## 3 Results and Discussions
Photoacoustic imaging (PAI) is a rapidly emerging, nondestructive imaging tool to detect metal defects, which can visualize multiple types of cracks. There have been studies on the detection of multi-scale damage in carbon fiber-reinforced plastic (CFRP) composites with ultrahigh-resolution detection [43]. Studies employing finite element model simulations of a structure with micro-cracks study the interaction of the micro-cracks with Lamb waves. The results indicate that the micro-cracks lead to the generation of second harmonics [44]. Figure 11: Comparison of the actual response and optimized mathematical model for the pipe with damage in axis 3. For internal defect detection, some studies extract and combine spatial and temporal features to estimate the physical properties such as Young's modulus [45]. Previous studies also suggest that faults can be detected by estimation of absolute acoustic nonlinearity parameter, beta, defined by the displacement amplitudes of the fundamental and second-order harmonic frequency components of the ultrasonic wave propagating through the material [46]. This study analyzed the effects of cracks in a steel pipe and their location on the properties of acoustic waves that travel through it. Figures 7-14 and Tables 3-5 present the results of this study. Figure 9 displays the actual response of the system and its corresponding numerical model for damage locations along the axis of wave propagation. There is a figure for each system response and its optimized model for damage at offsets of 12%, 25%, 37%, 50%, 63%, 75%, and 87% along the wave propagation axis, i.e., axis 1. The black curve is the actual data curve of the system, and the blue series is the numerical model curve. Figure 12: Residual error of the actual signal and model results for cracks at axis 1. There is quite good fit at most points, except on a few occasions, where there are small differences, although the differences are within a permissible tolerance limit. Table 3 shows optimized parameters and root mean square error (RMSE) for the wave propagating with cracks along axis 1. These parameters include amplitudes, frequencies, and phases for both sine components of the composite sine wave numerical model. The results of the study are consistent with previous studies that show that the non-linear components of ultrasonic waves generated by frequency mixing can be utilized to detect micro-cracks in metals [47]. Similarly, Figure 10 and Table 4 show the results for the system with cracks at locations along the axis at 90\({}^{\circ}\) from the wave propagation axis, i.e., axis 2. Finally, Figure 11 and Table 5 show the results for crack locations on the opposite edge axis of the wave propagation axis, i.e., axis 3. Figure 12 plots the residual error that occurs among the actual and numerical series for each crack location along the wave propagation axis, axis 1. The residuals appear randomly scattered around zero, which shows that our numerical model is a good fit to the actual simulation data. Figures 13 and 14 show the residual error plots for cracks along axis 2 and axis 3, respectively. Figure 13: Residual error of the actual signal and model results for cracks at axis 2. By increasing the simulation data with more damage locations, this output area can always be minimized, and more precision can be introduced.