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Modeling perforates in mufflers using two-ports

Article  in  Journal of vibration and acoustics · October 2010

DOI: 10.1115/1.4001510

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 T. Elnady
ASU Sound & Vibration Laboratory,
Faculty of Engineering,
Ain Shams University,
Modeling perforates in mufflers
1 Elsarayat Street, Abbaseya,
11517 Cairo, Egypt
e-mail: tamer@asugards.net
using two-ports
One of the main sources of noise of a vehicle is the engine where its noise is usually
M. Åbom damped by means of acoustic mufflers. A very common problem in the modeling of
The Marcus Wallenberg Laboratory for Sound automotive mufflers is that of two flow ducts coupled through a perforate. A new seg-
and Vibration Research, mentation approach is developed here based on two-port analysis techniques, in order to
AVE, model perforated pipes using general two-port codes, which are widely available. Ex-
KTH, amples are given for simple muffler configurations and the convergence of the technique
SE-10044 Stockholm, Sweden is investigated based on the number of segments used. The results are compared with
closed form solutions form the literature. Finally, an analysis of a complicated multi-
S. Allam chamber perforated muffler system is presented. The two-port simulation results show
Department of Automotive Technology, good agreement with both the measurements, and the simulations using the classical
Faculty of Industrial Education, four-port elements. 关DOI: 10.1115/1.4001510兴
Helwan University,
Elsawah Street-Elkoba,
11282 Cairo, Egypt

1 Introduction so that his results match Sullivan experiments. It was unclear how
he determined this end correction and on what basis he includes it
Perforated tubes are commonly found inside automotive muf- or not.
flers. They are used to confine the mean flow in order to reduce The distributed approach is mainly convenient for relatively
the back-pressure to the engine and the flow generated noise in- simple perforated mufflers. There are, however, many complicated
side the muffler. Ideally, perforates are acoustically transparent muffler configurations in which perforations are used in nonstand-
and permit acoustic coupling to an outer cavity acting as a muffler. ard ways. The discrete approach is more convenient to analyze
Being able to theoretically model these mufflers enables car advanced muffler systems because of numerical simplicity and
manufacturers to optimize their performance and increase their flexibility. It is also straightforward to model gradients in mean
efficiency in attenuating engine noise. Therefore, there has been a flow and temperature using this approach, and as demonstrated in
lot of interest to model the acoustics of two ducts coupled through this paper, arbitrary complex perforated systems can be also
a perforated plate or tube. Generally, the modeling techniques can handled.
be divided into two main groups, the distributed parameter ap- A new version of the segmentation approach was developed. It
proach and the discrete or segmentation approach. uses a two-port transfer matrix to model both the perforated
In the distributed approach 关1–7兴, the perforated tube is seen as branches and the intermediate hard pipe segments. A convergence
test is presented for the new method based on a simple perforated
a continuous object, and the local pressure difference over the
muffler configuration. This new technique is more flexible as it
tube is related to the normal particle velocity via surface averaged
facilitates the modeling of perforated pipes with an arbitrary con-
wall impedance. The main challenge facing this approach is the figuration within a muffler. The calculations were done using an
decoupling of the equations on each side of the perforate. in-house two-port code 共SIDLAB兲 for modeling low frequency
The discrete or segmentation approach was first developed by sound propagation in complex duct networks. The new approach
Sullivan 关8,9兴. In this approach, the coupling of the perforate is was also applied to the complicated multichamber muffler.
divided into several discrete coupling points with straight hard
pipes in-between. Each segment consists of two straight hard
2 Two-Port Segmentation Model
pipes and a coupling branch. The total 4 ⫻ 4 transmission matrix
of the perforated element is found from successive multiplication The basic element to be considered is two parallel flow ducts,
of the transmission matrices of each segment. Kergomard et al. which are joined together by a perforated section of length L and
关10兴 used this concept and presented, for the case of two specific acoustic impedance Zw. The impedance can vary along
waveguides communicating via single holes, a model for wave the perforate. The main idea of the segmentation approach is to
transmission in a periodic system. Dokumaci 关11兴 presented an- divide the perforated section into a number of segments along the
length of the ducts. Each segment consists of two parts. The first
other discrete approach based on the scattering matrix formula-
part is two parallel hard ducts where convective plane wave mo-
tion. He discussed several possibilities for modeling the connect-
tion is assumed. The second part is discrete impedance, which can
ing branch. He proposed a continuous viscothermal pipe model, a be seen as an open branch representing a parallel coupling of all
continuous isentropic pipe model with end corrections, and a the holes in the segment.
lumped impedance model 共as in Sullivan兲. The conclusion was Consider a short segment ⌬x of a perforated wall as in Fig. 1,
that the lumped element model is the most appropriate for this the pressure and normal volume velocity on each side are related
problem. He sometimes added a correction to the segment length by
U1 = U2 共1兲
Contributed by the Technical Committee on Vibration and Sound of ASME for
publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 6, p1 − p2
2009; final manuscript received March 4, 2010; published online October 8, 2010. =Z 共2兲
Assoc. Editor: Jeffrey Vipperman. U2

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x/2
x/2
U1
p2

p1

x = L / (N
1)

