Wiener–Hopf approach for predicting the transmission loss of a circular silencer with a locally reacting lining

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International Journal of Engineering Science 43 (2005) 398–416

Wiener–Hopf approach for predicting the transmission lossof a circular silencer with a locally reacting lining

Ahmet Demır, Alinur Buyukaksoy *

Department of Mathematics, Gebze Institute of Technology, P.O. Box 141, Gebze, Kocaeli 41400, Turkey

Received 11 October 2004; received in revised form 24 December 2004; accepted 24 December 2004

(Communicated by H. DEMIRAY)

Abstract

The propagation of sound in an infinite rigid circular cylindrical duct with an inserted expansion cham-

ber whose walls are treated with an acoustically absorbent, locally-reacting material is investigated rigor-

ously through the Wiener–Hopf technique. The expansion chamber is separated from the central airwaywhich contain a uniform main gas flow by a perforated cylindrical screen which also increases the silencing

performance. The influence of the expansion chamber radius, lining impedances, the mean flow and the

acoustical impedance of the central perforated tube on the transmission loss are displayed graphically.

� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

The acoustic characteristics of sudden area changes in ducts, such as expansions and contrac-tions are important in noise reduction applications. For example, to reduce the unwanted exhaustnoise produced by internal combustion engines one usually introduces expansion chambers tomuffle the noise as it travels along the duct. A wave propagating through ducts with rapid changesin the cross-sectional area can experience significant reflections which reduce the energy in the

0020-7225/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijengsci.2004.12.003

* Corresponding author. Tel.: +90 262 653 84 97; fax: +90 262 653 84 90.

E-mail address: [email protected] (A. Buyukaksoy).

mailto:[email protected]

A. Demır, A. Buyukaksoy / International Journal of Engineering Science 43 (2005) 398–416 399

propagating transmitted wave. This, together with cavity resonance mechanisms, is the method bywhich silencer box helps to reduce noise in the car exhaust system [1].The treatment of the duct walls with a thin acoustically absorbent lining is another effective

method that has been proved useful in reducing unwanted noise [2].As a sound attenuator, the acoustic performance of an expansion chamber in a duct can be in-

creased significantly by lining its walls by an acoustically absorbent material [3,4]. The acousticmodelling of simple silencer geometries, such as circular or rectangular cross-sections, has beenestablished and classified depending on the assumption of locally- or bulk-reacting lining. Morse[5] used a locally-reacting model to investigate the sound transmission in pipes with absorbingmaterial on the inner walls. By using bulk-reacting model for linings, Scott [6] studied the trans-mission of sound in infinite rectangular and circular ducts. Ko [7] derived a governing eigen equa-tion for annular and circular ducts containing a bulk-reacting liner and mean gas flow. Later, thepropagation of sound in cylindrical ducts with mean flow and bulk-reacting lining has been trea-ted in a series of papers by Nilsson and Brander [8–11].The acoustic performance of circular silencers have been subjected to numerous past investiga-

tions. For example, Cummings and Chang [12], Kirby [13] and Peat [14] used the well knownmode matching technique in order to calculate the transmission loss of a finite length dissipativecircular silencer. Numerical techniques such as the finite element method [15–17], the boundaryelement method [18,19], and the boundary integral equation method [20] have also been used suc-cessfully to model dissipative silencers.In the present work the propagation of sound in an infinite rigid circular cylindrical duct with

an inserted expansion chamber whose walls are treated with an acoustically absorbent material(characterized by a constant surface impedance) is investigated rigorously through the Wiener–Hopf technique. The expansion chamber is separated from the central airway which contain a uni-form main gas flow by a perforated cylindrical screen. The effect of the perforated center tube is toregulate the main flow and increase the silencing performance (see [20]). The simpler case, wherethe mean flow and central perforated duct are absent, has been recently investigated by the presentauthors [21]. By introducing the Fourier transform for the scattered field and applying the bound-ary conditions in the transform domain, the problem is reduced into a modified Wiener–Hopfequation. The solution involves four sets of infinitely many constants satisfying four infinite sys-tems of linear algebraic equations which are solved by means of numerical procedures. The effectsof the expansion chamber radius, lining admittances, the mean flow and the acoustical impedanceof the central perforated tube on the transmission loss are displayed graphically. For some specialcases, the results are compared with those reported previously in [20,22,23] and it is found that theagreement is excellent.Notice that the results reported in this work may also be used as a benchmark solution for

approximate and numerical treatment of circular silencer problems.

