Research on Conditions and Inﬂuence Factors of an Acoustic Wave Acting as a Plane Wave in Tire Acoustic Cavity

: Tire acoustic cavity resonance noise (TACRN) contributes signiﬁcantly to the interior noise of electric cars and passenger cars with lower powertrain noise, which affects the comfort of the ride. To suppress TACRN effectively, it is crucial to clarify the characteristics of TACRN. In previous studies, the acoustic wave in the tire acoustic cavity is straightforwardly assumed to be the plane wave for convenience. In fact, there exist strict conditions for the acoustic wave propagating in the pipeline to act as a plane wave. The aim of this paper is to make the characteristics and evolution of acoustic waves in tire acoustic cavities clear. To do so, a simpliﬁed model of the tire cavity is established, and the sound ﬁeld distribution and the acoustic wave propagation characteristics in the tire cavity are analyzed based on the theory of acoustic waveguide. Then, the existence ranges of higher-order waves (non-plane waves), the conditions of an acoustic wave evolving into a plane wave, and the frequency range of a plane wave are investigated. Finally, the characteristics and evolution law of an acoustic wave in a tire acoustic cavity are obtained. The work in this paper may deepen the understanding of the characteristics and mechanism of acoustic waves in the tire cavity and be helpful and meaningful for analyzing and suppressing TACRN. Therefore, it is of practical signiﬁcance to reduce TACRN transmitted to the vehicle and improve the sound quality inside the vehicle.


Introduction
The investigation of tire acoustic cavity resonance noise (TACRN) has absorbed the attention of the industry and academia since its contribution to the interior noise of a car.When a car is running, the interaction between the road surface and tire tread with pattern generates the excitation with wide frequency bandwidth enough to cover the first-order natural frequency of the tire acoustic cavity, thus resulting in TACRN [1][2][3][4].
Grasping the characteristics of TACRN is a key step in investigating the attenuating strategy.To this aim, many researchers have performed a series of investigations, especially in revealing the mechanism of TACRN, predicting its modal characteristics, and analyzing the influences of load and velocity [5][6][7].Sakata et al. [8] first clarified the modal characteristics of tire cavity resonance by means of the finite element simulation and road test and put forward the calculation formula of the cavity resonance frequency.Thompson [9] developed a closed-form solution to predict the resonance frequencies of the deflected tire and performed experimental verification.It was also shown that the cavity resonance of the deflected tire generated forces acting at the spindle simultaneously along vertical and fore-aft directions with slightly different frequencies.
Molisani and Burdisso et al. [10,11] included the effect of the tire cavity resonance in a more refined structural-acoustic tire model, and a finite cylindrical cavity with rigid boundary conditions was employed.The model could help to predict the effect of the cavity resonance on the spindle forces as well as identify the potential noise control design.paid attention to the influences of the coupling of tire, cavity, and rim on the TACRN and verified the simulation models by using different analysis approaches.O'Boy and Walsh [15,16] showed a numerical model of the tire cavity with passive resonators to reduce the TACRN, and the larger the number of resonators was, the better the ability to attenuate the cavity noise over a wider frequency range was.Liu et al. [17][18][19] researched the mechanism and characteristics of the frequency split phenomenon and novel modal of TACRN from different aspects, and the obtained function could help to determine the frequency variation range under different conditions, and the theoretical calculation results were verified by the experiment and simulation.The above works showed that the researchers regarded the acoustic wave in the tire cavity as a plane wave, and then analyzed the characteristics of tire acoustic cavity resonance.
According to the existing literature, it can be concluded that, in the past modeling and analysis, TACRN was straightforwardly assumed to be the plane wave or was qualitatively described as a plane wave without analyzing the feasibility.However, there exist strict conditions for the acoustic wave propagating in the pipeline to act as a plane wave, and TACRN is a typical acoustic wave propagating in the pipeline.Therefore, it is necessary to investigate TACRN's evolution law and the conditions of TACRN being regarded as a plane wave.However, to the authors' knowledge, there is not yet a paper that completely describes the conditions for an acoustic wave in a tire cavity to evolve into a plane wave.
Aiming at this problem, in this paper, firstly, the tire acoustic cavity is simplified under the premise of maintaining its essential characteristics, and its acoustic model is established.Then, the propagation and attenuation characteristics of an acoustic wave in the cyclic tire cavity are quantitatively researched based on acoustic waveguide theory, and the existence range of a higher-order wave (non-plane wave) is analyzed.Next, the conditions of evolving into a plane wave and the frequency range of a plane wave are verified and revealed through the finite element method.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 crosssection 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.In Figure 1d, a coordinate frame is established, and y l and x l represent the width and height of the rectangular cross-section, respectively.Therefore, the starting position of the half-infinite straight tube is set as 0 z = , and the acoustic wave travels along the z- axis from 0 z = 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]: And the boundary conditions are The conditions suggest a wave consisting of standing waves in the transverse directions ( x and y ) and a trav- eling wave along the z direction.For the rectangular cross-section and rigid walls, the acceptable solutions are as follows: where = 0 , 1 , 2 , 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]: 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: where For convenience, we define the following expression: Therefore, k z can be written as 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 • e iωt , according to Equation (2), we have 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 Obviously, the Order (0, 0) wave in the semi-infinite duct with a rectangular crosssection is a plane wave.Only when k z is real does the wave with a propagating mode advance in the +z direction.From Equation (4), for the Order (n x , n y ) wave, we may see that the cut-off angular frequency is The minus sign must be taken on the physical grounds so that p n x ,n y → 0 as z → ∞ .Therefore, we have where This is an evanescent wave [17] that attenuates exponentially with z.
From Equation (8), we may observe ω 00 = 0.If ω < min(ω 01 , ω 10 ), then all order of waves except Order (0, 0) are the evanescent waves, that is, the waves will disappear after traveling a certain distance.This also suggests that a sound source definitely generates some evanescent wave appearing in a certain range.
Taking the tire acoustic cavity of a 185/60R15 tire as an example, according to the actual tire size, the parameters in the simplified tire acoustic cavity model are as follows: outer radius of 290 mm, inner radius of 200 mm, width of 155 mm, and circumference of the tire acoustic cavity cross-section center of L c = 1539.4mm (see Figure 1b).Therefore, the tire cross-section is approximately a rectangular cross-section of 90 × 155 mm (l x = 0.09 m, l y = 0.155 m).The sound velocity in the tire acoustic cavity is set as c = 340 m/s, and the sound pressure amplitude of sound source is assumed to be P 0 = 1 Pa in order to facilitate calculation in the formula.According to Equation (8), l y > l x leads to ω 10 > ω 01 = 6891 rad/s, which means that if the angular frequency (ω) of the sound source is less than ω 01 (ω < ω 01 ), then only an Order (0, 0) wave (plane wave) can propagate down the waveguide and other waves are evanescent.ω < ω 01 leads to λ > 2l y .That means that, as long as the wavelength is more than twice the cross-section characteristic size, only the plane wave is in the propagating mode.
Figure 2 shows the amplitude distributions of sound pressures on the cross-sections away from the sound source in the tire acoustic cavity when f = 250 Hz (ω < ω 01 ), n x = 1, and n y = 2. Since the tire acoustic cavity resonance frequency ranges from 200 to 250 Hz generally, f = 250 Hz is chosen for the sake of calculation result reliability (shown in Section 3).
propagate down the waveguide and other waves are evanescent. .That means that, as long as the wavelength is more than twice the cross-section characteristic size, only the plane wave is in the propagating mode.
Figure 2 shows the amplitude distributions of sound pressures on the cross-sections away from the sound source in the tire acoustic cavity when

