## 3 Conditions for an Acoustic Wave in a Tire Acoustic Cavity to Act as a Plane Wave
The distributions of SPL at these three cross-sections are calculated and shown in Figure 7, and the maximum and average SPL are shown in Table 2. Obviously, the acoustic wave at the cross-section 160 mm away from the sound source is not a plane wave, while the acoustic wave is very close to the plane wave at the cross-section 165 mm away from the sound source, but the acoustic wave can be undoubtedly recognized as the plane wave. \begin{table}
\begin{tabular}{c c c} \hline
**Distances Away from the Sound Source/mm** & **Maximum/dB** & **Average Value/dB** \\ \hline
155 & 55 & 48.4 \\
160 & 47 & 46.1 \\
165 & 46 & 46.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Maximum and average SPL at different cross-sections in Figure 6.  
Figure 6: SPL distribution at the different cross-sections for the acoustic wave of 250 Hz. Obviously, the above research verifies the conditions of the acoustic wave in the tire acoustic cavity acting as a plane wave. Then, we may perform further investigation based on the approach in Section 3. Influence of Medium Type and Temperature in Tire Cavity on the Characteristics of Interior Plane Wave  
In the above discussion, the sound velocity in the tire cavity is assumed to be 340 m/s. In practice, the tire temperature is not constant, and the medium in the tire cavity may be air, nitrogen, or helium. The change in sound velocity caused by these factors may generate changes in the resonant frequency of the tire acoustic cavity, the cut-off frequency, and the frequency range in which the wave in the tire cavity can be regarded as the plane wave. According to Equation (12) and the parameters provided by [15], for four kinds of media with temperatures of \(-20\)-\(50\)\({}^{\circ}\)C in the tire cavity, the tire acoustic cavity resonant frequency, the cut-off frequency, and the frequency range in which the wave in tire cavity is regarded as the plane wave are listed in Table 3. The change of tire pressure will change the air density inside the tire, and the different types of asphalt will change the tire temperature, and then change the sound velocity inside the tire. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distances Away from the Sound Source/mm** & **Maximum/dB** & **Average Value/dB** \\ \hline
160 & 53 & 46.5 \\
165 & 46 & 44.2 \\
170 & 44 & 44.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum and average SPL at different cross-sections in Figure 7.  
Figure 7: SPL distribution at the different cross-sections for the acoustic wave of 350 Hz. From Table 3, it can be observed that for different media and tire temperatures, there exist great differences in the tire cavity resonant frequency, the cut-off frequency, and the frequency range in which the wave in the tire cavity is considered to be a plane wave. Then, to clarify the effect of medium and temperature on the existing range of evanescent waves when the tire acoustic cavity resonance occurs, the attenuation characteristics of evanescent waves in the tire cavity are discussed. The space range of plane waves in the tire cavity depends on the attenuation characteristics of evanescent waves. It can be seen from Equation (12) that the attenuation of evanescent wave is determined by  
\[\alpha_{n_{x},n_{y}}=\sqrt{\left[\left(\frac{n_{x}}{l_{x}}\right)^{2}+\left( \frac{n_{y}}{l_{y}}\right)^{2}\right]\pi^{2}-\frac{\omega^{2}}{c^{2}}} \tag{13}\]  
When tire acoustic cavity resonance occurs, the frequency satisfies \(f=\frac{c}{L_{c}}\). Therefore, for \(f=\frac{c}{L_{c}}\), we have  
\[\alpha_{n_{x},n_{y}}=\sqrt{\left[\left(\frac{n_{x}}{l_{x}}\right)^{2}+\left( \frac{n_{y}}{l_{y}}\right)^{2}\right]\pi^{2}-\frac{4\pi^{2}}{{L_{c}}^{2}}} \tag{14}\]  
From Equation (14), we may observe that under the condition of tire acoustic cavity resonance, the space range with evanescent wave has nothing to do with the sound velocity, i.e., the type and temperature of the medium, but is related to the tire size parameters.