## 3 Conditions for an Acoustic Wave in a Tire Acoustic Cavity to Act as a Plane Wave
For the acoustic wave of 250 Hz, according to Equation (12), \(\eta<5\%\) can be reached after traveling 161 mm. While for the acoustic wave of 200 Hz, \(L\) is calculated as 159 mm. When \(f<f_{01}=\frac{\omega_{01}}{2\pi}=\) 1096.77 Hz, for \(\eta<5\%\), the relationship between the traveling distance of the evanescent waves and the acoustic wave frequency is obtained as shown in Figure 3.  
For the 185/60R15 tire, the circumference of the tire acoustic cavity cross-section center is about 1539.4 mm, and the existing distance of high-order waves for acoustic wave  
Figure 3: Variation of traveling distance with the acoustic wave frequency for \(\eta<5\%\). frequency of 200 to 250 Hz in the above unidirectional propagation is about 159-161 mm, accounting for 10.33-10.46% of the circumference. Considering that the acoustic waves traveling along the forward or backward directions have similar characteristics, it is concluded that, beyond the range of about 20% of the circumference around the contact patch, the high-order evanescent waves disappear, and the sound field may approximately be regarded as a plane wave. For the acoustic wave of 1076-1077 Hz, the range of high-order waves in the tire cavity is about half the circumference of the tire cavity according to Equation (12). Considering the backward propagation of high-order waves, it can be concluded that when the acoustic wave frequency is greater than 1076-1077 Hz, there exists no plane wave in the tire cavity. Verification of Plane Wave Existing Conditions in Tire Acoustic Cavity Based on Finite Element Simulation  
To verify the plane wave existing condition in the tire acoustic cavity obtained in the above section, the finite element model of a pipe with a dimension of \(90\times 155\times 300\) mm is established as shown in Figure 4. The cross-section of the pipe and the position of the sound source are the same as that mentioned in Section 3. The sound pressure amplitude of the sound source is assumed to be \(P_{0}=1\) Pa, and the finite element mesh element type is hexahedron. The exit cross-section (opposite to the sound source cross-section) has full sound absorption property in order to simulate a semi-infinite pipe, which is realized by defining the acoustic impedance of the fluid. These processes can be set in Virtual.Lab 13.10 software. Figure 5 shows that the distributions of sound pressure level (SPL) on the cross-sections away from the sound source in the finite element model when \(f=250\) Hz. According to the results shown in Figure 3, for the acoustic wave with a frequency of 250 Hz, the traveling distance of the evanescent waves is 161 mm, that is, at the position being over 161 mm away from the sound source, the plane wave appears. To verify the conclusion, three cross-sections 155 mm, 160 mm, and 165 mm away from the sound source  
Figure 4: A semi-infinite waveguide with a rectangular cross-section. Figure 5: Distribution of SPL in the semi-infinite waveguide. are selected from the model in Figure 5, and the distribution of SPL is shown in Figure 6a-c. It can be seen that the sound pressure at the cross-section being 155 mm away from the sound source is obviously different, while the sound pressure is basically the same at the cross-section being 160 mm away from the sound source, and the sound pressure is completely the same at the cross-section being 165 mm away from the sound source. For the acoustic waves with other frequencies, similar conclusions can also be obtained. Therefore, the correctness of the theory can be verified. According to the SPL distribution at different cross-sections in Figure 6, the maximum and average SPL at different cross-sections are calculated and shown in Table 1.  
Similar to the analysis of the acoustic wave with a frequency of 250 Hz shown in Figure 3, three cross-sections corresponding to the acoustic wave of 350 Hz in Figure 3 are also selected from the model in Figure 5. The distances away from the sound source are, respectively, 160 mm, 165 mm, and 170 mm.