## 3 Experimental Setup and FEM Theory
### Layout of the FEM Model  
To understand the propagation of sound waves in the graphite rod, we established a steady-state finite element model in the frequency domain in COMSOL. We used this model to analyze the transmission characteristics of ultrasonic waves with different frequencies between 25 kHz and 100 kHz. The model was established based on the sound transmission loss measurement standard [31]. The finite element model consists of a source room, a receiving room, and a graphite rod, as shown in Figure 6. The left is the source room, and the right is the receiving room. Both the source room and the receiving room are filled with air. We set the surroundings of the sound source room and the receiving room as a perfect matching layer (PML) to simulate the infinite air domain. The thickness of the perfect matching layer was set as the maximum wavelength. We used pressure acoustics, solid mechanics, and acoustic-structure coupling boundaries in the model. The sound source room and the receiving room were set as pressure acoustics. The graphite rod was set as the solid mechanic. The graphite rod material was set to conform to linear elastic behavior. The attenuation of ultrasonic waves in graphite was simulated by adding an isotropic loss factor in the material properties. We used the cantilever beam resonance method [32] to measure the isotropic loss factor of graphite, which is 0.012. The boundary between the fluid and solid domains was set as the acoustic-structure coupling boundary, which simulates the coupling between pressure acoustics and solid mechanics. For pressure acoustics, we used the wave equation shown in Equation (1). We selected \(1\times 10^{-6}\) Pa as the reference pressure of the sound pressure level. We set the model to atmospheric pressure and temperature and the transient pressure acoustic model to linear  
Figure 6: Model setup of a graphite electrode. elasticity. We set the velocity of sound in the air as 346 m/s and the density of the air as 1.205 kg/m\({}^{3}\). \[\
abla\cdot\left(-\frac{1}{\rho_{c}}(\
abla p_{t}-q_{d})\right)-\frac{\omega^{2} p_{t}}{c_{c}^{2}\rho_{c}}=Q_{m} \tag{1}\]  
where, \(\rho_{c}\) is the density of the air, \(p_{t}\) is the total pressure, \(q_{d}\) is the dipole domain source, and \(Q_{m}\) is the unipolar domain source. To simulate the excitation of sound sources with different frequencies between 25 kHz and 100 kHz, we defined the sound field of the sound source room as the sum of \(N\) uncorrelated plane waves moving in random directions. Using \(N\) waves in different directions to excite sound waves can avoid the necessity of modeling the transducer, which can reduce the calculation amount of the model. If the transducer is used, it will not be easy to define the model because it has different structural dimensions because of different frequencies. The pressure field of the sound source room was determined by the following formula:  
\[ \begin{array}{l}p_{emit}=\frac{1}{\sqrt{N}}\sum\limits_{n=1}^{N}\exp(-i(k_{ n,x}x+k_{n,y}y+k_{n,z}z))\exp(i\Phi_{n})\\ k_{n,x}=\cos(\theta_{n})\\ k_{n,y}=\sin(\theta_{n})\cos(\varphi_{n})\\ k_{n,z}=\sin(\theta_{n})\sin(\varphi_{n})\end{array}  \tag{2}\]  
where, \(\theta_{n}\), \(\varphi_{n}\), and \(\Phi_{n}\) are the direction angles of the plane wave, and they are independent random numbers. Each plane wave has a different direction angle. The \(1/\sqrt{N}\) term is used to ensure that any \(N\) has a constant strength. Therefore, the incident sound intensity in the \(z\)-axis direction of the sound source room can be obtained as follows:  
\[ \begin{array}{l}I_{in}=\frac{1}{4}\text{Re}(p_{emit},v_{emit})\\ v_{emit}=\frac{-1}{i\omega\rho}\frac{\partial p_{emit}}{\partial z}\end{array}  \tag{3}\]  
where, \(p_{emit}\) is the pressure field of the sound source chamber, \(v_{emit}\) is the velocity in the \(z\)-axis direction, \(\rho\) is the density of air, and \(\text{Re}(a,b)\) is the dot product of complex numbers \(a\) and \(b\).