PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Cellular automata modeling of propagation and absorption ofacoustic waves in impedance tube
Meng WANG1; Bo ZHANG1,2 Qiqi CHEN1; Liheng WANG1; Yutian BAI1
12

School of Mechanical Engineering, Ningxia University, China

CAE Key Laboratory for Intelligent Equipment of Ningxia, China

ABSTRACT
A one-dimensional Cellular Automata (CA) model for acoustic wave’s propagation and absorption inimpedance tube with and without porous metals is studied to explore acoustic absorbing properties of porous
metal in multiple physical fields. In cellular space, wave motion equation for small amplitude travellingwaves at room temperatures and normal sound pressure levels is reconstructed by using finite differential
method; accordingly, local evolutionary rules of each cell are deduced out. Provided that a specific soundsource is definitely given, the sound pressure distribution inside impedance tube can be obtained by solving
this CA model. Based on above CA theory and classical acoustic models of porous materials, theevolutionary rules of cell for the case that porous material is placed at the end of impedance tube areformulated. And sound absorption coefficient of porous materials may be calculated as well. Finally, theeffects of temperature and sound pressure level on nonlinear propagation properties of acoustic travellingwaves with finite amplitude are analysed by rewriting Westervelt wave motion equation in the form of finite
differential. In this way, one may predict sound absorption of porous material at different temperatures andsound pressure levels. The results from CA model agree well with experimental data.
Keywords: Cellular Automata, Sound Pressure Distribution, MATLAB

1. INTRODUCTION
Porous metal materials are widely used in extreme environments such as high temperature, highpressure, and strong air flow. The interior of the material is composed of metal skeleton and many
voids, which can effectively absorb sound energy and reduce noise. Porous metal materials have aseries of excellent properties such as high specific strength, high specific stiffness and good corrosion
resistance. Therefore, porous metal materials can be used a sound absorbing material in aircraft engine
cabin (1). Noise is essentially a mechanical wave, and as it propagates in high temperature orimperfect media, energy transformation may occur to attenuate the sound waves and achieve soundabsorption. However, when we study the problem of sound transmission in extreme environment, we
often need to solve high-order differential equation. At the same time, the three theoretical models forstudying the internal structure and sound absorption properties of porous materials also involve manyfactors affecting the structural complexity of material, which all make the research difficult. Jiuhui Wuet al.used transfer-function method to measure the sound absorption coefficient of materials at
different temperatures (2). Bo Zhang et al. used wave equation method to study the sound absorption
properties of porous metal materials at high temperatures (3). However, their results are not in good
agreement with the experimental results.
Cellular Automata (CA) as a numerical algorithm is very suitable for the simulation of acomplicated physical system. It is based on the principle of local action. It is a process of advancing
and adjusting repeatedly to reach the system balance (4).

2. ACOUSTIC MODELING OF CELLULAR AUTOMATA
We take the impedance tube as the research object, and place the impedance tube in two dimensions,and divide it into cells according by the time step and space step. There are four different types of cells.
1
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zhangb@nxu.edu.cn
12712580@qq.com

The cells in the first column belong to sound source cells, and the source excitation is a given plane
wave; note that the state variables of the source remain constant. The neighboring cell is the air
medium cell, which is next to the sound source. Cell state will be affected by the sound source at theinitial time, the sound absorbing medium cell of porous metal material is followed by air medium cell.
The number of the cell is determined according to the material thickness. The cells in the last column
stand for rigid media. There are different local transfer rules between cells at different locations. The
state variables of each cell in different time steps can be obtained by MATLAB programming.
2.1 Rules in Air MediaWe assume that the waves in impedance tube do not contain non-planar wave components, and allsound waves are planar waves. Therefore, there is no reflection at the upper and lower boundaries ofeach column.
The space step is 1 mm and the impedance tube length is set to be 1m, so there are 1002 cells in total
in the model. The first and 1002 th cells stand for sound source and rigid wall, respectively, and thesecond to 1001 cells are sound medium parts. We define that each cell has two state variables: excess
sound pressure (i.e. sound pressure) and change of sound pressure. At normal pressure and temperature,
we use the one-dimensional wave equation of small amplitude sound wave.
w2 p

1 w2 p2c 02 wt 2wx(1)