x
L
Fig. 1 A segment of the perforated wall
Fig. 3 Location of the lumped elements. Note: A lumped ele-
ment is put at the beginning and end. From the figure, it follows
that ⌬x = L / „N − 1…, where N is the number of lumped elements.
where p is the acoustic pressure, U is the volume velocity incident
on the perforated section, and Z = Zw / Sw is the segment impedance
共Sw is the wall area of the perforated segment兲, 1 denotes the inlet
side, and 2 denotes the outlet side. Only plane waves are assumed lumped elements will be placed as in Fig. 3. The area of each
to propagate on each side of the perforate and parallel to it. element will be the lateral surface area of the corresponding seg-
A code to analyze low frequency sound propagation in complex ment 共␲DLi兲, where Li is equal to the segment length ⌬x except at
duct networks 关12兴 called SIDLAB has been developed at KTH. It is the edge elements where it is equal to ⌬x / 2, and D is the diameter
based on the representation of a duct network as a network of two of the perforated pipe.
ports. The two-port elements are then joined and analyzed using
the method described in Ref. 关13兴. The present version of SIDLAB 2.1 Examples for Different Configurations. Due to the flex-
couples the elements at each node using the continuity of pressure ibility of using two-ports, it is easy to set up a network of any
and volume velocity. In order to model perforated tubes using this configuration involving two or more pipes connected through a
scheme, they are divided into discrete lumped impedance ele- perforated section. A few examples are given in Table 1 as a guide
ments separated by hard segments on both sides of the perforate in to set up muffler networks including perforations. As mentioned
a similar way to that suggested by Sullivan 关8兴. The main differ- earlier, each perforated tube section is divided into lumped ele-
ence here is that the perforated tube is described by a number of ments 共shaded rectangles兲 separated by hard-walled straight pipes
two-port elements instead of the four-port elements used by Sul- 共unshaded rectangles兲. A split into three lumped elements 共two
livan 关8兴. The representation in the form of two-ports is attractive segments兲 is used in these examples.
since two-port codes are commonly used for muffler analysis, and
2.2 Convergence Test. A critical parameter to the success of
the proposed method makes it possible to model arbitrary com-
the segmentation technique is to divide the perforated tube into a
plex perforated systems.
sufficient number of segments. A test case for a simple perforated
Assuming that the perforated section ⌬x is much shorter than
resonator muffler was chosen to test the convergence of the ap-
the acoustic wavelength ␭, then it can be represented as a lumped
proach 共Fig. 4兲. The exact solution was taken from the closed
element as in Fig. 2. The two-port of such an element is found by
form solution presented by Sullivan and Crocker 关1兴 for the case
rearranging Eqs. 共1兲 and 共2兲 in matrix form to give

冋 册 冋 册冋 册
without flow. The test case was designed to have similar dimen-
p1 1 Z p2
= · 共3兲
U1 0 1 U2
The perforated tube is divided into several segments and the per- Table 1 Examples of possible perforated tube configurations
forate effect is lumped at discrete points separated by hard pipes. and the associated two-port network
The number of segments must be large enough to obtain accept-
able accuracy. The length of each segment should be much
smaller than the wavelength 共say, less than ␭ / 8兲. To determine the
Tube

number of necessary lumped elements, N, we can use the follow-

ing criteria:
L ␭
⌬x = ⱕ 共4兲
Associated Network

N−1 r
where r is the required resolution, and L is the length of the
perforated section of the tube. The wavelength will be determined
by the maximum frequency of interest, normally determined by
the plane wave 共one-dimensional 共1D兲兲 limitation. In terms of
frequency, the relation becomes
L·r·f
Nⱖ1+ 共5兲
c
N should be rounded off to the nearest larger integer. It is better to
put one lumped element at the beginning and one at the end of the
perforate in order to model the lengths of any attached devices
accurately. For a perforated section divided into N segments, the
144 57

pin pout
Uin Uout
144

Fig. 2 The representation of a two-port element relating two

pairs of state variables „p and U… Fig. 4 The resonator muffler test case

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 (a) N = 2 (b) N = 3

(c) N = 4 (d) N = 5

Fig. 5 Different networks for different cases

sions to the longest perforated tube in the automotive muffler

which will be considered later in this paper. Different cases with Fig. 7 Convergence analysis for the short resonator. d1 = 50,
d2 = 75, and L = 65 mm.
different numbers of segments, N, were considered. The SIDLAB
network for each case is shown in Fig. 5.
The result for this convergence test is shown in Fig. 6. The
solid line is the exact solution presented in Ref. 关1兴. The frequency
range of interest for the advanced muffler considered later is its Bauer 关16兴. The resistance consists of four terms due to viscous
plane wave region, which is limited by 566 Hz. To cover this losses inside the hole, radiation resistance to the vibrating piston
region, it is sufficient to divide the perforated tube into three seg- of air inside the orifice, grazing flow term, and bias flow term. The
ments. This agrees with the criteria given in Eq. 共5兲, which re- reactance is assumed to be only caused by the mass plug and any
flow effects on the reactance are neglected here. The proposed
quires N to be bigger than 2.9 共with r = 8兲 for this frequency range.
model can be summarized in the following equations for the nor-
Another test case was considered to demonstrate the ability of the
malized perforate resistance and reactance, respectively:

再 冋 册冎 冋 册
present approach to handle high frequencies. A so called “short
resonator” was designed to match one of the cases presented in jk t ␦re 1 2J1共kd兲 0.3
Ref. 关1兴. The result is shown in Fig. 7. In this case, at least seven ␪ = Re + f int + 1− + Mg
␴CD F共␮⬘兲 F共␮兲 ␴ kd ␴
segments are required to capture the correct behavior up to 3500
Hz. 1.15
+ Mb 共6兲
2.2.1 The Used Perforate Impedance Model. The key design ␴CD
parameter of perforates is the acoustic impedance. The impedance
is what determines their efficiency to absorb sound waves. Several
studies have been conducted to develop impedance models, in-
再 冋
␹ = Im
jk t
+
0.5d
␴CD F共␮⬘兲 F共␮兲
f int 册冎 共7兲
cluding the effect of different configurations and surrounding con- where t is orifice thickness, d is the orifice diameter, ␴ is the
ditions. In spite of the large number of published research, a single porosity, k is the wavenumber ␻ / c, c is the speed of sound, CD is
verified global model does not exist. The objective of an earlier the orifice discharge coefficient, J is the Bessel function, ␷
work done at KTH 关14兴 was to review all available models in the = ␮ / ␳0 is the kinematic viscosity, ␳0 is the fluid density, ␮ is the
literature and include all effects in a single global model. A model
adiabatic dynamic viscosity, ␮⬘ = 2.179 ␮, M g is the grazing flow
based on Crandall’s theory 关15兴 was developed for the no flow
Mach number, and M b is the bias flow Mach number inside the
case. Some parameters were included from later models, which
holes of the perforate. The rest of the parameters are defined as
were not in the original model. Other parameters were determined
follows:
from experiments and semi-empirical formulas were derived.
Measurements were performed to verify the model. The grazing
and bias flow impedance values were calculated according to K= 冑 −
j␻
␷
, K⬘ = 冑−
j␻
␷⬘
共8兲

4J1共Kd/2兲
F共Kd兲 = 1 − 共9兲
Kd · J0共Kd/2兲

␦re = 0.2d + 200d2 + 16000d3 共10兲

f int = 1 – 1.47冑␴ + 0.47冑␴3 共11兲

Equation 共11兲 is a correction factor for the orifice interaction ef-
fects.