2. Analysis

The silencer studied in the present work is assumed to have a uniform circular cross-sectionwith radius b and to contain a uniform mean gas flow in the central channel of radius a, The veloc-ity of the mean gas flow is v0 =Mc where c is the velocity of sound andM is the Mach number. In

Fig. 1. Circular silencer with a locally reacting lining.

400 A. Demır, A. Buyukaksoy / International Journal of Engineering Science 43 (2005) 398–416

the silencer chamber, a perforate screen separates the central channel (region 1) from the outercavity (region 2). The inlet and outlet pipes with radius a, are rigid walled (see Fig. 1). FromSullivan�s experiment [24], it can be understood that most of the medium flows straight troughthe center tube and only a little mass flows into the outer cavity at the fore stage of the perforatedsurface and flows out at the end part. It is therefore, assumed that the main flow only exists in thecenter tube [13,20].From the symmetry of the geometry of the problem and of the incident field, the acoustic field

everywhere will be independent of /, where (q,/,z), are the usual cylindrical polar coordinates.We shall therefore introduce a scalar potential uj(q,z) which defines the displacement vj, the veloc-ity vj and the acoustic pressure pj, in the regions j = 1 and j = 2 by vj = graduj, vj = gradDjuj andpj ¼ �.0D

2j uj, respectively, with D1 ¼ �ix þMc o

oz and D2 = �ix. Here .0 is the density of theundisturbed medium and a time factor exp(�ixt) has been assumed and suppressed.The walls of the expansion chamber defined by {q = b,z 2 (0, l)}, {q 2 (a,b), z = 0} and

{q 2 (a,b), z = l} are assumed to be coated by a thin acoustically absorbent material. The acousticimpedance of the lining is defined by the ratio Z = p2/(v2 Æ n) [25], where the normal n is directedinto the lining. Thus, in terms of the velocity potential u2, the boundary condition on the absor-bent surface is given by

n � gradu2 �ikf‘u2 ¼ 0;

where k = x/c denotes the wave number, and f‘ (=Z/.0c) is the specific impedance of the lining.Consider the problem of a plane wave, given by

uiðq; zÞ ¼ expð�in�0 zÞ ð1aÞ

with

n�0 ¼ � k

1þMð1bÞ

incoming from the left of the junction at z = 0, shown in Fig. 1.For analysis purposes it is convenient to express the total field uT(q,z) as follows:

uTðq; zÞ ¼u1ðq; zÞ þ uiðzÞ; q < a; z 2 ð�1;1Þ;u2ðq; zÞ; q 2 ða; bÞ; 0 < z < l:

�ð2Þ


In the region q < a, the incident and the scattered field u1 satisfies the following partial differentialequation (see for example [9]):

1

qo

oqqo

oq

� �þ o2

oz2� �ik þM

o

oz

� �2" #u1ðq; zÞ ¼ 0: ð3Þ

By taking the Fourier transform of (3) with respect to z 2 (�1,1), we obtain
1
qd

dqqd

dq

� �þ H 2ðaÞ

� �F ðq; aÞ ¼ 0 ð4Þ

with

HðaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk þMaÞ2 � a2

q: ð5Þ

H(a) given by (5) is the square-root function which is defined in the complex a-plane cut as shownin Fig. 2.In (4), F(q,a) stands for

F ðq; aÞ ¼Z 1

�1u1ðq; zÞeiaz dz ð6Þ

with a being the Fourier transform variable. F(q,a) defined by (6) can also be written as

F ðq; aÞ ¼ F �ðq; aÞ þ F 1ðq; aÞ þ eialF þðq; aÞ ð7Þ
with F+(q,a), F�(q,a) and F1(q,a) being defined by
F �ðq; aÞ ¼Z 0

�1u1ðq; zÞeiazdz; ð8aÞ

F þðq; aÞ ¼Z 1

lu1ðq; zÞeiaðz�lÞdz ð8bÞ

Fig. 2. Branch-cuts and integration lines in the complex plane.


and

F 1ðq; aÞ ¼Z l

0

u1ðq; zÞeiazdz; ð8cÞ

respectively. Note that the half range Fourier transforms, F+(q,a) and F�(q,a) are regular func-tions of a in the regions Ima > Im{�k/(1 +M)} (upper half-plane) and Ima < Im{k/(1 �M)}(lower half-plane), respectively, whereas F1(q,a) is an entire function.Solving (4) by taking into account the finite energy condition for q = 0, it is found that