By algebraic transformation, as follows,p( x  dx, t )  2 p( x, t )  p( x  dx, t )dx

1 p( x, t  dt )  2 p( x, t )  p( x, t  dt )
dx

c02

(2)

the local evolutionary rules of sound wave propagation in air medium can be obtained:
dt 2 p ix  1, jt  2 p ix, jt  p ix  1, jt- p ix, jt  1  2 p ix, jt
dx 2
The first cell on the left is the sound source cell, so the plane wave is :p ix, jt  1S

pP0 ej˄Zt  ˅
2(3)

(4)

Starting from step 1002, standing wave arises due to the superposition of incident and reflectedsound pressures in tube, which is expressed as:(5)p 2 p  cos˄kx˅e jtZ
Acoustic pressure distribution of the standing wave in 2000Hz
2ta=2800
ta=3000
ta=3400
ta=3800

The total sound pressure(pa)

1.5
1
0.5
0-0.5
-1-1.5-20100

200

300

400
500
600
700
The location of the cell

800

900

1000

Figure 1 – Sound Pressure Distribution
It can be seen from the figure that the position of the standing wave crest and trough does notchange with time, and the maximum value of the amplitude is 2Pa and the minimum value is -2Pa. The
result is reasonable.
We further add some porous sound-absorbing materials, and the flow resistance of the materials canbe determined by the definition below,

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σ0Δp
νφd(6)

ǡᶭ corresponds to the variation of sound pressure, and then deduces as follow:

p ix, jt -p ix, ji  1
νφdσ0(7)The local rules of the impedance tube having sound absorbing medium can be obtained also:
p x, t  1

VI1
Zc]
p( x, t )  'p( x, t  1)[VI1Zc(8)It can be seen from the formula that the sound pressure at this moment is related to the properties ofthe material itself and the sound pressure at the cell at the previous moment. Furthermore, we need tocalculate the characteristic impedance of metal materials. There are three sound absorption models:
empirical model (D-B model), phenomenological model (J-C-A model) and flow around model. In thiswork, we use phenomenological models to calculate sound absorbing of porous metal.
2.2 CA Simulation Based on J-C-A Phenomenological Model
J-C-A model needs not only two parameters of porosity and flow resistance, but also other fiveparameters such as tortuosity factor, thermal characteristic length and viscous characteristic length.
The bulk modulus Keff and effective density ² eff are given in the model (5):K

 1ºª
§
·«
¨iZL2th B 2 ρ ¸ »8η0«¸ »γP γ  γ  1 ¨1 10«16η¨¸ »iZL2 B 2 ρ¨¸ »«
th0©
¹
¬¼eff

ρeffªºσ φ4iZτ 2 ηU 0 »1τρ ««1  0»
0iZρ 0 τ
σ 2 L 2I 2 »«¬0 v¼-1(9)(10)

In Eqn. (9) and Eqn. (10), ² 0 is the static density of air,¨is the viscosity coefficient of air, ¤isthe specific heats of air , B is the Prandtl constant, ¶is porosity, ³ 0 is flow resistivity, Lv is viscouscharacteristic constant, L th is thermal characteristic constant. Characteristic impedance of material is:ZcK eff  U eff(11)It is noticed that in this paper, the sound absorption coefficient of materials is evaluated by using
standing wave method:P maxDP
1 - ( minP maxP min-1)2(12)1

In the formula, we need to get the maximum and minimum of sound pressure. Since we haveobtained the distribution of the sound pressure, we can easily extract the maximum and minimum ofthat to get the sound absorption coefficient.

3. MODEL OPTIMIZATION AT HIGH TEMPERATURE
In a complicated high environment, in theory, the internal structure of porous materials may be
modified accordingly, thus some structural or acoustic parameters more or less change with theincrease of temperature. Considering non-uniform high temperature fields, one way use thefollowing wave equation in this subsection (2).

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d2p2

 2D (O , O t )dp2
 k0 O, O p
dz

0(13)

J  J  1 F Ot

(14)

dz
And two functions in Eqn. (13) are as follows:

k 0 O , Ot

2

DfZ2

1c FO
20

1 ª dF O dO 1 ªD f F Ot  B 2 F O º º 1 dT0
 «(15)«» » 0.5
1  B2F O ¬ dO dT0 T0 ¬
¼ ¼ D f dz
In Eqn. (15), dT 0/dz is temperature gradient and if the temperature is constant, its value is equal
to zero. Local rules with temperature gradient and without temperature gradient can be gained bydifferential treatment of the equation.