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 , other orders of waves as evanescent waves also exist in a certain range.

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 ω < ω 01 or λ > 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.
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 cuto f f = 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 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 (x, y, z, t) to that of the plane wave is η after traveling a distance of L, we have 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.
For the acoustic wave of 250 Hz, according to Equation (12), η < 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 = ω 01 2π = 1096.77Hz, for η < 5%, the relationship between the traveling distance of the evanescent waves and the acoustic wave frequency is obtained as shown in Figure 3. 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 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, 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 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 × 155 × 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.
Appl.Sci.2023, 13, x FOR PEER REVIEW 8 of 15 conclusion, three cross-sections 155 mm, 160 mm, and 165 mm away from the sound source 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.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.
Appl.Sci.2023, 13, x FOR PEER REVIEW 8 of 15 conclusion, three cross-sections 155 mm, 160 mm, and 165 mm away from the sound source 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 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 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.The distributions of SPL at these three crosssections 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.
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.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.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.

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 °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.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 • 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.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 When tire acoustic cavity resonance occurs, the frequency satisfies f = c L c .Therefore, for f = c L c , we have 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.When tire acoustic cavity resonance occurs, the existing distance L of the evanescent wave can be calculated by setting η = 5% according to Obviously, similar to α n x ,n y , L does not depend on the type and temperature of the medium, but on the tire size parameters.
For the tire described in this paper, according to Equation (15), the relationship between η and L is calculated and shown in Figure 8 when tire acoustic cavity resonance occurs.Since the evanescent wave is recognized to disappear completely when η < 5% based on the assumption in Section 2, it can be observed that the evanescent wave only exists within a range of 160 mm away from the sound source no matter what kind of gas is inflated and no matter what the temperature of the tire is.It can also be seen from Equation ( 15) that the larger the tire cross-section size is, the greater the existing range of the evanescent wave is, and the larger the circumference c L of the tire is, the smaller the existing range of the evanescent wave is.
The conclusion of this research is universal and applicable to other types of tires.When the tire size and cross-section change, the relevant frequency range and cut-off frequency also change.