3 Example of a Realistic Muffler System

3.1 Muffler Description. To test if the coupling condition is
of importance for more realistic automotive mufflers, the system
shown in Fig. 8 has been studied. The details of the baffles 共3–6兲
are shown in Figs. 9–11. This muffler was manufactured for re-
search purposes and was built to resemble an automotive producer
muffler but with a modular and simplified setup. The inlet duct 共1兲
has 45 mm internal diameter and 1.5 mm thickness. The outlet
Fig. 6 Convergence analysis for the test case „long resona- pipe 共2兲 has 57 mm internal diameter and 1.5 mm thickness. A
tor…. d1 = 57, d2 = 144, and L = 144 mm. stainless steel sleeve 共7兲 of length 180 mm is located above the

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 89

1 R1
R2

3 4 5 R6 R5 R4
7 6
R9 R8
2 R7
R3

y
Fig. 12 Flow resistances inside the muffler considered as 1D
x 227 130 100 100 162 flow network. The pressure drop or flow loss across each re-
1.5 sistance is ⌬p = RQ円Q円, where Q is the volume flow.

Fig. 8 Sketch of the muffler internal scheme „dimensions are

in mm…. The encircled numbers refer to the inlet/outlet pipe „1–
2…, the four baffles „3–6… and the sleeve „7…. When analyzing the low frequency propagation in muffler sys-
tems, 1D models are usually used and only plane waves are as-
sumed to propagate in all elements. The direction of propagation
is normally chosen in the direction of the longest cross dimension
288 holes perforated area in outlet duct. It has a density of
of the muffler. The x-direction is therefore chosen as the direction
340 kg/ m3 and a weight of 76 g.
of propagation inside the muffler shown in Fig. 8 except between
It is worth noting that the following analysis is limited to low
baffle 6 and the right-hand side end plate. For this section, the
frequencies where only plane waves propagate throughout the sys-
y-direction is chosen as the propagation direction since the longest
tem and the problem is considered 1D. For the muffler under
dimension is in the y-direction. The holes in baffle 6 共Fig. 11兲 are
consideration, the width is about 30 cm and the first cross mode
also grouped at the opposite side of the cavity compared with the
will cut-on when the wavelength is approximately double this
inlet duct indicating that sound propagation will mainly take place
distance, which happens at around 566 Hz 共f = c / ␭ = 340/ 0.6兲. A
in the y-direction.
possible continuation to this work would be to model higher order
mode effects. But the complex geometry will then imply the use 3.2 Flow Distribution Calculation. When flow is introduced
of finite element or boundary element methods. through the muffler, its transmission properties are affected in
three ways. The first is through the convective effects, which af-
fect the propagation inside straight pipes. This effect was ac-
counted for in the formulation of the transfer matrices for different
pipe elements. The second is the introduction of extra losses at the
area expansion, which takes place at the end of the inlet pipe. The
third and most important effect of flow is introduced by the
change of the perforate impedance. The flow can be either grazing
to the perforate, through the perforate, or both. The bigger effect
comes from the flow through the perforate, which increases the
resistance considerably.
In order to simulate the acoustic properties of the muffler with
flow, it is crucial to know the flow distribution through different
paths in order to correctly estimate the change in the impedance of
Fig. 9 A detailed drawing of the perforated baffle 3 „50 holes… perforates and to model the convective effects. The muffler is
modeled as an electric circuit. The pressure drop resembles the
voltage drop, the volume flow resembles the electric current, and
the flow resistance resembles the electric resistance. There are two
types of flow resistances in this muffler. The first is the resistance
across pipe open ends. The second is the resistance across perfo-
rate elements. Figure 12 shows these flow resistances in the muf-
fler. R1 and R7 represent the open end flow resistances of the inlet
and outlet pipes, respectively. R3, R4, R5, and R6 represent the
perforated baffles flow resistances of the baffles 6, 5, 4, and 3 in
Fig. 8, respectively. R2 represent the tube perforation flow resis-
tance in the inlet pipe. R8 and R9 represent the tube perforation
Fig. 10 A detailed drawing of the perforated baffles 4 and 5 „72 flow resistance in the outlet pipe. R9 consists of two parts, one due
holes… to the perforations and the other due to the sleeve 共item 7 in Fig.
8兲.
3.2.1 Equivalent Electric Circuit. Figure 13 shows the equiva-
lent electric circuit of the muffler, which was used to calculate the
flow distribution. Q0 is the input volume flow, which is usually
measured outside the muffler. It is related to the Mach number by
Q0 = u0S0 = 共Mc兲S0 = 0.541 M m3/s 共12兲
where c = 340 m / s for ambient temperature and pressure, and S0
= 0.001591 m2. Assuming incompressible flow, Kirchoff’s first
law can be applied. This implies that the volume flow into a junc-
tion 共or node兲 must equal the volume flow out of the junction. For
Fig. 11 A detailed drawing of the perforated baffle 6 „42 holes… the circuit above, this gives

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 Q0 Q1 Table 2 Open end flow resistance values „␳0 = 1.21 kg/ m2…
In R1
Pipe diameter Area Resistance
Q2 No. 共m兲 共m2兲 共kg/ m7兲
R2
Q7 Q5 Q3
R1 0.045 0.001591 23.72⫻ 104
R6 R5 R4 R3 R7 0.057 0.002552 4.61⫻ 104