F �ðq; aÞ þ F 1ðq; aÞ þ eialF þðq; aÞ ¼ �AðaÞ J 0ðHqÞHðaÞJ 1ðHaÞ

: ð9Þ

In (9), A(a) is a spectral coefficient to be determined, while J0 and J1 are the zeroth and first orderBessel functions of the first kind, respectively.Consider now the boundary condition satisfied by u1(q,z) on the rigid walls of the inlet and out-

let pipes of radius q = a, namely

o

oqu1ða; zÞ ¼ 0; f�1 < z < 0 [ l < z < 1g: ð10Þ

In Fourier transform domain (10) takes form

_F �ða; aÞ ¼ 0; _F þða; aÞ ¼ 0; ð11Þ
where (Æ) denotes the derivative with respect to q.The derivative of (9) with respect to q yields
_F �ðq; aÞ þ _F 1ðq; aÞ þ eial _F þðq; aÞ ¼ AðaÞ J 1ðHqÞJ 1ðHaÞ

: ð12Þ

By setting q = a in (12) and considering (11), it is obtained that

AðaÞ ¼ _F 1ða; aÞ: ð13Þ
The substitution of (13) in (9) gives
F �ðq; aÞ þ eialF þðq; aÞ þ F 1ðq; aÞ ¼ � _F 1ða; aÞJ 0ðHqÞHJ 1ðHaÞ

: ð14Þ

Now consider the outer cavity region q 2 (a,b) and z 2 (0, l) where the scattered field u2(q,z) sat-isfies the Helmholtz equation

1

qo

oqqo

oq

� �þ o2

oz2þ k2

� �u2ðq; zÞ ¼ 0: ð15Þ

The Fourier transform of (15) in the finite range z 2 (0, l) reads
1
qd

dqqd

dq

� �þ K2ðaÞ

� �G1ðq; aÞ ¼

o

ozu2ðq; 0Þ � iau2ðq; 0Þ

� �

� eial o

ozu2ðq; lÞ � iau2ðq; lÞ

� �ð16Þ


with

KðaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � a2

p:

In (16), G1(q,a) which is defined by

G1ðq; aÞ ¼Z l

0

u2ðq; zÞeiaz dz ð17Þ

denotes an entire function, regular in the whole complex a-plane.Considering the boundary conditions

ikf‘þ o

oz

� �u2ðq; 0Þ ¼ 0; a < q < b; ð18aÞ

ikf‘� o

oz

� �u2ðq; lÞ ¼ 0; a < q < b ð18bÞ

which are satisfied on the lined ends of the expansion chamber, Eq. (16) can be reduced into

1

qd

dqqd

dq

� �þ K2ðaÞ

� �G1ðq; aÞ ¼ � ik

f‘þ ia

� �f ðqÞ � eial ik

f‘� ia

� �gðqÞ; ð19Þ

where f(q) and g(q) stand for

f ðqÞ ¼ u2ðq; 0Þ; ð20aÞ

gðqÞ ¼ u2ðq; lÞ: ð20bÞ
Now, a particular solution to the non-homogeneous ordinary differential equation in (19) will beobtained by using the Green�s function method. The Green�s function satisfies the equation
1

qd

dqqd

dq

� �þ K2ðaÞ

� �Gðq; t; aÞ ¼ 1

tdðq � tÞ; q; t 2 ða; bÞ ð21Þ

with the following conditions

Gðt þ 0; t; aÞ ¼ Gðt � 0; t; aÞ; ð22aÞ

o

oqGðt þ 0; t; aÞ � o

oqGðt � 0; t; aÞ ¼ 1

t; ð22bÞ

o

oqGða; t; aÞ ¼ 0 ð22cÞ

and

ikf‘� o

oq

� �Gðb; t; aÞ ¼ 0: ð22dÞ


The solution of (21) satisfying (22a–d) is

Gðq; t; aÞ ¼ 1

KðaÞLðaÞQðq; t; aÞ ð23Þ

with

Qðq; t; aÞ ¼ p2KðaÞ

J 0ðKqÞY 1ðKaÞ � J 1ðKaÞY 0ðKqÞ½ ;J 0ðKtÞY ðb; aÞ � Jðb; aÞY 0ðKtÞ½ ; a 6 q 6 t;