Influence of Tire Rotation on the Existing Range of Plane Wave in Tire Cavity
In a previous analysis, a static tire is considered; however, the research aiming at the rotating tire is of greater significance.Here, the influence of tire rotation on the existing range of plane waves in the tire cavity is studied.
When the tire is steadily rotating at a rotation speed of Ω (Figure 1c), the medium in the tire moves with the tire.Then, the speed of the medium may be approximately regarded as t v R = Ω⋅ (16) where Ω is the angular velocity of tire rotation; t R denotes the effective radius of the tire cavity, which is generally the radius of the cavity cross-section center.
Obviously, the speed of the forward traveling wave increases by v, while that of the backward traveling wave decreases by v.According to Equations ( 12) and ( 13), the wave speed may change the attenuation characteristics and the existing range of higher-order evanescent waves.For the tires with speeds of 10, 20, and 30 m/s, respectively, the variations of the existing range of the high-order waves with acoustic wave frequency for the forward and backward traveling waves are calculated and shown in Figure 9.It can also be seen from Equation ( 15) that the larger the tire cross-section size is, the greater the existing range of the evanescent wave is, and the larger the circumference L c of the tire is, the smaller the existing range of the evanescent wave is.
The conclusion of this research is universal and applicable to other types of tires.When the tire size and cross-section change, the relevant frequency range and cut-off frequency also change.

Influence of Tire Rotation on the Existing Range of Plane Wave in Tire Cavity
In a previous analysis, a static tire is considered; however, the research aiming at the rotating tire is of greater significance.Here, the influence of tire rotation on the existing range of plane waves in the tire cavity is studied.
When the tire is steadily rotating at a rotation speed of Ω (Figure 1c), the medium in the tire moves with the tire.Then, the speed of the medium may be approximately regarded as where Ω is the angular velocity of tire rotation; R t denotes the effective radius of the tire cavity, which is generally the radius of the cavity cross-section center.
Obviously, the speed of the forward traveling wave increases by v, while that of the backward traveling wave decreases by v.According to Equations ( 12) and ( 13), the wave speed may change the attenuation characteristics and the existing range of higherorder evanescent waves.For the tires with speeds of 10, 20, and 30 m/s, respectively, the variations of the existing range of the high-order waves with acoustic wave frequency for the forward and backward traveling waves are calculated and shown in Figure 9.
With the increase of tire speed, as far as the backward traveling wave is concerned, the existing range of higher-order waves becomes larger, while for the forward traveling waves, the existing range of higher-order waves decreases.The reason is that the tire rotation results in the increase of the velocity of the backward traveling wave, further in the decrease of the attenuation of the higher-order wave (which can be seen from Equation (13)), and vice versa for the forward traveling wave.
By including the combined effects of forward and backward traveling waves, we can obtain the existing range of high-order waves with tire speed, as shown in Figure 10.We may observe that the range of high-order waves gradually increases with tire speed, but within the frequency range (200-250 Hz) of tire cavity resonance; tire speed has a slight effect on the existing range of high-order waves.Therefore, when studying the tire acoustic cavity resonance noise, the influence of tire rotation on the change of acoustic characteristics of the tire acoustic cavity may be ignored.With the increase of tire speed, as far as the backward traveling wave is concerned, the existing range of higher-order waves becomes larger, while for the forward traveling waves, the existing range of higher-order waves decreases.The reason is that the tire rotation results in the increase of the velocity of the backward traveling wave, further in the decrease of the attenuation of the higher-order wave (which can be seen from Equation (13)), and vice versa for the forward traveling wave.
By including the combined effects of forward and backward traveling waves, we can obtain the existing range of high-order waves with tire speed, as shown in Figure 10.We may observe that the range of high-order waves gradually increases with tire speed, but within the frequency range (200-250 Hz) of tire cavity resonance; tire speed has a slight effect on the existing range of high-order waves.Therefore, when studying the tire acoustic cavity resonance noise, the influence of tire rotation on the change of acoustic characteristics of the tire acoustic cavity may be ignored.