R9 Q6 R8 R7 Q4

Q0
Out where S is the area of the pipe. The values of R1 and R7 are listed
in Table 2.
Fig. 13 The equivalent electric circuit The second type of flow resistance results from the pressure
drop across a perforate. This pressure drop is given by 关16兴
␳0
⌬p = ␳0c␪vu + ⬘
u兩u兩 = Rviscous Q + RperforateQ兩Q兩 共24兲
Q0 = Q1 + Q2 共13兲 2 ␴ 2C D
2

where u here is the velocity incident on the perforate, ␴ is the

Q1 = Q3 + Q4 共14兲 perforate porosity 共percentage open area兲, CD is the orifice dis-
charge coefficient, and ␪v is the perforate viscous resistance. For a
Q3 = Q5 + Q6 共15兲 perforate, the viscous part is usually neglected compared with the
nonlinear part of the resistance. For R9, the viscous part is in-
Q7 = Q2 + Q5 共16兲 cluded to account for the linear resistive part from the sleeve. The
where Qi is the volume flow in path i. Kirchoff’s second law perforate flow resistance Rperforate is thus given by
states that for any closed loop path around a circuit, the sum of the
␳0 ␳0
voltage gains and voltage drops equals zero. This was applied as Rperforate = = 共25兲
follows: 2 ␴ 2C D
2 2
S 2 2
2CD Sholes
R1Q1兩Q1兩 + R3Q1兩Q1兩 + R4Q3兩Q3兩 + R5Q5兩Q5兩 − R2Q2兩Q2兩 = 0 For the sleeve, R⬘ is obtained from the measurement and is given
by
共17兲
␳ 0c ␪ v
R7Q4兩Q4兩 − R8Q6兩Q6兩 − R4Q3兩Q3兩 = 0 共18兲 R⬘ = 共26兲
S
R8Q6兩Q6兩 − R5Q5兩Q5兩 − R6Q7兩Q7兩 − R9Q7兩Q7兩 − R9⬘Q7 = 0 R⬘ is the resistance of the metallic sleeve. Measurements of its
flow resistance showed that it is very low and therefore its effect
共19兲 is neglected in the following calculations. For an orifice of thick-
Equations 共13兲–共19兲 represent a set of nonlinear equations. This ness 1.5 mm and diameter 5 mm, the discharge coefficient is equal
system has seven equations and seven unknowns. The “fsolve” to 0.817. The values of the perforate flow resistances are listed in
built-in function in MATLAB was used to solve this system in the Table 3.
form of f共x兲 = 0, where 3.2.3 Numerical Results. The simulations were performed at

冤 冥
Q1 + Q2 − Q0 three flow speeds corresponding to Mach 0.1, 0.2, and 0.3, and
assuming air at standard temperature and pressure. The flow dis-
Q4 + Q3 − Q1 tribution inside the muffler is given in Table 4. If the sleeve resis-
Q5 + Q6 − Q3 tance is included, the equations have to be solved for each flow
f̄共Q̄兲 = Q2 + Q5 − Q7 speed, but we have neglected its resistance here; therefore, the
distribution becomes independent of the inlet flow spend.
共R1 + R3兲Q1兩Q1兩 + R4Q3兩Q3兩 + R5Q5兩Q5兩 − R2Q2兩Q2兩
R7Q4兩Q4兩 − R8Q6兩Q6兩 − R4Q3兩Q3兩
R8Q6兩Q6兩 − R5Q5兩Q5兩 − 共R6 + R9兲Q7兩Q7兩 − R9⬘Q7
Table 3 Perforate flow resistance values „␳0 = 1.21 kg/ m2…
共20兲
Area Resistance
and
No. Type No. of holes 共m2兲 共kg/ m7兲
Q̄ = 关Q1 Q2 Q3 Q4 Q5 Q6 Q7 兴⬘ 共21兲 R2 Tube 154 0.003023 9.831⫻ 104
R3 Baffle 42 0.000825 132.17⫻ 104
3.2.2 Calculation of the Resistances. As mentioned earlier, R4 Baffle 72 0.001413 44.976⫻ 104
there are two types of flow resistances. The first results from the R5 Baffle 72 0.001413 44.976⫻ 104
flow losses across an open end, either in the inlet or outlet ducts R6 Baffle 50 0.000982 93.263⫻ 104
connected to a large volume. This loss across an outlet duct is R8 Tube 144 0.002827 11.244⫻ 104
given by 关17兴 R9 Tube 288 0.005655 2.811⫻ 104
1
⌬p = ␧ 2 ␳0u兩u兩 = RopenQ兩Q兩 共22兲
Table 4 Flow distribution inside the muffler, in percentage of
where u is the flow speed in the pipe and ␧ is a constant, which is
Q0
equal to 1 for an outlet and 0.5 for an inlet 关18兴. The open end
flow resistance Ropen is therefore Q1 Q2 Q3 Q4 Q5 Q6 Q7
1 ␳0
Ropen = ␧ 2 共23兲 28.86 71.14 ⫺9.70 38.55 ⫺41.06 31.36 30.08
2S

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 06

01 04 05

F1 07

25 22 21 20 17 16 15 12 11 10 09 08

F3 F2

32 29 26

Quarter wavelenght resonator Straight hard pipe

Perforated baffle Segmented four-port element

Fig. 14 The breakdown of the muffler into a network of two- and four-ports

4 Classical Solution Using Four-Ports unknowns and 36 equations that can be written for the network in
Fig. 14. We will start by writing the equations of the nine two-
A breakdown of the muffler into two-port and four-port ele-
ports, which gives 18 equations:
ments is shown in Fig. 14. The coupling between the ducts on
both sides of the tube perforations are modeled by the generalized
segmentation four-port transmission model described in Sec. 2.
The four-port matrices are denoted by F1, F2, and F3. There are
冋 p2
U2
册 = 关T1兴 · 冋 册
p4共1兲
U4共1兲
共27兲

冋 册
two quarter wavelength resonators 共elements 6 and 25兲 at both
ends of the muffler. Each perforated baffle 共elements 8, 11, 16,
and 21兲 is modeled as a single lumped two-port element described
by Eq. 共3兲. All connecting pipes are modeled as two-port ele-
p4共2兲
U4共2兲
= 关T2兴 · 冋册p3
0
共28兲

冋 册 冋 册
ments. Element 5 represents the acoustic losses due to the sudden
expansion of flow at the open end of the inlet pipe, and is also p4共3兲 p5共3兲
= 关T3兴 · 共29兲
represented by a lumped two-port element. A further simplified U4共3兲 U5共3兲
network is shown in Fig. 15 where all in-line two-ports are
grouped together. These two-ports have the same area and the
same flow velocity; hence, the pressure and volume velocity are
continuous. A special coupling matrix needs to be applied when
冋 p5共4兲
U5共4兲
册 = 关T4兴 · 冋 册
p6
U6
共30兲

冋 册 冋 册
calculating the transfer matrix of the two-port T3 because ele-
ments 7 and 8 have different areas. The quarter wavelength reso- p7 p8
nators 共elements 6 and 25兲 are modeled as straight hard pipes = 关T5兴 · 共31兲
U7 U8
where the volume velocity at the dead end is assumed to be equal

冋 册 冋 册
to zero.
p9 p10
4.1 Muffler Analysis. The variables chosen for the analysis = 关T6兴 · 共32兲
U9 U10
of this network are the pressure and volume velocity at each node.