J 0ðKqÞY ðb; aÞ � Jðb; aÞY 0ðKqÞ½ ;J 0ðKtÞY 1ðKaÞ � J 1ðKaÞY 0ðKtÞ½ ; t 6 q 6 b

8>>><>>>:

ð24aÞ

and

LðaÞ ¼ J 1ðKaÞY ðb; aÞ � Jðb; aÞY 1ðKaÞ½ : ð24bÞ
J(b,a) and Y(b,a) appearing in the latter expressions are defined by
Jðb; aÞ ¼ ikf‘J 0ðKbÞ þ KJ 1ðKbÞ; ð25aÞ

Y ðb; aÞ ¼ ikf‘Y 0ðKbÞ þ KY 1ðKbÞ: ð25bÞ

Now the general solution of (19), which satisfies the impedance boundary condition

ikf‘� d

dq

� �G1ðb; aÞ ¼ 0; 0 < z < l ð26Þ

on the lined lateral wall q = b, 0 < z < l can be written as

G1ðq; aÞ ¼�1

KðaÞLðaÞ

�_G1ða; aÞ J 0ðKqÞY ðb; aÞ � Y 0ðKqÞJðb; aÞ½

þZ b

a

ikf‘þ ia

� �f ðtÞ þ eial ik

f‘� ia

� �gðtÞ

� �Qðt; q; aÞtdt

�: ð27Þ

In (27) _G1ða; aÞ is an entire function to be determined while f(t) and g(t) are given by (20a) and(20b), respectively.Consider now the boundary conditions characterizing the central perforated cylinder at q = a,

0 < z < l, which are given by

u2ða; zÞ � 1þ iMk

o

oz

� �2u1ða; zÞ � i

fpk

o

oqu1ða; zÞ ¼ 1þ iM

ko

oz

� �2uiðzÞ; 0 < z < l; ð28aÞ

o

oqu1ða; zÞ ¼

o

oqu2ða; zÞ; 0 < z < l ð28bÞ

(see for example [8]). In (28a), fp stands for acoustical impedance of perforated screen which re-lates acoustic pressures between two regions through the interface. For stationary media and graz-ing flow, empirical formulae of specific acoustical impedance fp proposed by Sullivan and Crocker[26] are used.


For the case of perforates in stationary media [26]

fp ¼ 0:006� ikðtw þ 0:75dhÞ½ =r; ð29aÞ

where tw is the wall thickness, dh perforate hole diameter, r the porosity. For the case of perforateswith grazing flow [27]

fp ¼ 7:337� 10�3ð1þ 72:23MÞ � i2:2245� 10�5ð1þ 51twÞð1þ 204dhÞf� �

=r ð29bÞ

with f being the sound frequency.By multiplying both sides of (28a) and (28b) by eiaz and integrating over the interval z 2 (0, l)

one obtains that

G1ða; aÞ � 1þMak

� �2F 1ða; aÞ � i

fpk

_F 1ða; aÞ ¼ 1þMn�0

k

� �21� e�iðn�0 �aÞl

iðn�0 � aÞ ; ð30aÞ

_F 1ða; aÞ ¼ _G1ða; aÞ: ð30bÞ
The substitution of (30b) in (27) yields
G1ðq; aÞ ¼�1

KðaÞLðaÞ_F 1ða; aÞ J 0ðKqÞY ðb; aÞ � Y 0ðKqÞJðb; aÞ½

�

þZ b

a

ikf‘þ ia


f‘� ia

� �gðtÞ

� �Qðq; t; aÞtdt

�: ð31Þ

The left-hand side of (31) is an entire function, then its right-hand side must also be an entire func-tion. The regularity of the right-hand side of (31) may be violated by the presence of simple poleslying at the lower (Ima < Imk) and upper (Ima > Im(�k)) halves of the a-plane, namely ata = � am(m = 1,2, . . . , Im � am < Im � k) and a = am, (m = 1,2, . . . , Imam > Imk), respectively.These poles are the simple zeros of the function L(a) and satisfy the equation

Lð�amÞ ¼ J 1ðKmaÞY ðb;�amÞ � Y 1ðKmaÞJðb;�amÞ ¼ 0 ð32aÞ
with
Km ¼ Kð�amÞ: ð32bÞ
These poles can be removed by enforcing that their residues are zero, that is
_F 1ða;�amÞ ¼ � p2