Conclusions
In the existing literature on TACRN, the wave in the tire cavity is usually assumed to be a plane wave for convenience.But up to now, no literature researched the real sound field in the tire cavity systematically.Aiming at this problem, the characteristics and  With the increase of tire speed, as far as the backward traveling wave is concern the existing range of higher-order waves becomes larger, while for the forward travel waves, the existing range of higher-order waves decreases.The reason is that the tire tation results in the increase of the velocity of the backward traveling wave, further in decrease of the attenuation of the higher-order wave (which can be seen from Equat (13)), and vice versa for the forward traveling wave.
By including the combined effects of forward and backward traveling waves, we c obtain the existing range of high-order waves with tire speed, as shown in Figure 10. may observe that the range of high-order waves gradually increases with tire speed, b within the frequency range (200-250 Hz) of tire cavity resonance; tire speed has a slig effect on the existing range of high-order waves.Therefore, when studying the tire aco tic cavity resonance noise, the influence of tire rotation on the change of acoustic char teristics of the tire acoustic cavity may be ignored.

Conclusions
In the existing literature on TACRN, the wave in the tire cavity is usually assumed be a plane wave for convenience.But up to now, no literature researched the real sou field in the tire cavity systematically.Aiming at this problem, the characteristics a

Conclusions
In the existing literature on TACRN, the wave in the tire cavity is usually assumed to be a plane wave for convenience.But up to now, no literature researched the real sound field in the tire cavity systematically.Aiming at this problem, the characteristics and evolution of traveling waves in tire acoustic cavities are investigated in this paper, and the following conclusions are reached: 1.
The acoustic wave close to the contact patch in the tire acoustic cavity contains the plane wave and the high-order evanescent waves, so it is not a real plane wave.And, the existing range of high-order waves depends on the frequency of acoustic waves, the sound velocity, and the cross-section size.

Figure 1 .
Figure 1.Tire acoustic cavity and simplified model.

Figure 1 .
Figure 1.Tire acoustic cavity and simplified model.
Since the tire acoustic cavity resonance frequency ranges from 200 to 250 Hz generally, 250 f = Hz is chosen for the sake of calculation result reliability (shown in Section 3).

Figure 2 .
Figure 2. Amplitude distributions of sound pressures on the cross-sections (the unit of numbers in the colour bar is Pa.).

Figure 2 .
Figure 2. Amplitude distributions of sound pressures on the cross-sections (the unit of numbers in the colour bar is Pa).

Figure 3 .
Figure 3. Variation of traveling distance with the acoustic wave frequency for

Figure 3 .
Figure 3. Variation of traveling distance with the acoustic wave frequency for η < 5%.

Figure 4 .
Figure 4.A semi-infinite waveguide with a rectangular cross-section.

Figure 4 .
Figure 4.A semi-infinite waveguide with a rectangular cross-section.
Appl.Sci.2023, 13, x FOR PEER REVIEW 9 of 15 (a) Cross-section at 155 mm from the sound source (b) Cross-section at 160 mm from the sound source (c) Cross-section at 165 mm from the sound source

Figure 6 .
Figure 6.SPL distribution at the different cross-sections for the acoustic wave of 250 Hz.

Figure 6 .
Figure 6.SPL distribution at the different cross-sections for the acoustic wave of 250 Hz.
(a) Cross-section at 160 mm from the sound source (b) Cross-section at 165 mm from the sound source (c) Cross-section at 170 mm from the sound source

Figure 7 .
Figure 7. SPL distribution at the different cross-sections for the acoustic wave of 350 Hz.

Figure 7 .
Figure 7. SPL distribution at the different cross-sections for the acoustic wave of 350 Hz.

Figure 9 .
Figure 9. Variations of the existing range of the high-order waves with acoustic wave frequency for different values of tire speed.

Figure 10 .
Figure 10.Range of the high-order waves when summing the effects of forward/backward traveling waves.

Figure 9 .
Figure 9. Variations of the existing range of the high-order waves with acoustic wave frequency for different values of tire speed.

Figure 9 .
Figure 9. Variations of the existing range of the high-order waves with acoustic wave frequency different values of tire speed.

Figure 10 .
Figure 10.Range of the high-order waves when summing the effects of forward/backward travel waves.

Figure 10 .
Figure 10.Range of the high-order waves when summing the effects of forward/backward traveling waves.

Table 1 .
Maximum and average SPL at different cross-sections in Figure6.

Table 1 .
Maximum and average SPL at different cross-sections in Figure6.

Table 2 .
Maximum and average SPL at different cross-sections in Figure7.

Table 3 .
Basic parameters and related frequencies of four gas media.

Table 2 .
Maximum and average SPL at different cross-sections in Figure7.

Table 3 .
Basic parameters and related frequencies of four gas media.