## 4 Experimental Results and Analyses  
### Deployment of the Experimental Environment  
The schematic diagram of elastic modulus measurement is shown in Figure 5, which includes experimental frame, tension sensor, air-coupled ultrasonic transducer, dual-channel signal generator and digital oscilloscope. In Figure 5, the experimental frame is the main platform for the measurement of elastic modulus. In the frame, one end of the sample is clamped by the left chuck, and is connected with the tension sensor T1 fixed on the frame, the other end is clamped by the right chuck, and is connected with the screw through the frame. The screw is fitted with a force nut. The nut adopts rotating force to apply external force, which is simple, intuitive and stable. The digital tension meter T2 shows the tension value of the sample under test. This method is simple, intuitive and stable to apply after-force. The air-coupled ultrasonic transducers Tx and Rx are fixed on the frame and the screw and kept horizontally aligned. Among them, the transducer Tx is connected with S1 channel on the double-channel signal generator, the transducer Rx is connected with the digital oscilloscope CH1, and the signal generator S2 channel is connected with the digital oscilloscope CH2. Figure 5: Schematic diagram of an instrument for measuring elastic modulus of materials. Specifically, sine continuous wave signals of 400 kHz and 3 Vpp are added to signal generator S1 and S2 channels at the same time, and the transmitting transducer Tx converts the sine wave signals from signal generator S1 channel into ultrasonic signals of the same frequency. After converting the received acoustic signal into sinusoidal signal by receiving transducer Rx, the results are displayed by digital oscilloscope CH1 channel. At the same time, the sine wave signal from S2 channel of signal generator is displayed through CH2 channel of digital oscilloscope. Next, the ultrasonic wave from Tx travels through the medium to Rx. At this point, there is a certain phase difference between the transmitted signal and the received signal. As shown in Figure 6, \(\Delta\varphi\) is the phase difference between the two. By changing the distance between Tx and Rx, the change of \(\Delta\varphi\) can be observed. The relationship between phase difference \(\Delta\varphi\) and angular frequency \(\omega\), propagation time \(\Delta t\), acoustic speed \(\upsilon\), distance \(l\) between Tx and Rx, and wavelength \(\lambda\) is as follows:  
\[\Delta\varphi=\omega\Delta t=2\pi f\times\frac{l}{\upsilon}=2\pi\frac{l}{\lambda} \tag{7}\]  
According to Equation (7), to change the phase difference \(\Delta\varphi\) by \(\pi\)phases, the distance \(l\) must change correspondingly by half wavelength \(\lambda/2\). For the phase difference \(\Delta\varphi\) to change by \(2\pi\) phases, the distance \(l\) must change by a corresponding wavelength \(\lambda\). Set the operation mode of the oscilloscope to 'X-Y', then the transmitted and received waves are added to the 'X' and 'Y' deflector plates of the oscilloscope respectively. In this way, the two harmonic oscillations are superimposed vertically to each other to form the Lissajous--Figure on the fluorescent screen. If we move Rx continuously in the same direction, then changing the distance between Tx and Rx will change the phase difference between the two vibrations. When the phase difference of the two vibrations changes from 0 to \(\pi\), it can be seen on the screen that the graph changes from a positive straight line to an ellipse and then to a negative straight line, as shown in Figure 7. Here, the elongation of the sample material \(\Delta L\) is equal to \(\lambda/2\). Figure 6: The phase difference pattern of two harmonic vibrations. Figure 7: Lissious—Figure for a specific phase difference. (**a**) \(\Delta\varphi=2\lambda\pi\); (**b**) \(\Delta\varphi=2\lambda\pi+\frac{\pi}{2}\); (**c**) \(\Delta\varphi=2\lambda\pi+\frac{\pi}{2}\); (**d**) \(\Delta\varphi=2\lambda\pi+\frac{\pi}{2}\); (**e**) \(\Delta\varphi=2\lambda\pi+\pi\).