KmJ 1ðKmaÞJðb; amÞ

Z b

a

ikf‘� iam

� �f ðtÞ þ e�iaml ik

f‘� iam

� �gðtÞ

� �� J 0ðKmtÞY ðb; amÞ � Y 0ðKmtÞJðb; amÞ½ tdt; m ¼ 1; 2; . . . ð33Þ

By substituting q = a in (14) and (31) and taking into account (30a), one obtains:

_F 1ða; aÞV ðaÞ � 1þMak

� �2F �ða; aÞ � 1þMa

k

� �2eialF þða; aÞ

¼ �1KðaÞLðaÞ

Z b

a

ikf‘þ ia


f‘� ia

� �gðtÞ

� ��

� J 0ðKtÞY ðb; aÞ � Y 0ðKtÞJðb; aÞ½ tdt�þ 1þM

n�0

k

� �2e�iðn

�0 �aÞl � 1

iðn�0 � aÞ ; ð34aÞ


where

V ðaÞ ¼ J 0ðKaÞY ðb; aÞ � Y 0ðKaÞJðb; aÞ½ KðaÞ J 1ðKaÞY ðb; aÞ � Y 1ðKaÞJðb; aÞ½ � 1þMa

k

� �2 J 0ðHaÞHðaÞJ 1ðHaÞ

þ i fpk

( )

¼ vðaÞKðaÞ J 1ðKaÞY ðb; aÞ � Y 1ðKaÞJðb; aÞ½ HðaÞJ 1ðHaÞ

; ð34bÞ

vðaÞ ¼ HðaÞJ 1ðHaÞ J 0ðKaÞY ðb; aÞ � Y 0ðKaÞJðb; aÞ½

� 1þMak

� �2J 0ðHaÞ � i

fpkHðaÞJ 1ðHaÞ

( )

� KðaÞ J 1ðKaÞY ðb; aÞ � Y 1ðKaÞJðb; aÞ½ : ð34cÞ

Since f(t) and g(t) appearing in (34a) are absolutely integrable functions satisfying Dini condi-tions, they can be expanded into the series of the following complete sets of orthogonal functions[28, pp. 453 and 449].

f ðtÞ ¼X1m¼1

fm½J 0ðKmtÞY ðb; amÞ � Y 0ðKmtÞJðb; amÞ ð35aÞ

and

gðtÞ ¼X1m¼1

gm½J 0ðKmtÞY ðb; amÞ � Y 0ðKmtÞJðb; amÞ ð35bÞ

with

fm ¼ p2

2

1

#m

Z b

af ðtÞ½J 0ðKmtÞY ðb; amÞ � Y 0ðKmtÞJðb; amÞ tdt; ð35cÞ

gm ¼ p2

2

1

#m

Z b

agðtÞ½J 0ðKmtÞY ðb; amÞ � Y 0ðKmtÞJðb; amÞ tdt; ð35dÞ

where

#m ¼ 1þ ikf‘Km

� �2� J 2ðb; amÞK2mJ

21ðKmaÞ

" #: ð35eÞ

By taking into account (33), together with (35c) and (35d), _F 1ða; amÞ can be expressed in terms offm and gm as follows:

_F 1ða;�amÞ ¼ � 1pKmJ 1ðKmaÞJðb; amÞ

#mikf‘� iam

� �fm þ e�iaml ik

f‘� iam

� �gm

� �: ð36Þ

By substituting (35a,b) into (34a) and evaluating the resulting integrals one obtains the followingmodified Wiener–Hopf equation (MWHE) of the third kind which is valid in the stripImðn�

0 Þ < ImðaÞ < Imk:


_F 1ða; aÞV ðaÞ � 1þMak

� �2F �ða; aÞ � 1þMa

k

� �2eialF þða; aÞ

¼ 2

ap

X1m¼1

Jðb; amÞKmJ 1ðKmaÞ

1

a2m � a2ikf‘þ ia

� �fm þ eial ik

f‘� ia

� �gm

� �

þ 1þMn�0

k

� �2e�iðn

�0�aÞl � 1

iðn�0 � aÞ : ð37Þ

2.1. Solution of the modified Wiener–Hopf equation

Consider first the MWHE in (37) and rearrange it as follows:

