## 5 Comparison of Simulation Results and discussion
### Uncorrected Nonlinear Parameter  
The material nonlinearity is frequently quantified by the \"absolute\" nonlinear parameter, \(\beta\), which is defined by Equation (3). However, in practical experiments, the \"uncorrected\" nonlinear parameter of Equation (18) is measured using finite size transducers. In this case, in order to determine the \(\beta\), corrections for diffraction effects and material attenuation are required. In pulse-echo mode experiments, additional correction for boundary reflection is required.  
The \(\beta^{\prime}\) was calculated using the data of Table 1 and is presented in Figure 17. Case 5 with \(R=-1\) shows a relatively high \(\beta^{\prime}\) because it yields the largest second harmonic amplitude. In fact, the calculated \(\beta^{\prime}\) is about three times larger than the reference \(\beta\), so a small correction value much lower than 1 is required. This case may be a possible option for an accurate measurement of \(\beta\), but it is not the best option. The focused transducer (Case 9, \(R=-1\)) may be a better choice because it gives a reasonable \(\beta^{\prime}\) resulting from the improved SHG and requires a correction value close to 1. When a phase shift is accompanied for the measurement of \(\beta^{\prime}\) only with \(R=-1\), it seems that Cases 3 and 6 are suitable choices. The optimal phase shift criterion for the reliable determination of \(\beta\) using the four-element transducer is to maximize SHG from the stress-free boundary and to minimize the dependence on the correction by making the correction value as close to one as possible. It is noted that the largest value of \(\beta^{\prime}\) occurs when \(R=1\) in Case 4 because the second harmonic amplitude is relatively large, and the fundamental wave amplitude is the smallest.  
The simulation results observed here show that the transmission phase shift can be a useful tool in the pulse-echo SHG of relatively thin solid samples with stress\(-\)free boundaries. The design of array transducers and the amount of phase shift to apply for a given material and frequency can be optimized in terms of maximum possible SHG or in terms of the more accurate determination of \(\beta\).  
The current work covers ideal cases, but the actual phenomenon will not differ much from the theoretical investigation presented here. Future work should be able to include laboratory verification and field verification. Some of the challenges associated with experimental validation are making such a multiple element transducer. It should be able to carry high power input signals with minimal source harmonics. All elements of the transducer should be able to act as both a transmitter and a receiver and need to have a wide bandwidth to cover both the fundamental and second harmonic frequency components. In addition, at least four channels of function generators and high-power amplifiers may be required.  
This study deals with the generation of second harmonic in relation to the evaluation of the properties of nonlinear solid materials. Just as the nonlinear image based on the second harmonic is superior to the linear image, the third harmonic image can have a better resolution in tissue characterization because the main lobe is narrower in the transverse beam pattern than in the second harmonic. A measure of the third harmonic is the nonlinear coefficient C/A in fluids and biological media. Other studies have shown that the third harmonic is more sensitive to microstructure changes than the second harmonic by measuring the amplitude of the third harmonic or measuring the relative third-order nonlinear parameter for the damaged solid medium [32].