## 3 Experimental Setup and FEM Theory
### Experimental Set-Up  
As shown in Figure 3, the ultrasonic measurement system used in this paper was composed of a Langevin transducer, signal generator, power amplifier, impedance matcher, filter circuit, data acquisition card, and PC. The ultrasonic measurement experiment adopts the discontinuous measurement method. In the experiment, we used 10 cycles of sine signal to excite the transducer. The amplitude of the excitation signal was 500 V and the pulse width was 0.04 ms.  
At room temperature, we placed the graphite electrode sample between two Langevin transducers (as shown in Figure 4). At the beginning of the measurement, the signal generator output a sinusoidal signal with a predetermined frequency and then amplified the signal through a power amplifier, which used an impedance matcher to drive the transmitting transducer to work. At the same time, the data acquisition card was triggered by the external clock to record the signal received from the receiving transducer and save it to the PC. The number of received signal sampling points was 512, and the sampling period was 0.4 microseconds. To reduce the interference noise in the received signal, we used a low-pass filter circuit to filter the output of the receiving transducer.  
Figure 3: Ultrasonic measuring system.  
To select the frequency of the transducer in the experiment, it is necessary to know the propagation velocity of each guided wave mode of the graphite rod. Therefore, we obtained the group velocity curve of the graphite rod with a diameter of 50 mm by using the semi-analytical finite element method [29], as shown in Figure 5. Please refer to Appendix A for the theory of guided wave mode in rod.  
The following can be seen from Figure 5:  
1. Different modes had different degrees of dispersion. The same mode had different degrees of dispersion in different frequency ranges;
2. The phase velocity of the \(T\) (0, 1) mode was frequency-independent, which indicates that the \(T\) (0, 1) mode was a non-dispersive mode;
3. Near the cutoff frequency of the \(L\) (0, 1) longitudinal mode, there was a frequency range with a small degree of dispersion, and the \(L\) (0, 1) mode had the largest group velocity in this frequency range. In summary, the group velocity of the \(L\) (0, 1) mode was larger than that of other modes in the range of its low-frequency dispersion frequency. This means that the \(L\) (0, 1) mode signal will reach the transducer first.  
The group velocity value of \(L\) (0, 1) mode was the largest in the frequency range of 25-60 kHz. Therefore, we selected two endpoints of this frequency range for experimental measurement. In addition, we added 47 kHz as the test point near the midpoint of 25-60 kHz. When the frequency was 100 kHz, the group velocity value of \(B\) (1, 1) mode was the largest, so we also took 100 kHz as the test point.  
The equipment list of the ultrasound system is as follows:  
The model of the signal generator is DG1022 Z;  
The model of the power amplifier is ATG-2021. The maximum output voltage is 170 Vpp, the maximum output power is 42.5 W, the input impedance is 50 \(\Omega/5\) K\(\Omega\), and the output impedance is 5 \(\Omega/50\)\(\Omega\);  
The model of the data acquisition card is PCI-1714. The highest sampling frequency is 30 MS/s, and the resolution is 12 bits;  
We used [PERSON] transducers with frequencies of 25 kHz, 47 kHz, 60 kHz, and 100 kHz. The beam angles of the transducers are 94\({}^{\circ}\), 58\({}^{\circ}\), 53\({}^{\circ}\), and 42\({}^{\circ}\) respectively;  
Figure 4: Illustration of the ultrasonic measuring system.  
Figure 5: Group velocity dispersion curve of the graphite rod.  
We used a low-pass filter circuit with a cutoff frequency of 250 kHz.