## 4 Evaluation Experiments
### Experimental Results  
Figure 8 shows the SPL measurement results for sideband waves (\(\mathrm{SPL}_{\mathrm{s}}\)) along the \(z\)-axis. Because a PAL produces a frequency-dependent response, the results are shown in separate panels according to sideband frequency \(f_{\mathrm{s}}\). These measurement results were used for parameter determination, as mentioned in Section 4.1. Figure 9 shows the simulation and measurement results for the SPL of demodulated sound (\(\mathrm{SPL}_{\mathrm{d}}\)) along the \(z\)-axis. Similarly to Figure 8, the results are shown in separate panels \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline Demodulated frequency \(f_{\mathrm{d}}\) (kHz) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline Sideband frequency \(f_{\mathrm{s}}\) (kHz) & 39 & 38 & 37 & 36 & 35 & 34 & 33 \\ \hline Ratio of sound pressure \(P_{\mathrm{s}}\) & 1.0 (reference) & 0.76 & 0.28 & \(8.2\times 10^{-2}\) & \(2.8\times 10^{-2}\) & \(1.5\times 10^{-2}\) & \(8.8\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calibration coefficients on sound pressure of sideband wave \(P_{\mathrm{s}}\). according to demodulated frequency \(f_{\rm d}\). The measurement results were also used for parameter determination, as mentioned in Section 4.1. The simulation results of the proposed method generally match the measurement results, especially for \(f_{\rm d}=2,3,\ldots,7\) kHz. However, it is difficult for the proposed method to express the SPL fluctuation as the distance from the PAL increases. When \(f_{\rm d}=1\) kHz, the error between simulation and measurement is larger than that at other frequencies. This can be explained by the low demodulation efficiency when the demodulated frequency \(f_{\rm d}\) is low, which can be demonstrated theoretically from Equations (4) and (5). The simulation results of demodulated sound pressure on the \(xz\)-plane in the experimental setup without reflection (as shown in Figure 7a) are shown in Figures 10 and 11, and the results for the experimental setup with reflection (as shown in Figure 7b) are shown in Figures 12 and 13. The simulation and measurement results are given in pairs for comparison. As a quantitative evaluation, we used Equations (35) and (37) (Section 4.1) to calculate the RMSE and average absolute error, respectively, of the demodulated sound pressures  
Figure 8: Measurement results for the sound pressure level of sideband waves along the \(z\)-axis for different sideband frequencies: (**a**) \(f_{\rm s}=39\) kHz, (**b**) \(f_{\rm s}=38\) kHz, (**c**) \(f_{\rm s}=37\) kHz, (**d**) \(f_{\rm s}=36\) kHz, (**e**) \(f_{\rm s}=35\) kHz, (**f**) \(f_{\rm s}=34\) kHz, and (**g**) \(f_{\rm s}=33\) kHz. The above results show that the proposed method can generally approximate the propagation of the demodulated sound of a PAL. The frequency-dependent demodulation efficiency [25; 28] can also be reflected in the simulation results of the proposed method because our method can include self-demodulation in the calculation of demodulated sound pressure. In the cases without reflection, the error between simulation and measurement is very small, especially when the demodulated frequency \(f_{\rm d}\) is in the range of 2 kHz \(\leq f_{\rm d}\leq 6\) kHz. When \(f_{\rm d}=1\) kHz, the demodulation efficiency is much lower so it is considered that the simulation results contain more errors. In the cases with reflection,the error between simulation and measurement is much larger than that in the cases without reflection, and the error increases obviously as the demodulated frequency \(f_{\text{d}}\) increases, which is different from the error in the cases without reflection. Two major reasons can be identified: one is that the reflection coefficients were not set properly, and the other is that the measurement results also lack accuracy. In summary, the experimental results demonstrated the effectiveness of the proposed method in approximating the demodulated sound of a PAL.