## 3 Simulation Results and Discussion
### Annular Array Transducer with Phase-Shifted Radiation  
1. Phase shift of \(\pi\) between adjacent elements. Figure 10 shows the unfolded schematic of longitudinal wave radiation from the four-element array transducer with the transmit phase shift of \(\pi\) radians between the elements and reception in the pulse-echo mode. A phase shift technique can be applied to array transducers to improve SHG from the stress-free boundary. Here, the linear phase shifting is used for simulation, i.e., \(\phi^{(m)}=(m-1)\phi\) where \(m\) denotes the element number and \(\phi\) represents the phase angle of the input signal. A schematic of the phase-shifted radiation is shown in Figure 10, where the phase difference of \(\phi=\pi\) between the elements is used and the simulation results are shown in Figure 11.  
Figure 11 presents the distribution of the received displacement amplitude of the fundamental and second harmonics for the four-element transducer. The received fundamental wave amplitude is about half of that of the single-element transducer as shown in Figure 11a. Figure 11b shows the received displacement amplitude of the second harmonic. The overall behavior looks much different from the single-element transducer of Figure 8b. The total second harmonic displacement amplitude decreases during backward propagation from the rigid boundary (\(R=1\)). On the other hand, it gradually increases during backward propagation from the stress-free boundary (\(R=-1\)) and exhibits weak cumulative behavior, achieving about 45% of the rigid boundary case of the single-element transducer at the source position. This is a big change achieved through the phase shift. As further studied below, one can optimize the amount of phase shift to apply to maximize second harmonic generation from the stress-free boundary. Figure 10: Phase\(-\)shifted radiation of \(\pi\) between the elements of the annular array transducer and reception in the pulse-echo mode. Figure 9: Phase analysis results of Figure 8: (**a**) Fundamental wave and (**b**) second harmonic wave. The results of the phase analysis for the received displacement in Figure 11 are shown in Figure 12. For the fundamental wave of Figure 12a, the phase behavior is the same as in the previous two cases, so no further discussion is made here. Figure 12b shows the phase analysis results of the second harmonic. The phase of the reflected second harmonic is continuous at the rigid boundary (\(R=1\)), and the phase difference with the newly generated one starts from about \(2\pi/5\) at the boundary and gradually increases to about \(7\pi/10\) at the source position. As a result, the total second harmonic displacement decreases as shown in Figure 11b. Conversely, the phase difference between the reflected and newly generated second harmonics gradually decreases from about \(3\pi/5\) to about \(3\pi/10\) for the stress-free boundary (\(R=-1\)). This is why the summed second harmonic amplitude increases slowly after reflection, as seen in Figure 11b. (2) Summary of phase\(-\)shifted simulation results. Since the received fundamental and second harmonic amplitudes are highly dependent on the applied phase shift, the simulation was performed using more phase-shift values. A series of phase shifts were applied, using the linear phase between the adjacent elements as before, \(\phi^{(m)}=(m-1)\phi\), where \(m\) denotes the element number and \(\phi\) represents the phase angle. Here, \(\phi\) ranges from 0 to \(\pi\) in \(\pi/20\) increments. The simulation results are presented in Figure 13. When there is no phase shift, the received amplitude of the four-element transducer is the same as the single-element transducer. The rigid boundary for the four-element transducer with zero phase shift  
Figure 11: Changes in the amplitude of displacement received from phase\(-\)shifted radiation of \(\pi\): (**a**) fundamental wave and (**b**) second harmonic wave. Figure 12: Phase analysis results of Figure 11: (**a**) fundamental wave and (**b**) second harmonic wave.