## 6 Conclusions  
For evaluation of spatial distributions of non-uniform elastic properties of thin plates and films the method based on measurement of local phase velocities of a sub-sonic A\({}_{0}\) mode Lamb wave was proposed. The phase velocities are found from the B-scans by two proposed signal processing methods--the zero crossing method and analysis of the 2D spatial-temporal spectrum of the B-scan. The sub-sonic A\({}_{0}\) mode is excited by the contact or air-coupled ultrasonic transducer made of a PMN-32%PT piezoelectric element with a small 5 \(\times\) 1 mm aperture. The off-plane displacements caused by the A\({}_{0}\) mode are measured by a POLYTEC laser interferometer at different distances from the excitation area and in this way the B-scan is obtained. The combination of air-coupled excitation and contactless reception, in principle, allows measurements of moving objects and hygroscopic materials. The efficiency of the proposed signal processing methods was evaluated theoretically and experimentally using thin mineral mica paper sheets with non-uniform in-plane properties. The theoretical evaluation was carried out using simulation results obtained by a finite element modelling for three hypothetical cases--the mica paper sample with two zones of rectangular shape with different thickness and elastic properties, the sample with a single circular non-uniformity, and the sample with multiple circular non-uniformities of different diameters. The processing of the simulated B-scans by the both proposed methods showed that they allow obtaining the spatial distribution of the local phase velocity, from which such elastic properties such as the Young's module can be evaluated. Experiments performed with the mica paper samples confirmed results of the theoretical investigations. The performed investigations also showed that, in the case of the non-uniformities with properties different from the rest of the sample under investigation instead of one basic A\({}_{0}\) mode, a few A\({}_{0}\) modes with different phase velocities originate. That can be rapidly found by the analysis of the 2D spatial-temporal spectrum. The velocities of those waves can be determined by both signal processing methods. The difference between them is that the zero-crossing method allows getting spatial distributions of the local phase velocities at one operation frequency and the 2D spectrum method enables the determination of the phase velocity at any temporal frequency; however, the spatial resolution in this case is lost. In the investigated mica sample two A\({}_{0}\) modes with different phase velocities were found. One mode propagates with the phase velocities \(c_{\rm ph1}=(95\pm\Delta 3)\) m/s and another one with the phase velocities \(c_{\rm ph2}=(106\pm\Delta 6)\) m/s at a frequency of 47 kHz. The reconstructed Young's modulus values are in the range of 9-13 GPa. The variations of the phase velocities \(\pm\Delta c\) are due to the random character of non-uniformities in the mica sample and depend on the area in the sample in which the measurements are performed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Zone in Figure 24** & **Mode Name** & \begin{tabular}{c} **Spatial Frequency** \\ **Range, mm\({}^{-1}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Frequency Range,** \\ **kHz** \\ \end{tabular} &
\begin{tabular}{c} **Phase Velocities \(c\),** \\ **m/s** \\ \end{tabular} \\ \hline a & \(A^{\prime}_{0},A^{\prime\prime}_{0}\) & 0.4–0.55 & 43.11–53.31 & 100–112 \\ \hline b & \(A^{\prime}_{0}\) & 0.45–0.51 & 46.02–52.35 & 95–103 \\ \hline c & \(A^{\prime}_{0}\) & 0.45–0.48 & 46.02–49.02 & 96–102 \\ \hline d & \(A^{\prime\prime}_{0}\) & 0.40–0.43 & 46.02–48.35 & 111–112 \\ \hline e & \(A^{\prime}_{0}\) & 0.48–0.51 & 51.02–52.35 & 104 \\ \hline f & \(A^{\prime}_{0}\) & 0.44–0.47 & 50.02–52.02 & 111–112 \\ \hline g & \(A^{\prime\prime}_{0}\) & 0.41–0.44 & 49.35–51.68 & 118–121 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Phase velocities of different modes.