冋 册 冋 册
Pressure and volume velocity are continuous for all the nodes
shown in the network because they all have the same inlet and p11 p12
= 关T7兴 · 共33兲
outlet areas. The exception to this rule is the coupling at nodes 4 U11 0
and 5 where three elements are connected to each node. Continu-
ity of energy and mass flow is assumed to model these nodes.
Each of these nodes is described by three pressure and three vol-
ume velocity variables, i.e., represents a three-port. There are 38
冋 p5共8兲
U5共8兲
册 = 关T8兴 · 冋 册
p13
U13
共34兲

T2
1 2
T1 4

F1
12 11
10 9 8 7 6 5
T7 T6 T5 T4 T3

F3 F2

15 14 13
T9 T8
16

Fig. 15 The simplified muffler network, with positive flow directions defined

061010-6 / Vol. 132, DECEMBER 2010 Transactions of the ASME

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 冋 册 p14
U14
= 关T9兴 · 冋 册
p15
U15
共35兲
Table 5 The subelements of each two-port element in Fig. 8

Two-port No. 共T兲 Constituting elements

共j兲 共j兲
where pi and Ui are the pressure and volume velocity, respec- 1 Pipe 共4兲 and expansion 共5兲
tively, at node i from the side of element j. Next, we write the 2 Pipe 共6兲
equations for the three four-ports representing the perforated tube 3 Pipe 共7兲, coupling, baffle 共8兲, and pipe 共9兲
sections. The flow through the perforated tube was assumed to be 4 Pipe 共10兲, baffle 共11兲, and pipe 共12兲
divided equally for each branch. The three four-ports give us an- 5 Pipe 共15兲, baffle 共16兲, and pipe 共17兲
other 12 equations: 6 Pipe 共20兲, baffle 共21兲, and pipe 共22兲
7 Pipe 共25兲

冤 冥 冤 冥
p1 p2 8 Pipe 共26兲
9 Pipe 共29兲
U1 U2
= 关F1兴 · 共36兲
p9 p8
− U9 − U8

冋冉 冊 冏 冏册
冤冥 冤冥
2 2
p6 p7 1 + M in zout t12 zint22
TL = 10 log t11 + + zint21 +
U6 U7 1 + M out 4zin zout zout
= 关F2兴 · 共37兲
p13 p14 共45兲
U13 U14 where M in, M out, zin, and zout are the flow Mach numbers and the
characteristic impedances at the inlet and outlet, respectively.

冤冥 冤冥
p10 p11
4.2 Description of the Muffler Elements. The network
U10 U11 shown in Fig. 15 consists of either four-port or two-port elements.
= 关F3兴 · 共38兲
p15 p16 The two-ports are divided into four categories, straight hard pipes,
U15 U16 perforated baffles, area expansion, and coupling elements. The
details of the two-ports can be extracted from Fig. 14, and is
In Eq. 共36兲, the negative signs of U8 and U9 are included because summarized in Table 5. Each element type will be described in
the direction of propagation inside the four-port F1 is opposite to detail and all geometric parameters will be given in Secs.
the direction of propagation inside the two-ports T5 and T6, which 4.2.1–4.2.5.
are both connected to it. The remaining six equations are the cou-
pling conditions at nodes 4 and 5. Mass and energy conservation, 4.2.1 Area Expansion. At the end of the inlet pipe, there is a
respectively, imply the following relations between the pressure sudden area expansion, which introduces acoustic losses in the
and volume velocity at each node: presence of mean flow. This can be modeled as a lumped element
whose two-port transfer matrix is described by

冋 册 冋 册冋 册
U4共1兲 + M 1 p4共1兲/z1 = 共U4共2兲 + M 2 p4共2兲/z2兲 + 共U4共3兲 + M 3 p4共3兲/z3兲
pin 1 Z pout
共39兲 = · 共46兲
Uin 0 1 Uout
p4共1兲 + z1M 1U4共1兲 = p4共2兲 + z2M 2U4共2兲 = p4共3兲 + z3M 3U4共3兲 共40兲 The flow induced loss across the open end is

U5共3兲 + M 3 p5共3兲/z3 = 共U5共4兲 + M 4 p5共4兲/z4兲 + 共U5共8兲 + M 8 p5共8兲/z8兲 ⌬p = ␳0u2/2 共47兲

共41兲 The acoustic loss across the opening can be obtained by differen-
tiating Eq. 共47兲 and assuming linear acoustics. The acoustic im-
p5共3兲 + z3M 3U5共3兲 = p5共4兲 + z4M 4U5共4兲 = p5共8兲 + z8M 8U5共8兲 共42兲 pedance of the opening is defined as the ratio between the pres-
sure drop and the velocity.
As mentioned earlier, there are 36 equations and 38 unknowns. If
we assume that the pressure p1 and volume velocity U1 at the inlet d共⌬p兲
are known, all other variables can be determined. The 36 equa- Zo = = ␳ 0u 共48兲
du
tions form a complete set of linear equations, which can be ar-
ranged in matrix form where the system matrix has 36⫻ 36 ele- The resistance defined in Eq. 共46兲 is then given by
ments. The objective is to calculate the equivalent two-port ␳0cM
transfer matrix between the inlet and outlet Z= 共49兲

冋 册 冋 册冋 册
S
p1 t11 t12 p16
= · 共43兲 where S is the area of the opening. For element 5 in Fig. 14, S is
U1 t21 t22 U16 equal to 0.001591 m2.
In order for the four elements of this matrix to be fully deter-
4.2.2 Hard-Walled Pipes. The four-pole transfer matrix of
mined, another load or source condition needs to be solved for.
hard-walled pipes is given by 关18,19兴
The elements are then found by solving the following set of equa-

