## 3 Results and Discussion
### Experimental Results  
We first investigate the optimal size for the sub-array. Figure 3 shows the RMSEs when we use full-array sizes of \(N_{\rm R}=28\) and \(32\) and change sub-array size. The measured specimen was bone-mimicking plate. The results show that the optimum sub-array size is \(N=N_{\rm R}/2\). Therefore, with \(N_{\rm R}=28\), the value of \(N_{\rm sub}=15\) is the optimal size. In the previous paragraph, we described the experimental investigation of the optimal size. Here, we describe the theoretical investigation of this size [29]. To recover the rank of the covariance matrix with \(M\) modes, a minimum of \(M\) times averaging, i.e., \(N=M\), is required. In addition, to measure \(M\) modes, a minimum of \(N_{\rm sub}=M+1\) is required. Under these two conditions, when we attempt to maximize \(M\) and \(N_{\rm sub}\), the number of receivers \(N_{\rm R}\) should be expressed using \(N_{\rm R}=N_{\rm sub}+N-1=2N\). Theoretically, we can confirm that the selection \(N=N_{\rm R}/2\) is reasonable. Figure 3: Dependence of the proposed method on sub-array size. The black solid line and the dotted line show the results for arrays composed of different numbers of receivers. The vertical black dotted lines show the horizontal values with \(N=N_{\rm R}/2\), i.e., \(N_{\rm sub}-1=N_{\rm R}/2\). We determined that \(N_{\text{sub}}=15\) is the optimal size. Therefore, the following results employed \(N_{\text{sub}}\) of 15. Figures 4-7 show the experimental results that were obtained for the copper plate and the bone-mimicking plate, respectively. The color maps show the results obtained using the conventional SVD method. The red dots represent estimated results obtained using the proposed method. The solid white lines show the theoretical curves. To remove any obvious false estimates, we removed all estimates with \(c_{\text{p}}<1200\) and \(c_{\text{p}}>50,000\) m/s. The proposed method thus successfully depicted the wavenumber without the need for peak search processes. The RMSEs of the estimation results when using the proposed method with the copper plate and bone-mimicking plate were 108 rad/m and 121 rad/m, respectively. RMSEs are shown in Table 1. The wavenumbers of the shear waves that had larger effects on the theoretical curve than the longitudinal wave at the center frequency (1.0 MHz) of the copper plate and the bone-mimicking plate were 2780 rad/m and 3879 rad/m, respectively. When compared with the wavenumber of the shear wave, the RMSE was less than 4%. Figure 4: Wavenumbers of the 4-mm-thick copper plate estimated using the conventional singular value decomposition (SVD) method. The color intensity map shows the results obtained using the conventional SVD method. Solid white lines show the theoretical curves. Figure 5: Wavenumbers of the 4-mm-thick copper plate estimated using the proposed method and the conventional SVD method. Red dots show the estimates obtained using the proposed method. The color intensity map shows the results obtained using the conventional SVD method. Solid white lines show the theoretical curves. The spectrum at the frequency that is indicated by the white dotted line in Figure 7 is shown in Figure 8. As Figure 8 illustrates, the proposed method has higher resolution than the conventional method because the method can depict two modes around 4000 rad/m with small error. The wavenumbers that were estimated using the proposed method were almost the same as those at the peak position in the SVD spectrum when the resolution of the SVD method was high enough or the resolution of the ESPRIT algorithm was insufficient to separate the signals. However, it is difficult to identify some of the peaks because of their small amplitudes and low prominence when using the SVD method, such as the peaks with wavenumbers of approximately 2800 and 4300 rad/m.