The underlying silicon half-space is of cubic symmetry specified by three independent elastic constants. The full set of Si[111] parameters used in the calculations is presented in Table 1.  
The first layer (coating) is isotropic (two independent elastic constants that are expected to be known, in such cases the values from Table 1 are used as input). However, due to the roughness of the top surface, the averaged elastic moduli of the top layer tend to be smaller than those of the pure CrPt alloy. Therefore, the effective material constants of the first layer and its thickness \(h_{1}\) are also generally unknown and, in many cases, it is determined simultaneously with the HSQ-NW parameters. The engineering constants are related to the elastic moduli through the connection between the \(6\times 6\) stiffness matrix \(C\), which elements \(C_{ij}\) are written in the Voigt notation, and the compliance matrix \(S\): \(C=S^{-1}\).