# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 4 Experimental Verifications
The critical refraction method is used to excite and receive the selected modes that near the phase velocity which satisfies [PERSON]'s theorem. Only the second harmonic generated by the modes pair satisfying the conditions (phase-matching and non-zero power flux) play a dominant role in the field of second harmonic signal [30]. The mode shapes of mode pairs 10 and 15 are shown in Figure 12.  
Figure 8: Physical experiment system: (**a**) the excitation and receiving angle probes for Lamb waves in the steel plate; (**b**) high-energy ultrasonic testing system. Figure 9: The experimental system’s diagram. Figure 11: Spectrum analysis diagram. Figure 12: Cont. \[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]0 Signal diagram. In order to verify that mode pair 10 and 15 can generate cumulative second harmonic, change the dip angle of transducers and gradually increase the distance between the excitation transducer and the receiving transducer from 95 to 145 mm. Calculate and observe the change of the relative nonlinear parameter, \(\beta\), according to Equation (23) in the process of distance increase. For every 10 mm increase in the distance, the test system samples 200 times, to calculate the mean value and variance of the relative nonlinear parameter at this distance. The final curve of the relative nonlinear parameter with the distance is shown in \[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]1  
Figure 13 elucidates that with the mode pairs 10 and 15, the relative nonlinear parameters increase gradually with the increase of propagation distance. It denotes that cumulative second harmonics can be generated by using the mode pairs 10 and 15 as the excitation and reception modes. Moreover, under the same propagation distance, the relative nonlinear parameter detected by mode pair 10 is larger than that detected by mode pair 15, which shows that second harmonic mode generated by mode pair 10 has higher amplitude. To check the changes of second harmonic modal amplitudes generated by the mode pairs 10 and 15 with the change of stress, the steel plate is stressed by the hydraulic tensile machine, which is shown  
\[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]2 The mode shapes of mode 10 and 15: (**a**) the fundamental mode of mode pair 10; (**b**) the second harmonic mode of mode pair 10; (**c**) the fundamental mode of mode pair 15; and (**d**) the second harmonic mode of mode pair 15. \[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]3 Nonlinear parameter versus propagation distances. in \[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]4 The distance between the excitation transducer and the receiving transducer is 145 mm, and other experimental settings remain unchanged. The tensile stress of 0 to 25 MPa is applied to the steel plate by the hydraulic tensile machine, with each increase of 5 MPa. The relative nonlinear parameters under different stress conditions are calculated. The experimental results are shown in \[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]5  
Figure 15 shows that the relative nonlinear parameter detected by mode pair 10 increases with the increase of tensile stress, while the relative nonlinear parameter detected by mode pair 15 decreases with the increase of tensile stress. Compared with Figure 7d,i, it can be found that the experimental results are consistent with the simulation results.