## 5 Comparison of Simulation Results and discussion  
### Fundamental and Second Harmonic Amplitudes  
Table 1 lists the case numbers for various types of transducers and their received displacement amplitudes calculated in the pulse-echo (PE) mode, using two boundary conditions. The single element transducer is listed as Case 1. The single element transducer in the PE mode with the rigid boundary is the same as the through-transmission (TT) mode of thickness \(z=2z_{0}\). Four representative cases (2 to 5) are listed for the four-element array transducer with three different phase shifts applied. Three representative cases (6 to 8) are listed for the dual-element transducer with three different phase shifts applied. Finally, the focused transducer is included as Case 9, where the focal distance is set at the reflecting boundary of the sample. The single element transducer with the rigid boundary will be used as a reference in the comparison of received amplitudes in various cases. Equation (11), for calculating the propagating sound beam fields, is called the Rayleigh-Sommerfeld (RS) integral and serves as the exact solution to the linear wave equation, Equation (9). Equation (12) also serves as an exact solution to the second harmonic wave equation in the quasilinear theory since the exact linear solution is used in the right hand-side of Equation (12). Therefore, all of the sound beam field equations derived in this \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Case No.**} & \multicolumn{3}{c}{**Phase Shift**} & \multirow{2}{*}{**Fundamental**} & \multicolumn{2}{c}{**Second Harmonic (10\({}^{-13}\) m)**} \\ \cline{3-4}  & & **Disk** & & **Annular Elements** & & **(10\({}^{-10}\) m)** & **Rigid** & **Stress-Free** \\ \hline Single element & 1 & - & - & - & - & 8.95 & 8.82 & 0.31 \\ \hline \multirow{4}{*}{Four element} & 2 & 0 & \(\pi/4\) & \(2\pi/4\) & \(3\pi/4\) & 3.78 & 1.39 & 0.16 \\  & 3 & 0 & \(\pi/3\) & \(2\pi/3\) & \(\pi\) & 1.62 & 1.31 & 0.34 \\  & 4 & 0 & \(\pi/2\) & \(\pi\) & \(3\pi/2\) & 1.33 & 4.51 & 0.71 \\  & 5 & 0 & \(\pi\) & \(2\pi\) & \(3\pi\) & 4.51 & 2.02 & 3.92 \\ \hline \multirow{4}{*}{Dual element} & 6 & 0 & \(\pi/2\) & - & - & 2.93 & 7.23 & 0.88 \\  & 7 & 0 & \(2\pi/3\) & - & - & 5 & 4.18 & 0.67 \\  & 8 & 0 & \(\pi\) & - & - & 8.04 & 7.57 & 1.74 \\ \hline Focused & 9 & 0 & \(1.16\pi\) & \(3.10\pi\) & \(5.78\pi\) & 4.94 & 4.09 & 1.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of received displacement amplitudes in various simulation cases. Figure 16: Phase analysis results of Figure 15: (**a**) fundamental wave and (**b**) second harmonic wave. study can be treated as exact solutions. Numerical calculations for the simulation were performed using MATLAB program. The amplitudes of the received displacements given in Table 1 can also be considered exact, although it may contain inherent errors that cannot be avoided in general numerical calculations. First, it is noticed that the received fundamental amplitude does not depend on the type of boundary used (Case 1). This is because the magnitude of the received displacement, which is always positive, is calculated. The received fundamental and second harmonic amplitudes show a strong dependence on the applied phase shift (Case 2 to 8). The received amplitude also varies with the focal position [31], although the results of only one focal position are shown in this study (Case 9). The single-element transducer with the rigid boundary (Case 1, \(R=1\)), which is equivalent to the through-transmission setup of propagation distance \(2z_{0}\), provides the largest SHG. This is why such a pulse-echo setup is used to successfully measure the nonlinear parameter of fluids [7]. Table 1 shows the improvement of SHG through the phase shift of the multi-element array transducer, and the maximum second harmonic amplitude is obtained when the phase shift between the elements is \(\pi\) (Case 5, \(R=-1\)).