冤 冥
tions: 1 ik L −ik L ␳0c ik L −ik L
共e + + e − 兲 共e + − e − 兲

冤 冥冤 冥 冤 冥
pload1 Uload1 0 0 t11 pload1 2 2S
16 16 1 Thwp = 共50兲
0 0 pload1 Uload1 t12 Uload1 S 1 ik L −ik L
16 16
· =
1
共44兲 共eik+L − e−ik−L兲 共e + + e − 兲
pload2 Uload2 0 0 t21 pload2 2 ␳ 0c 2
16 16 1
0 0 pload2
16 Uload2
16 t22 Uload2
1
where S is the area of the pipe, L is the length of the pipe, ␳0 is the
density of air, and c is the speed of sound. k+ and k− are the axial
The transmission loss 共TL兲 is defined as the logarithmic ratio be- wavenumbers and are given by
tween the power incident toward the system and the power trans-
mitted in the case of a reflection free termination, and is given by k⫾ = k/共1 ⫾ M兲 共51兲

Journal of Vibration and Acoustics DECEMBER 2010, Vol. 132 / 061010-7

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 Table 6 The dimensions of the hard-walled pipe elements Table 8 The properties of the perforated tube elements „D is
the tube diameter, L is the perforated length, Nh is the number
L S of holes, and Sopen is the open area of the perforated tube…
Element No. 共m兲 共m2兲
Four-port D L
1 0.247 0.001591 No. 共m兲 共m兲 Nh Mg Mb
4 0.293+ 0.013 0.001591
6 0.082 0.011 F1 0.045 0.090 154 共Q0 + Q1兲 / 共2cS p兲 Q2 / 共cSopen兲
7 0.208 0.011 F2 0.057 0.072 144 共2Q4 + Q6兲 / 共2cS p兲 Q6 / 共cSopen兲
9 0.095 0.02097 F3 0.057 0.144 288 共2Q0 − Q7兲 / 共2cS p兲 Q7 / 共cSopen兲
10 0.005 0.01814
12 0.014 0.01814
15 0.014 0.01814
17 0.018 0.01814
20 0.022 0.01814 4.2.4 Perforated Baffles. All the perforated baffles are mod-
22 0.044 0.01814 eled as lumped elements as in Eq. 共3兲. All perforates 共also tube
25 0.027 0.01814 perforations兲 have 5 mm diameter holes with 9 mm spacing that
26 0.026+ 0.017 0.002552 gives 24% porosity. The thickness is 1.5 mm. The properties of
29 0.197 0.002552 the lumped elements representing the perforated baffles are listed
32 0.069 0.002552 in Table 7. Air flow, either grazing or bias to the perforate, has a
considerable effect on the value of the perforate impedance.
Therefore, the flow distribution inside the muffler should be cal-
culated in order to estimate the values of the flow at each perfo-
rate. M g and M b in Eq. 共6兲 for each baffle are listed in Table 7. All
where k = ␻ / c and M is the Mach number.
␴ in Eqs. 共6兲 and 共7兲 are the actual porosities, which are listed in
The dimensions of all the pipes in the muffler network are listed
Table 7 for the baffles. The interaction factor in Eq. 共11兲 depends
in Table 6. An end correction is added to the open ends in pipes
on the spacing between the holes, which is constant for all tube
共4兲 and 共26兲 and is equal to 0.6 of the radius.
perforations. The holes in the baffles are not uniformly distrib-
4.2.3 Coupling Element. When two two-ports are connected at uted; therefore, the interaction factor should be calculated based
a node, the pressure and volume velocity at the node are continu- on the spacing porosity not on the actual one.
ous if both elements have the same area connected to the node.
4.2.5 Perforated Tubes. All the tube perforations have the
This condition applies to all two-ports connected together except
same hole configuration. The difference between different perfo-
between pipe 共7兲 and baffle 共8兲. A coupling two-port element is
rated pipes is in the additional impedance caused by either grazing
needed to fulfill the general coupling conditions between them,
or bias flow effects because the flow Mach number is different at
see Appendix. The four-pole transfer matrix of such an element is
each four-port element. The properties of the elements represent-
given by

冋 册
ing the tube perforations are listed in Table 8. M g and M b in Eq.
pin 共6兲 for each perforated tube are also listed in Table 8. For simplic-
Uin ity, the flow is assumed to pass equally through all branches and

冋 册冋 册
therefore M b is the same for all of them. M g is taken from inside
1 1 − ␣ M inM outzin/zout ␣zoutM out − ␣zinM in pout the tube 共higher Mach number兲 and also assumed constant for all
= · branches and equal to the average value of the element inlet and
1 − ␣ M in
2
M out/zout − M in/zin 1 − ␣ M inM outzout/zin Uout
outlet Mach numbers.
共52兲
5 Application of the Two-Port Segmentation Model
A breakdown of the muffler shown in Fig. 8 into a two-port
Table 7 The properties of the perforated baffle elements
system with lumped elements describing the perforations is shown
Sb Sopen
in Fig. 16. The network has 41 elements and 33 nodes. All perfo-
Element Porosity
No. 共m2兲 共m2兲 共%兲 Mg Mb rated pipe sections were modeled using three lumped elements
according to the conclusion of the investigation of the conver-
8 0.02097 0.000825 3.9 0 Q1 / 共cSopen兲 gence. The perforated baffles are modeled as single lumped ele-
11 0.01814 0.001413 7.8 0 Q3 / 共cSopen兲 ments 共Nos. 8, 11, 16, and 21兲. All the elements have the same
16 0.01814 0.001413 7.8 0 Q5 / 共cSopen兲 properties and dimensions as in the previous analysis. The new
21 0.01814 0.000982 5.4 0 Q7 / 共cSopen兲 elements in this network are the tube perforation lumped elements

6
06
3
1 2 4 5
01 02 03 04 05 7

07
33

34

35

26 25 24 23 22 21 17 16 15 14 13 12 11 10 9
25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 09 08 8
20 19 18
41

40

39

38

37

36

32 31 30 29 28 27 26
33 32 31 30 29 28 27

Fig. 16 The muffler network model in SIDLAB. The circuit symbols are defined in Fig. 14
and Table 1

061010-8 / Vol. 132, DECEMBER 2010 Transactions of the ASME

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 Table 9 The dimensions of the hard-walled pipe elements 60

Measurements
L S Simulations
Four in sec
port simulations
50
Element No. 共m兲 共m2兲 SID Simulations
SIDLAB simulations