_F 1ða; aÞV ðaÞ � RðaÞ ¼ eialSðaÞ ð38aÞ
with
RðaÞ ¼ 1þMak

� �2F �ða; aÞ þ

2

ap

X1m¼1


ðik=f‘ þ iaÞa2m � a2

fm � 1þMn�0

k

� �21

iðn�0 � aÞ

ð38bÞ
and
SðaÞ ¼ 1þMak

� �2F þða; aÞ þ

2

ap

X1m¼1


ðik=f‘ � iaÞa2m � a2

gm þ 1þMn�0

k

� �2e�in

�0l

iðn�0 � aÞ :

ð38cÞ
Now, factorize the kernel function V(a) given by (34b,c) in the Wiener–Hopf sense, that is
V ðaÞ ¼ V þðaÞV �ðaÞ: ð39Þ
Here, V+(a) and V�(a) are the split functions regular and free of zeros in the upper ðIma > Imn�
0 Þand lower (Ima < Imk) halves of the complex a-plane, respectively. For M 5 0, V(a) is not aneven function of a, hence V�(a)5 V+(�a).The zeros of V(a) lying in the upper and lower halves of the complex a-plane are denoted by bþ

mand b�

m and satisfy

vðb�mÞ ¼ 0; m ¼ 0; 1; . . . ; ð40Þ

while the poles ±am and n�m of V are the roots of (32a) and
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðk þMn�mÞ2 � ðn�

mÞ2

qJ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk þMn�

mÞ2 � ðn�

mÞ2

qa

� �¼ 0; m ¼ 0; 1; . . . ; ð41Þ

respectively. n�m ; m ¼ 0; 1; 2; . . . lying in the upper and lower half planes are defined by

n�m ¼ kM

1�M2� iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�M2p jm

a

� �2� k

1�M2

" #1=2; J 1ðjmÞ ¼ 0; m ¼ 1; 2; . . . ð42aÞ


with

n�0 ¼ �k

1�M: ð42bÞ

Now, the meromorphic function V(a) can be represented in terms of its zeros and poles as an infi-nite product. Hence, the explicit expressions of the split functions V+(a) and V�(a) reads

V �ðaÞ ¼ffiffiffiffiffiffiffiffiffiffiV ð0Þ

p ð1þ a=b�0 Þ

ð1þ a=n�0 Þ

Y1m¼1

ð1þ a=b�mÞ

ð1þ a=a�mÞð1þ a=n�

mÞ: ð43Þ

Multiplying (38a) first by (1/V�(a)) and then by (e�isl/V+(a)), and applying the Wiener–Hopf

decomposition procedure together with the Liouville�s theorem, we get the following pair ofsimultaneous Fredholm integral equations of the second kind:

RðaÞV �ðaÞ

¼ 1

2pi

ZL�

SðsÞeislV �ðsÞðs� aÞ dsþ

1

ap

X1m¼1

Jðb;amÞKmJ 1ðKmaÞ

ðik=f‘� iamÞfmamV �ð�amÞðamþ aÞ�

1þM n�0

k

� �2iV �ðn�

0 Þðn�0 � aÞð44aÞ

and

SðaÞV þðaÞ

¼ �12pi

ZLþ

RðsÞe�islV þðsÞðs � aÞ ds þ

1

ap

X1m¼1


ðik=f‘ � iamÞgmamV þðamÞðam � aÞ : ð44bÞ

The paths of integration Lþ and L� are depicted in Fig. 2.The integrals I1(a) and I2(a) in (44a) and (44b), defined by

I1ðaÞ ¼1

2pi

ZL�

V þðsÞSðsÞeislV ðsÞðs � aÞ ds; ð45aÞ

I2ðaÞ ¼ � 1

2pi

ZLþ

V �ðsÞRðsÞe�islV ðsÞðs � aÞ ds; ð45bÞ

respectively, can be evaluated easily by taking into account (38b) and (38c). Consider first theevaluation of I1(a). According to the Cauchy theorem I1(a) is equal to the sum of the residuesat the poles occurring at the zeros of v(a), lying in the upper half-plane, namely at s ¼ bþ

n wherebþn ðn ¼ 0; 1; 2; . . .Þ are defined by Eq. (40).The result is

I1ðaÞ¼X1n¼0

J 1ðKðbþn ÞaÞY ðb;b

þn Þ�Y 1ðKðb

þn ÞaÞJðb;b

þn Þ

� �v0ðbþ

n ÞHðbþ

n ÞJ 1ðHðbþn ÞaÞV þðbþ

n ÞKðbþn Þ

ðbþn �aÞ

Sðbþn Þeib

þn l

� �:

