## 3 Results and Discussions  
Figure 3 shows the z-direction displacement due to the surface wave propagating in the xy-plane. Comparing Model 1, Model 2, and Model 3, we can see that a wavefront showing constructive superposition appears first in Model 3. The color bar used in each subfigure in Figure 3 is the same, ranging from \(-1\times 10^{-6}\) (dark blue) to 0 (light blue) to \(3\times 10^{-6}\) mm (red). It can be seen that due to the time delay, the wavefront in Model 3 lags behind Model 1 and Model 2, but its intensity is significantly enhanced. Figure 3: Displacement in the z-direction due to the surface wave propagating in the xy-plane at different times. (**a**) Model 1; (**b**) Model 2; (**c**) Model 3. The corresponding times are 0.6, 0.8, 1, 1.2, and 1.4 \(\upmu\)s (from top to bottom). In Figure 4, we plot the displacement for the three models at points at which the distance to the \(x\)-axis and \(y\)-axis are both 1, 1.5, 2, and 2.5 mm, referred to as point 1, point 2, point 3, and point 4, respectively. It can be seen that the peak displacement for Model 3 is much larger than that for Model 1 and Model 2 because of the constructive superposition of the acoustic wavefronts. The simulation results agree closely with the theoretical model, but the noise of Model 3 is larger than that of Model 1 and Model 2, and this problem needs to be considered in the subsequent data processing. MATLAB was used to fit the wavefront data to confirm that the enhanced wavefront continues to propagate in the form of a constructively superposed wave. A surface composed of points of equal amplitude is considered a wavefront surface. For the purpose of choosing a fitting method with a simple form and high accuracy, we used second to fifth-order polynomial fitting with the Zernike polynomial. Rayleigh waves propagate in the form of cylindrical wavefronts along the free surface of the medium [30], so only the first three orders were considered when using Zernike polynomials for fitting. The first three orders of the Zernike polynomial are shown in Table 1.  
Due to symmetry, only the region with x \(>\) 0 was considered when fitting. Due to acoustic absorption, the surface wave decays exponentially inside the material. The surface wave on the \(z\)-axis far from the heat source is very weak and can barely be discriminated from noise, so only data within 0.1 mm of the heat source was used for fitting. Figure 5 depicts the wavefronts of Model 1, Model 2, and Model 3 fitted by polynomials of different orders. The fitting error is displayed in Table 2. For higher-order  
Figure 4: Displacement in the z-direction as a function of time of Model 1, Model 2, and Model 3 for (**a**) point 1, (**b**) point 2, (**c**) point 3, and (**d**) point 4.  
polynomials, the fitting error is reduced. The results using third-order Zernike polynomial fitting are similar to third-order polynomial fitting. In both Model 1 and Model 3, fitting using fourth-order and fifth-order polynomials offers significantly better accuracy compared with second-order and third-order fitting, and it is recommended to adopt the fourth-order polynomial fitting as the difference between fourth- and fifth-order polynomial fitting is marginal. Regardless of the order of fitting for Model 2, the fitting error is larger than that for Model 1 and Model 3, and the fitted results are also not ideal. It can be seen from the original data of Model 2 in Figure 5 that weaker wavefronts appear distinct from the main wavefront. This implies that the use of multiple sources without time delay for simultaneous excitation not only cannot form a single enhanced wavefront, but the weaker wavefronts also lead to difficulties in interpreting ultrasonic images and extracting data. For further verification, we conducted numerical calculations based on the parametric indirect microscopic imaging (PIMI) method [31], which is a method that can optically visualize the sound field wavefront by measuring the phase retardation and polarization ellipse orientation angle of the probe light.