D O , Ot

0.5

2  dz 2 k 0 O , Ot

p z 1p z 1

2p z  p z 1

2  2D O, Ot dz  dz 2 k 0 O, Ot 2 p z  2D O, Ot dz  1 p z  1

The relationship between acoustic constants and temperature is as follows˖
U 0 T1U1 T0

V (T ) V (Tc )(T / Tc )3/ 2 (Tc  Ts ) /(T  Ts )B

20.5658T

0.04

(16)
(17)

(18)
(19)
(20)

Among them, ² 0 is the air density at room temperature,² 1 is the air density at high temperature,T 0 is the absolute temperature at room temperature, T1 is the absolute temperature at hightemperature. In Eqn. (18), Tc=300 K, and Ts=110.4 K.

4. MODEL OPTIMIZATION UNDER HIGH SOUND PRESSURE
When the sound pressure rises to a certain value, particle velocity is no longer a negligible
quantity compared to the sound velocity, so the problem at this time belongs to nonlinear acoustics.
In non-linear acoustics, the trace in the equation will be retained to the second order trace andWestervelt equation is considered as:
E w2 p2
1 w2 p G w3 p
2 p 2
0(21)U 0 c 04 wt 2c 0 wt 2 c 0 wt 3
After rewriting each term of the equation above by using the difference method , the cell soundpressure value at a certain position at time t+1 can be obtained˖
p t 12º1 ª 2 dt
p t , x  1  2 p t , x  p t , x  1  P 2 p t , x  P3 p t  1, x » «c 02
P1 «¬ dx»¼P4>34 p t  2, x  24 p t  3, x  8 p t  4, x  p t  5, x @P4E
2Ep t, x p t  1, x
2U0c0
U 0 c 02

(23)

3G
6E
4E

p t, x p t  1, x22
U 0 c 02c0 dt U 0 c 0(24)23G
2c02 dt(25)P1 1 
P22(22)

P3 1 2987

P4G(26)

2c02 dt

With the increase of sound pressure, not only the wave equation of sound wave transmission
changes, but the sound absorption property of materials does, the most obvious of which is thechange of material flow resistance. In the case of finite amplitude, t he flow resistance rate no longer
obeys Darcy's linear law. We adopt Hongyuan Jiang’s formula (6) for calculating flow resistance.
1.42(1  I )V V0 
U0 u(27)
2I 3 daIn Eqn. (26), ³ 0,¶ , da, ² 0 , u are static flow resistivity, porosity, pore size, equivalent density,and particle velocity. Starting from the one-dimensional non-linear motion equation, one may get theone-dimensional non-linear continuity equation of porous materials at high sound pressure levels in
the following form:w 2Xwx

2Z2

U eff
KX  jZ

8P 0XX
3SK(28)

Figure 2 – Sound Pressure Distribution
The Fig. 2 shows the distribution of material flow resistance with respect to incident sound pressurelevel at frequencies of 1000Hz, 2000Hz and 3000Hz. It can be seen that the material flow resistanceincreases nonlinearly with the increase of sound pressure level.

5. SIMULATION RESULTS AND ANALYSIS
Based on the theory in previous sections, we employ J-C-A model to conduct a morecomprehensive simulation analysis of the system. Considering the changes of some parameters ofmaterials relative to those at room temperature, the sound absorption coefficient of the system issimulated at high temperature and high sound pressure.
Table (1) gives the physical and acoustical parameters of samples under consideration. The Cellular
Automaton model based on J-C-A model is used to simulate the sound absorption coefficient o f thematerial at room temperature; the result is shown in Fig. 3.
Table 1 – Physical and acoustical parameters of samples
Flow
Materials

Porosity

Thickness,

Tortuosity

4mm

factorresistance,
N.s/m

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ThermalViscouslength,length,μmμm

