## 3 Conditions for an Acoustic Wave in a Tire Acoustic Cavity to Act as a Plane Wave  
The interaction between the tire and road surface is the excitation source of acoustic waves in the tire acoustic cavity. According to the literature [8; 9], the frequency band of excitation is wide enough to cover the first-order tire acoustic cavity resonance frequency. Although only the plane wave (Order \((0,0)\)) is in the propagating mode for \(\omega<\omega_{01}\) or \(\lambda>2l_{y}\), other orders of waves as evanescent waves also exist in a certain range. Obviously, within this range, the acoustic wave in the tire acoustic cavity is the superposition of all orders of waves, and it is definitely not a plane wave but a complex mode. For a long duct, since the length of space where the evanescent waves exist is much smaller than that of the duct, the wave in this duct can be regarded as a plane wave. But, for the tire acoustic cavity with the circular shape, although the wave may travel inside with no barrier just like in a semi-infinite duct, the circumference length of the tire acoustic cavity is limited. If the length of space where the evanescent waves exist is greater than or equal to the circumference length, we say that a higher-order wave fills the whole tire acoustic cavity. Obviously, the wave in the tire acoustic cavity is not a real plane wave even if at the position far away from the sound source. But, for the convenience of research, the wave in the tire acoustic cavity is often assumed to be a plane wave. To evaluate whether the wave can be regarded as a plane wave or not, the following assumptions are made:  
1. On a cross-section of tire cavity, if the ratio of maximum amplitude of summed evanescent waves to that of the plane wave is less than or equal to 5%, the contribution of the evanescent waves is ignored, that is, the wave in the cavity is approximately regarded as the plane wave. 2. If the ratio of the length of space where only a plane wave exists to the circumference at the cross-section center of the whole tire acoustic cavity is greater than or equal to 80%, the acoustic wave in the whole tire acoustic cavity is approximately regarded as the plane wave. Figure 2: Amplitude distributions of sound pressures on the cross-sections (the unit of numbers in the colour bar is Pa). Since the characteristics of plane waves in tire cavities are the only concern of this paper, the discussion is concentrated on the excitation frequency of \(f<f_{cutoff}=\min(f_{01},f_{10})\). At one position (\(x,y,z\)) in the tire cavity, the superposition of all higher-order waves except Order (0, 0) wave is expressed as  
\[p^{\prime}(x,y,z,t)= \sum P_{0}\cos\biggl{(}\frac{n_{x}\tau}{l_{x}}x\biggr{)}\cdot\cos \biggl{(}\frac{n_{y}\tau}{l_{y}}y\biggr{)}\cdot e^{-n_{x}n_{x}n_{y}z}\cdot e^{ i(\omega t)} \tag{11}\] \[except\ n_{x}=n_{y}=0\]  
From Equation (11), we may observe that all high-order waves are in phase, and the maximum sound pressure amplitude at any cross-section of the tire cavity occurs at position (\(x=0,y=0\)). Assuming that the ratio of the maximum amplitude of \(p^{\prime}(x,y,z,t)\) to that of the plane wave is \(\eta\) after traveling a distance of \(L\), we have  
\[\eta=\frac{P(x=0,y=0,L)}{P_{0}}= \sum_{ \begin{array}{c}n_{x},n_{y}\\ except\ n_{x}=n_{y}=0\end{array} }e^{-\alpha n_{x}n_{y}\cdot L} \tag{12}\]  
Obviously, the higher the order is, the faster the wave attenuation is. For the example, as shown in Section 2, since the tire acoustic cavity resonance frequency generally ranges from 200 to 250 Hz, the frequency of the sound source is assumed to be 250 Hz for the sake of calculation result reliability, and the Order (20, 20) evanescent wave is analyzed. According to Equation (12), we may obtain by calculating that the amplitude of a single Order (20, 20) evanescent wave reduces to 0.0312% of the original amplitude after traveling 1 cm, which is small enough to be ignored. Therefore, the maximum of \(n_{x}\) and \(n_{y}\) is chosen as 20 in this paper.