ð46aÞ
By proceeding similarly, we get
I2ðaÞ¼X1n¼0

J 1ðKðb�n ÞaÞY ðb;b

�n Þ�Y 1ðKðb

�n ÞaÞJðb;b

�n Þ

� �v0ðb�

n ÞHðb�

n ÞJ 1ðHðb�n ÞaÞV �ðb�

n ÞKðb�n Þ

ðb�n �aÞ Rðb�

n Þe�ib�n l

� �;

ð46bÞ
where v 0(a) denotes the derivatives of v with respect to a.


Using these results, (44a) and (44b) takes the following form:

RðaÞV �ðaÞ

¼X1n¼0


þn Þ�Y 1ðKðb

þn ÞaÞJðb;b

þn Þ

� �v0ðbþ

n ÞHðbþ


n ÞKðbþn Þ

ðbþn �aÞ

Sðbþn Þeib

þn l

� �

þ 1

ap

X1m¼1


ðik=f‘� iamÞfmamV �ð�amÞðamþaÞ�

1þM n�0

k

� �2iV �ðn�

0 Þðn�0 �aÞ ð47aÞ

and

SðaÞV þðaÞ

¼X1n¼0


�n Þ�Y 1ðKðb

�n ÞaÞJðb;b

�n Þ

� �v0ðb�

n ÞHðb�


n ÞKðb�n Þ

ðb�n �aÞ Rðb�

n Þ� �

e�ib�n l

þ 1

ap

X1m¼1


ðik=f‘� iamÞgmamV þðamÞðam�aÞ : ð47bÞ

Now, the solution of the modified Wiener–Hopf equation reads:

_F 1ða;aÞ¼1

V þðaÞ

�X1n¼0


þn Þ�Y 1ðKðb

þn ÞaÞJðb;b

þn Þ

� �v0ðbþ

n ÞHðbþ


n ÞKðbþn ÞSðb

þn Þ

ðbþn �aÞ

eibþn l

� �

þ 1

apV þðaÞX1m¼1


ðik=f‘� iamÞfmamV �ð�amÞðamþaÞ�

1þM n�0

k

� �2iV �ðn�

0 ÞV þðaÞðn�0 �aÞþ

eial

V �ðaÞ

�X1n¼0


�n Þ�Y 1ðKðb

�n ÞaÞJðb;b

�n Þ

� �v0ðb�

n ÞHðb�


n ÞKðb�n ÞRðb

�n Þ

ðb�n �aÞ e�ib

�n l

� �

þ eial

apV �ðaÞX1m¼1


ðik=f‘� iamÞgmamV þðamÞðam�aÞ : ð48Þ

2.2. Determination of the expansion coefficients

The expression of _F 1ða; aÞ in (48) involves the unknown constants fm, gm, Sðbþn Þ and Rðb

�n Þ. fm

and gm are related to _F 1ða;�amÞ through (36). To determine the constants fm, gm, Sðbþn Þ and Rðb

�n Þ

we substitute a = ±a1,±a2, . . . ,±aN in (48), a ¼ b�0 ;b

�1 ; b

�2 ; . . . ;b

�N�1 in (47a) and a ¼ bþ

0 ;bþ1 ; b

þ2 ;

. . . ;bþN�1 in (47b), then we get the following infinite systems of linear algebraic equations:

