## 1 Introduction
The research results indicate that in the tire acoustic cavity, there not only exists a plane wave but also higher-order waves, and only when the distance away from the sound source is over a certain value can the contribution of higher-order evanescent waves be ignored and the acoustic wave be regarded as the plane wave. The research in this paper is helpful for understanding the characteristics and mechanism of an acoustic wave in the tire cavity. Simplified Model of Tire Acoustic Cavity and Propagation Characteristics of Interior Acoustic Waves  
Figure 1a shows the tire cross-section of a passenger car. We may observe that the cross-section of the tire acoustic cavity is approximately rectangular as shown in Figure 1b, and the acoustic wave can propagate down the ring cavity along both forward and backward directions without any barrier (Figure 1c). The tire acoustic cavity is enclosed by one tire tread, two sidewalls, and the rim, which may be assumed to be rigid in the study TACRN [10; 11; 12; 13; 14]. Thus, in order to facilitate the analysis without deviating from the essence of the problem, the tire acoustic cavity may be simplified as an infinite straight tube with a rectangular cross-section and rigid walls. Considering that the acoustic waves traveling along the forward or backward directions have similar characteristics, only the traveling wave along one direction is investigated. Additionally, the tire contact patch is the sound source of TACRN. Thus, the tire acoustic cavity may be further simplified as a semi-infinite waveguide with a rectangular cross-section and with the sound source being the starting point, as shown in Figure 1d. Although the theory about a semi-infinite waveguide with a rectangular cross-section is well known [11; 12; 13; 14], for the sake of descriptive integrality, some results will also be simply narrated. In Figure 1d, a coordinate frame is established, and \(l_{y}\) and \(l_{x}\) represent the width and height of the rectangular cross-section, respectively. Therefore, the starting position of the half-infinite straight tube is set as \(z=0\), and the acoustic wave travels along the \(z\)-axis from \(z=0\) to infinity. Generally, the sound pressure in the tube is uneven in \(x\), \(y\), and \(z\) directions, so the acoustic wave equation is expressed in three dimensions as follows [20]:  
\[\frac{\partial^{2}p}{\partial x^{2}}+\frac{\partial^{2}p}{\partial y^{2}}+ \frac{\partial^{2}p}{\partial z^{2}}=\frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{ \partial t^{2}} \tag{1}\]  
And the boundary conditions are \(v_{x}|_{x=0,l_{x}}=0;v_{y}|_{y=0,l_{y}}=0\). The conditions suggest a wave consisting of standing waves in the transverse directions (\(x\) and \(y\)) and a traveling wave along the \(z\) direction. For the rectangular cross-section and rigid walls, the acceptable solutions are as follows:  
\[p_{n_{x},n_{y}}(x,y,z,t)=A_{x,y}\cos(k_{x}x)\cdot\cos(k_{y}y)\cdot e^{i(\omega t -k_{z}z)} \tag{2}\]  
where  
\[k_{x}=\frac{n_{x}\tau}{l_{x}}\ \ \ n_{x}=0,1,2,\cdots \tag{3a}\] \[k_{y}=\frac{n_{y}\tau}{l_{y}}\ \ \ n_{y}=0,1,2,\cdots\] (3b) \[k_{z}=\sqrt{\left(\frac{\omega}{c}\right)^{2}-\left(k_{x}^{2}+k_{y}^{2}\right)} \tag{3c}\]  
For convenience, we define the following expression:  
\[\beta_{n_{x},n_{y}}^{2}=\left(k_{x}\right)^{2}+\left(k_{y}\right)^{2} \tag{4}\]  
Figure 1: Tire acoustic cavity and simplified model. Therefore, \(k_{z}\) can be written as  
\[k_{z}=\sqrt{\left(\frac{\omega}{c}\right)^{2}-\beta_{n_{x},n_{y}}^{2}} \tag{5}\]  
Assuming that the sound pressure of the acoustic source located at \(x=y=z=0\) is expressed as \(p(0,0,0,t)=P_{0}\cdot e^{i\omega t}\), according to Equation (2), we have  
\[A_{x,y}=P_{0} \tag{6}\]  
Actually, Equation (2) is the generalized expression of the Order \((n_{x},n_{y})\) wave. For the Order \((0,0)\) wave, the sound pressure is expressed as  
\[p_{0,0}(x,y,z,t)=P_{0}\cdot e^{i\left(\omega t-k_{z}z\right)} \tag{7}\]  
Obviously, the Order \((0,0)\) wave in the semi-infinite duct with a rectangular cross-section is a plane wave.