## 1 Introduction
### Multilayered Structures  
Coming back to the complex layered structure of current devices such as acoustic resonators (Figure 1), one way to achieve a complete knowledge of the elastic properties of the stack is to prepare and analyze a series of single films with the different materials, studying each of them independently according to the procedure described in the previous paragraphs. Alternatively, one can even directly measure the entire stack in a surf-BLS experiment and compare the frequencies of the detected modes to those calculated solving acoustic propagation in the multilayer by a proper numerical approach. The case of a bilayer on a substrate was successfully analyzed, for instance, in Refs. [39; 70]. The difficulty that may arise is that in this case the number of fitting parameters is so large (a few elastic constants for each constitutive layer) that, likely, it will not be possible to achieve a meaningful set of best fit parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symmetry** & **Isotropic** & **Cubic** & **Hexagonal** \\ \hline Independent constants & \(C_{11}\) C\({}_{44}\) & \(C_{11}\) C\({}_{12}\) C\({}_{44}\) & \(C_{11}\) C\({}_{13}\) C\({}_{22}\) C\({}_{44}\) C\({}_{66}\) \\ \hline Thin transparent film * & \(C_{11}\) and C\({}_{44}\) from the dispersion & \(C_{11}\) C\({}_{12}\) and C\({}_{44}\) from the dispersion of the RW and SW & \(C_{11}\) C\({}_{13}\) C\({}_{33}\) and C\({}_{44}\) from the dispersion of RW and SW \\  & & & dispersion of RW and SW \\ \hline Thick transparent film ** & \(C_{11}\) from the LM and/or LB C\({}_{44}\) & \(C_{11}\) from the LM, C\({}_{44}\) from the RW, from the RW, from the RW, \(C_{12}\) from the LB & \(C_{12}\) and C\({}_{33}\) from the LB \\ \hline Thin opaque film * & \(C_{11}\) and C\({}_{44}\) from the dispersion & \(C_{11}\) C\({}_{12}\) and C\({}_{44}\) from the dispersion of the RW and SW & \(C_{11}\) C\({}_{13}\) C\({}_{33}\) and C\({}_{44}\) from the dispersion of RW and SW \\  & & & dispersion of RW and SW \\ \hline Thick opaque film ** & \(C_{44}\) from the RW and C\({}_{11}\) from the LT & \(C_{11}\) from the LT, C\({}_{12}\) and C\({}_{44}\) & \(C_{44}\) from the RW and C\({}_{11}\) from the LT \\  & & & from the RW and TT \\ \hline \hline \end{tabular}  
* It is necessary to know the thickness and the mass density of the film; ** It is necessary to know the mass density of the film. Moreover, in order to exploit the information coming from the LB mode in transparent films, it is also necessary to know the refractive index.  
\end{table}
Table 1: Methods of determination of the elastic constants for films of different symmetry, thickness and transparency. “Thin” and “thick” refer to films whose thickness is below about 500–600 nm and above about 1 micron. RW: Rayleigh wave; LM: longitudinal mode; LB: longitudinal bulk; LT: Longitudinal Threshold.  
Figure 13: Experimental Brillouin spectra (dots) taken from a InGaAs/InP superlattice with periodicity 3.1 nm and total thickness 201.5 nm with Qs along [100] (**a**) and along [110] (**b**), for an angle of incidence \(\theta=67.5^{\circ}\). The solid lines correspond to the calculated perpendicular components of the surface phonon power spectrum, convoluted with a Gaussian function of width 0.6 GHz (the instrumental resolution). The Rayleigh wave peak (RW) the dip corresponding to the longitudinal threshold (LT) and the shoulder starting at the transverse threshold (TT) are indicated. Reprinted with permission from [68], IOP Publishing, 1996.