## 4 Results and Discussions
### Numerical Simulation Results  
Figure 10 shows the calculated and measured group velocity. From Figure 10, we know that when the excitation frequencies were 25 kHz, 47 kHz, and 60 kHz, and the measured value is the group velocity of \(L\) (0, 1) mode. When the excitation frequency is 100 kHz, the measured value is the group velocity of \(B\) (1, 1) mode. The measured value has the same trend as the theoretical calculated value.  
As shown in Figure 11, the transmission loss of ultrasonic waves with different frequencies was calculated by Equation (5). The blue curve in Figure 11 results from numerical simulation, and the green curve is the result of experimental measurement. Although the simulated transmission loss (shown by the blue curve in Figure 11) exhibited fluctuations, it does not mean that the solutions did not converge. This is because we defined the sound field of the sound source room as the sum of \(N\) uncorrelated plane waves moving in random directions. As shown in Figure 12, there was an expected significant fluctuation at low frequencies where the wavelength was comparable to the rod diameter, because at low frequencies, sound waves cannot diffuse effectively. This means that the sound field will not diffuse under the measurement conditions under these low frequencies.  
Figure 8: The time and frequency response of the received signal in a 400 mm sample.  
Figure 9: The time and frequency response of the received signal in two samples.  
As shown in Figure 11, we know that the numerical simulation and experimental measurement results have the same change trend. Comparing the numerical simulation with the experimental data, the two agree. It shows that the numerical simulation method is reliable for calculating the ultrasonic transmission loss in graphite rods.  
From Figure 11, we know that when the frequency is low, the transmission loss of ultrasonic waves through the graphite rod is slight. When the ultrasonic wave frequency is 47 kHz, the transmission loss is only 1.837 dB/m. This is beneficial for reducing the attenuation of the sound wave amplitude and increasing the detection distance. As the ultrasonic wave frequency increases, the transmission loss will increase exponentially with the frequency. A higher ultrasonic wave frequency will cause the ultrasonic wave to decay quickly.  
In the frequency range of 25-55 kHz, the transmission loss changed relatively smoothly, and the transmission loss almost did not change with the increased acoustic frequency. In this frequency range, the transmission loss was insensitive to the ultrasonic wave frequency. The frequency in this range can be used as the transducer's resonant frequency.  
The incident intensity and transmitted intensity in the graphite rod surface are shown in Figure 13. The ultrasonic wave frequencies were 25 kHz, 47 kHz, 60 kHz, and 100 kHz. It can be seen from Figure 13 that the incident intensity distribution was only related to the randomness and the number of plane waves \(N\) in Equation (2). The transmitted intensity was determined by the sound pressure and displacement at the receiving end of  
Figure 11: Transmission loss of different frequency ultrasonic waves.  
Figure 10: The calculated and measured group velocity.  
the graphite rod. The displacement is related to the transferable modes in the graphite rod. The sound pressure and displacement at the receiving end of the graphite rod are shown in Figure 14.  
Figure 14: The displacement and sound pressure at the receiving end.  
Figure 13: The incident intensity and transmitted intensity in the graphite rod surface.