## 4 Time Delay Focusing and Received Amplitudes  
An annular array transducer that emits the longitudinal wave consists of a number of source elements and the radiation beam can be focused by applying an appropriate time delay to each element of the array transducer [29], as shown in Figure 14. A delay law for each element of the array transducer along the symmetry axis of the transducer with zero steering angle can be derived by considering the time-of-flight (TOF) from the centroid of each element to the specified focal distance \(F\). The TOF difference between the \(m\)th element and the center element can be calculated as follows [30]:  
\[\Delta t_{m}=\frac{r_{m}^{2}+r_{m+1}^{2}}{4 cF} \tag{22}\]  
where \(c\) is the longitudinal velocity of the sample, and \(r_{m}\) and \(r_{m+1}\) are the inner and outer radii of the \(m\)th element, respectively.  
Figure 13: Amplitude of the received displacement as a function of the phase shift between the array transducer elements: (a) fundamental wave, and (b) second harmonic wave.  
The delay of the time-domain signal is equivalent to multiplying the frequency domain signal by a phase term that is linear in frequency and proportional to its delay. If \(F(\omega)\) is the Fourier transform (FT) of the time-domain signal \(f(t)\), then the FT of the time shifted signal \(f(t-\Delta t)\) can be obtained as \(\exp(i\omega\Delta t)F(\omega)\), where \(\Delta t\) is the time delay.  
The array transducer used for a focused beam generation is the same as the four-element transducer, described before, used for phase-shifted radiation. The focal distance is set at the reflecting boundary of the sample (\(F=15\) mm) and the steering angle is \(0^{\circ}\). In the simulation, the transmission time delay was calculated using Equation (22) and applied to focus the radiation beam at the specified focal distance. When calculating the received acoustic fields, the reception time delay was applied in the same way.  
Figure 15 shows the received displacement amplitude of the fundamental and second harmonic waves when the radiation beam was focused at the reflecting boundary. Note that the received displacement amplitude of the fundamental wave fluctuates and increases very little in Figure 15a for both boundary conditions. This is due to the beam focusing of the array transducer. The boundary condition does not affect the calculated results of the fundamental wave. Figure 15b shows the received displacement amplitude of the second harmonic. In the back-propagation region, the rigid boundary yields a much larger SHG than the stress-free boundary. The stress-free boundary provides a relatively strong SHG, but the received amplitude initially tends to decrease due to the canceling effect and then accumulates during back propagation to the source location.  
Figure 16a shows the phase analysis results of the fundamental wave, which are the same as before. Figure 16b 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 0 at the boundary and gradually increases to about \(\pi/5\) at the source position. As a result, the total second  
Figure 14: Conventional time delay focusing of the four-element array transducer and reception in the pulse-echo mode.  
Figure 15: Changes in the amplitude of displacement received from conventional time delay focusing: (**a**) fundamental wave and (**b**) second harmonic wave.  
harmonic amplitude increases continuously as shown in Figure 15b. On the other hand, the phase difference between the reflected and newly generated second harmonics gradually increases from about \(\pi\) to about \(6\pi/5\) for the stress-free boundary (\(R=-1\)). This is why the summed second harmonic amplitude exhibits a sudden decrease at first and then a continuous increase, thereafter, as seen in Figure 15b.