2 0.045 0.001591

Transmission loss, dB
40
3 0.045 0.001591
13 0.036 0.01814
14 0.036 0.01814 30
18 0.045 0.01814
19 0.045 0.01814
23 0.072 0.01814 20
24 0.072 0.01814
27 0.036 0.002552
10
28 0.036 0.002552
30 0.072 0.002552
31 0.072 0.002552 0
0 100 200 300 400 500 600
Frequency, Hz

and the straight hard pipes between them. The dimensions of the Fig. 18 Comparison between the two simulations at Mach 0.15
new pipe elements and the new lumped elements are listed in
Tables 9 and 10, respectively. The analysis was performed using
the two-port code SIDLAB developed at KTH 关11兴. The lumped ments were used to obtain the results by the generalized segmen-
elements for the perforations and area expansion are not included tation model whereas two segments 共three lumped elements兲 were
as standard elements in SIDLAB. A MATLAB code was developed to used in the SIDLAB network. This proves that the convergence of
calculate the transfer matrix of such elements and store them in the new transfer matrix approach is quite good at low frequencies
the appropriate format to be imported into SIDLAB as a user- but can cause a small shift at higher frequencies similar to the
defined two-port element. shift in the third peak in Figs. 17 and 18. The main advantage of
A comparison between the transmission loss results from SID- using the new two-port segmentation approach is the simplicity of
LAB and the four-port simulations is shown in Figs. 17 and 18 setting up the model of the muffler using a general two-port code,
without flow and at Mach number 0.15, respectively. It is clear whereas the conventional method requires analysis of a large
that both simulations give very similar curves. Note that 50 seg- number of equations and solution of a large number of unknowns
in a process, which is likely subject to error in computations.
The present version of SIDLAB can only apply the classical cou-
Table 10 The properties of the tube perforated elements „Sin
pling conditions of the continuity of pressure and volume velocity
and Sout are the cross-sectional areas of the inlet and outlet
pipes, respectively… between different elements. The SIDLAB simulations are therefore
compared with the generalized segmentation simulations for the
Lp D Sw case when ␣ = 0. There is a small shift in the third peak, which is
Element No. 共m兲 共m兲 共m2兲 Mg Mb due to the limited number of segments considered. Otherwise, the
agreement is excellent with and without flow, except at the third
33 0.0225 0.045 0.003182 Q0 / 共cSin兲 Q2 / 共cSopen兲 peak, between the generalized segmentation model with 50 seg-
34 0.0450 0.006364 共Q0 + Q1兲 / 共2cSin兲 ments and the new two-port segmentation model with only two
35 0.0225 0.003182 Q1 / 共cSin兲 segments per perforated tube. The reason for the difference at the
36 0.018 0.057 0.003225 Q4 / 共cSout兲 Q6 / 共cSopen兲 third peak is due to the difference in the spatial resolution between
37 0.036 0.006449 共2Q4 + Q6兲 / 共2cSout兲 the two models. For the no flow case, where strong resonances are
38 0.018 0.003225 共Q4 + Q6兲 / 共cSout兲 seen, the levels of the resonance peaks are off because the damp-
39 0.036 0.057 0.006449 共Q0 − Q7兲 / 共cSout兲 Q7 / 共cSopen兲 ing is underestimated. At the high frequency end, the 1D model
40 0.072 0.012898 共2Q0 − Q7兲 / 共2cSout兲 gives systematic under prediction because of neglecting the 3D
41 0.036 0.006449 Q0 / 共cSout兲
effects. This can be clearly seen when compared with the finite
element method 共FEM兲 simulations, see Fig. 19. All simulations
60 are 3–4 dB too low compared with the measurements around 170–
Measurements
Measurements 300 Hz and the reason for this is not known. For the flow case, the
Simulations
Four in sec tion VI,
=0
port simulations main reason for the deviations is that the flow distribution is cal-
50 SID Simulations
SIDLAB simulations
culated using 1D models.
Transmission loss, dB

40
6 Comparison With FEM Simulations
In order to further investigate the validity of the new segmen-
30 tation model presented here, a comparison with 3D finite element
calculation for the same muffler was performed. The Mach num-
20
ber in an exhaust pipe is normally less than 0.3, which means that
inside a mufflers where the flow has expanded the average Mach
number is normally much smaller than 0.1. Therefore, one can
10 expect mean flow or convective effects on the sound propagation
to be small and possible to neglect. The main effect of the flow for
a complex perforated muffler is the effect on the perforate imped-
0 ances. For the predictions the 3D FEM software COMSOL MULT-
0 100 200 300 400 500 600
Frequency, Hz IPHYSICS has been used. In this software, assuming a negligible
mean flow, the sound pressure p will satisfy the modified Helm-
Fig. 17 Comparison between the two simulations without flow holtz equation:

Journal of Vibration and Acoustics DECEMBER 2010, Vol. 132 / 061010-9

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 60 existence of 3D effects that are not captured by the 1D simula-
Measurements
tions, for further comments, see Sec. 6.
SIDLAB Calculations
50 FEM Simulations
7 Conclusions
The modeling of the acoustic behavior of automotive mufflers
Transmission loss, dB