Sample 1

90.89

20610

25

1.664

123.19289.54

Sample 290

2100

30

1.305

206.10210.20Sample 380

3095

30

1.170

116.66313.34

Sample 490

1993

20

1.380

210.73215.36

Sample 5802140

30

1.324

242.85325.96

10.9

Absorption coefficient0.80.70.6
0.5
0.4
0.30.2cell
analyze

0.10500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 3 –Sound Absorption Coefficient Curves
Figure 4 shows the sound pressure level distribution and sound absorption coefficient curves ofporous sample 1 at 100,200,300 degrees Celsius, respectively, in literature (2).
30Sound Pressure Level

20100℃
200℃
300℃

100-10-20-30

0100

200300400500the Location of the Cell

600700800(a) Sound Pressure Level Distribution(b) Sound Absorption Coefficient Curves at high temperatures
Figure 4 –Simulation Result at Different Temperatures
It can be seen that the sound absorption performance of the material decreases with the increaseof temperature, because the flow resistance of the material increases further with the increase oftemperature. And, it can be seen that the error from cellular automata model is quite large, thepossible reason of which is from the choice of step size in simulations and the error of experiment
itself.
4000

The total sound pressure(pa)

3000

2000

1000

0-1000

-2000

-3000

0204060

80

100

120

140

160

180

200

The location of the cellFigure 5 – Sound Pressure Distribution

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10.9

Absorption coefficient

0.8
0.7
0.6
0.5
0.4
0.3
SPL=113.9dB
SPL=153.9dB
SPL=160dB

0.2
0.1
00500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 6 – Sound Absorption Coefficient Distribution
Figure 5 shows the sound pressure distribution in impedance tube, roughly a standing wave with
an amplitude of 2000 Pa and a frequency of 1500Hz. Because of the nonlinear distortion, thewaveform inside impedance tube is no longer a theoretical sinusoidal wave, but more than a shockwave finally. Fig. 6 shows the distribution of sound absorption coefficients of sample# 1 withamplitudes of sound source up to 1000Pa, 2000Pa, and 3000Pa, respectively.
0.8

Absorption coefficient

0.7
0.60.50.40.3
CA
Text

0.2
0.1

0500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 7 – Sound Absorption Coefficient Curves of Sample 21Absorption coefficient

0.8

0.60.4

CA
Text
Text

0.2

00500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 8 – Sound Absorption Coefficient Curves of Sample 3
0.8

Absorption coefficient

0.7
0.60.50.4
0.30.2CA
Text

0.100500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 9 – Sound Absorption Coefficient Curves of Sample 42990

0.8Absorption coefficient

0.7
0.6
0.5
0.4
0.3
0.2
CA
Text

0.1
00500

1000

1500

2000 2500 3000
Frequency /Hz

3500

4000

4500

5000

Figure 10 – Sound Absorption Coefficient Curves of Sample 5Figures 7 to 10 show that by introducing the parameters in Table 1 into the presented Cellular
Automata model, the sound absorption coefficients of materials at 125dB can be obtained. It can befound out that as a whole, the simulated and experimental results fit quite well, but there still exists
relatively large errors at some frequencies.
0.8

absorption coefficient

0.7
0.6
0.5
0.4
100dB
105dB
115dB
120dB
125dB

0.3
0.2
0.1

0500

1000

1500

2000 2500 3000
Frequency (Hz)

3500

4000

4500

5000

Figure 11 – Sound Absorption Coefficient Curves of Sample 5 at Different Sound Pressure Levels

6. CONCLUSIONSIn this work, the acoustic model of one-dimensional Cellular Automata is built up by modifying
linear acoustic wave equation for different environment conditions. It can be seen that, normally, as thetemperature increases, the viscosity and the density of air change accordingly, which makes the soundabsorption coefficient of material decrease at high temperature. According to the experiment al andsimulation results, the change of sound absorption coefficient is not obvious as the sound pressurelevel increases. To our knowledge, the tendency above may be due to that the strength of sound sourcein simulation or experiment is not large enough to apparently affect sound absorption of materials.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge supports for this work from the Graduate Innovative
Educational Project of Ningxia (Grant No. YKC201606), Research Project of CAE Key Laboratoryfor Intelligent Equipment of Ningxia, Project of National Natural Science Foundation of China(Grant No. 51365046), and Graduate Innovative Project of Ningxia University (Grant No.
GIP2018055).

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