40
incorporating perforated elements has been discussed in this pa-
per. The basic problem is to model two ducts coupled through a
30 perforate. Two main approaches have previously been used to
model this configuration, the distributed and segmentation meth-
ods. The segmentation approach is believed to be more convenient
20
and flexible for more complicated configurations.
In this paper, the segmentation approach has been studied fur-
10 ther. Different coupling conditions for the acoustic fields through
the perforate have been considered. These are the continuity of
pressure and volume velocity, the continuity of mass and acoustic
0 energy, and the continuity of mass and acoustic momentum. The
0 100 200 300 400 500 600
Frequency, Hz continuity of pressure and volume velocity is only correct without
flow. With flow, earlier experiments showed that the correct cou-
Fig. 19 Comparison with the FEM simulations without flow pling is something in-between the other two conditions 关10兴. In
order to evaluate the importance of the choice of the coupling
condition, the two extremes were used to calculate the transmis-
sion loss of a through flow and a plug flow muffler. A generalized
ⵜ. 冉 1
␳0
ⵜp−q + 冊
k2 p
␳0
=0 共53兲
segmentation model, which is capable of incorporating any cou-
pling condition, has been derived. It was found that the effect of
the choice of the coupling increases with the Mach number, and is
where k = 2␲ f / co is the wave number, ␳0 is the fluid density, and significant for the simple muffler configurations studied for Mach
c0 is the speed of sound. The q term is a dipole source term numbers typical for internal combustion 共IC兲 engines 共M ⬇ 0.3兲. A
corresponding to acceleration/unit volume, which here can be put more complicated muffler system was studied and a complete
to zero. Using this formulation one can compute the frequency analysis of the modeling procedure was presented in detail. The
response using a parametric solver to sweep over a frequency effect of the choice of the coupling condition was less significant
range. Within the COMSOL software, different boundary conditions in the results for this case.
are available and here continuity of normal un velocity combined A new version of the segmentation approach was also devel-
with 共p1 − p2兲 / Z = un, where Z is the perforate impedance and 1 oped where the perforated branches and the intermediate straight
and 2 denote the acoustic pressures on each side of the perforate, pipes were modeled as two-ports. This new technique is more
was used. It can be noted that the use of continuity of normal flexible as it facilitates the modeling of perforated pipes within
velocity is consistent with our assumption that mean flow effects any muffler system with an arbitrary configuration. It has also
are small and can be neglected. proved to converge quite fast, which means a small number of
The impedance model and flow distribution used in the FEM segments are needed to model a given problem.
model are the same ones as described earlier in Secs. 2.2 and 3.2,
respectively. For the COMSOL simulation, the “fine mesh” alterna-
tive with 73,739 elements was used. Tests showed that this level
Acknowledgment
of resolution was sufficient to resolve the 3D effects in the fre- Financial support from the EU Fifth Framework project ARTE-
quency range of interest. In Figs. 19 and 20, the FEM simulations MIS 共Contract No. G3RD-CT-2001-0511兲 is gratefully acknowl-
are compared with the 1D model and the measurements. It is edged. The authors wish to thank the Polytechnical University of
obvious that the FEM simulations are better than the SIDLAB Valencia for designing the complicated muffler presented in this
simulations especially at the high frequency end because of the paper, and Faurecia for manufacturing it. The authors would also
like to thank the Laboratoire de Mecanique Physique at the Uni-
versite Pierre & Marie Curie for providing the experimental data
of the muffler.
60
Measurements
SIDLAB Calculations Appendix
50 FEM Simulations
When two two-ports are connected at a node, the pressure and
volume velocity at the node are continuous if both elements have
the same area connected to the node. When the elements have
Transmission loss, dB

40
different areas, a coupling two-port element must be added in-
between in order to fulfill the coupling conditions of the continu-
30 ity of energy and mass flow,
Uin + M inpin/zout = Uout + M outpout/zout 共A1兲
20
pin + ␣ · zinM inUin = pout + ␣ · zoutM outUout 共A2兲
10 Where ␣ varies between 1 and 2. If ␣ is equal to 1, conservation
of energy applies, whereas if ␣ is equal to 2, conservation of
momentum applies. With ␣ equal to zero, continuity of pressure,
0
0 100 200 300 400 500 600
which is commonly used for the no flow case, is obtained. This
Frequency, Hz will be used here since studies performed in Ref. 关20兴 show that
varying ␣ does not significantly affect the result. Substituting for
Fig. 20 Comparison with the FEM simulations at Mach 0.15 Uin from Eq. 共A1兲 into Eq. 共A2兲 and rearranging

061010-10 / Vol. 132, DECEMBER 2010 Transactions of the ASME

Downloaded 02 Nov 2010 to 130.237.39.121. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
 pin关1 − ␣ M in
2
兴 = pout关1 − ␣ M inM outzin/zout兴 + Uout关␣zoutM out Multiple Pipe Mufflers Carrying Mean Flow,” J. Sound Vib., 191共4兲, pp.
505–518.
− ␣zinM in兴 共A3兲 关7兴 Peat, K., 1996, “Comments on Matrizant Approach to Acoustic Analysis of
Perforated Multiple Pipe Mufflers Carrying Mean Flow,” J. Sound Vib.,
Then substituting for pin pin from Eq. 共A1兲 into Eq. 共A2兲 and 191共4兲, pp. 606–609.
rearranging 关8兴 Sullivan, J., 1979, “A Method for Modelling Perforated Tube Muffler Compo-
nents. I. Theory,” J. Acoust. Soc. Am., 66共3兲, pp. 772–778.
Uin关1 − ␣ M in
2
兴 = pout关M out/zout − M in/zin兴 + Uout关1 关9兴 Sullivan, J., 1979, “A Method for Modelling Perforated Tube Muffler Compo-
nents. II. Applications,” J. Acoust. Soc. Am., 66共3兲, pp. 779–788.
− ␣ M inM outzout/zin兴 共A4兲 关10兴 Kergomard, J., Khettabi, A., and Mouton, X., 1994, “Propagation of Acoustic
Waves in Two Waveguides Coupled by Perforations. I. Theory,” Acta Acous-
Combining Eqs. 共A3兲 and 共A4兲, the two-port transfer matrix of the tica, 2共1兲, pp. 1–16.
coupling element can be given by 关11兴 Dokumaci, E., 2001, “A Discrete Approach for Analysis of Sound Transmis-

冋 册
sion in Pipes Coupled With Compact Communicating Devices,” J. Sound Vib.,
pin 239共4兲, pp. 679–693.
关12兴 Elnady, T., and Åbom, M., 2006, “SIDLAB: New 1D Sound Propagation
Uin

冋 册冋 册
Simulation Software for Complex Duct Networks,” Proceedings of ICSV13,
1 1 − ␣ M inM outzin/zout ␣zoutM out − ␣zinM in pout Vienna.
= · 关13兴 Glav, R., and Åbom, M., 1997, “A General Formalism for Analysing Acoustic
1 − ␣ M in
2
M out/zout − M in/zin 1 − ␣ M inM outzout/zin Uout 2-Port Networks,” J. Sound Vib., 202共5兲, pp. 739–747.
关14兴 Elnady, T., 2004, “Modelling and Characterization of Perforates in Lined
共A5兲 Ducts and Mufflers 共Paper III兲,” Ph.D. thesis, Department of Aeronautical and
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Sweden.
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