� 1pKrJ 1ðKraÞJðb; arÞ

ðik=f‘ þ iarÞ#rfr ¼1

apV þðarÞX1m¼1


ðik=f‘ � iamÞamV �ð�amÞðam þ arÞ

fm

� 1

V þðarÞX1n¼0

J 1 Kðbþn Þa

� �Y ðb; bþ

n Þ � Y 1 Kðbþn Þa

� �Jðb; bþ

n Þ� �

v0ðbþn Þ

�

� Hðbþn ÞJ 1ðHðbþ

n ÞaÞV þðbþn ÞKðb

þn ÞSðb

þn Þ

ðbþn � arÞ

eibþn l

��

1þM n�0

k

� �2iV �ðn�

0 ÞV þðarÞðn�0 � arÞ

; ð49aÞ


� 1pKrJ 1ðKraÞJðb; arÞ

ik=f‘ þ iarð Þ#rgr ¼1

apV �ð�arÞX1m¼1


ðik=f‘ � iamÞamV þðamÞðam þ arÞ

gm

þ 1

V �ð�arÞX1n¼0

J 1 Kðb�n Þa

� �Y ðb;b�

n Þ � Y 1ðKðb�n ÞaÞJðb;b

�n Þ

� �v0ðb�

n Þ

�

� Hðb�n ÞJ 1ðHðb�

n ÞaÞV �ðb�n ÞKðb

�n Þ

ðb�n þ arÞ

Rðb�n Þe�ib

�n l

�; ð49bÞ

Rðb�r Þ

V �ðb�r Þ

¼X1n¼0

½J 1ðKðbþn ÞaÞY ðb; b

þn Þ � Y 1ðKðbþ

n ÞaÞJðb; bþn Þ

v0ðbþn Þ

�

� Hðbþn ÞJ 1ðHðbþ

n ÞaÞV þðbþn ÞKðb

þn Þ

ðbþn � b�

r ÞSðbþ

n Þeibþn l

�

þ 1

ap

X1m¼1


ðik=f‘ � iamÞamV �ð�amÞðam þ b�

r Þfm �

1þM n�0

k

� �2iV �ðn�

0 Þðn�0 � b�

r Þ; ð49cÞ

Sðbþr Þ

V þðbþr Þ

¼X1n¼0

½J 1ðKðb�n ÞaÞY ðb; b

�n Þ � Y 1ðKðb�

n ÞaÞJðb; b�n Þ

v0ðb�n Þ

�

� Hðb�n ÞJ 1ðHðb�

n ÞaÞV �ðb�n ÞKðb

�n Þ

ðb�n � bþ

r ÞRðb�

n Þe�ib�n l

�

þ 1

ap

X1m¼1


ðik=f‘ � iamÞamV þðamÞðam � bþ

r Þgm: ð49dÞ

These infinite systems of equations are solved approximately by truncating the expansionseries.

3. The scattered field and computational results

The transmitted field in the region q < a, z > l can be obtained by taking the inverse Fouriertransform of F+(q,a). From (8b) and (14) we obtain

u1ðq; zÞ ¼ � 1

2p

ZL

_F 1ða; aÞJ 0ðHqÞ

HðaÞJ 1ðHaÞþ F �ðq; aÞ þ F 1ðq; aÞ

� �e�iaz da; ð50Þ

where L is a straight line parallel to the real a-axis, lying in the strip Imn�0 < Ima < Imk. By

using the residue theorem, we obtain easily

u1ðq; zÞ ¼i

a

X1m¼0

_F 1ða; n�mÞJ 0 Hðn�

mÞq� �

Mðk þMn�mÞ � n�

m

� �J 0 Hðn�

mÞa� � e�in�mz; ð51aÞ


where _F 1ða; aÞ is given by (48). It follows that the transmission coefficient is

Table

Ampl

N

1

2

3

4

5

6

7

8

9

10

11

12

T ¼ i_F 1ða; n�

0 ÞMðk þMn�

0 Þ � n�0

� �a¼ i

_F 1ða; n�0 Þ

ka: ð51bÞ

In the region q < a, z < 0 the inverse Fourier transform of F�(q,a) gives the reflected field, that is

u1ðq; zÞ ¼ � ia

X1m¼0

_F 1ða; nþmÞJ 0½Hðnþ

mÞq ½Mðk þMnþ

mÞ � nþm J 0½Hðnþ

mÞa e�in

þmz: ð52aÞ

The reflection coefficient is

R ¼ � i

½Mðk þMnþ0 Þ � nþ

0 a_F 1ða; nþ

0 ÞJ 0½Hðnþ

0 Þa ¼ i

_F 1ða; nþ0 Þ

ka: ð52bÞ

In what follows, some graphical results showing the effects of various parameters of the silenceron the transmission loss are presented. For numerical purposes the roots ±am of Eq. (32a) and b�

mof Eq. (40) should be determined. The Newton–Raphson algorithm is used to determine theseroots.Notice that, when 1/f‘ approaches zero, the roots n�

m tend to the values reported in [8].All the numerical results were derived by truncating the infinite series and the infinite systems of

linear algebraic equations after the first N terms. Table 1 shows the variation of the modulus oftransmission coefficient T against the truncation number N, for different values of the surfaceimpedance f‘. It is seen that the infinite series converge rapidly enough to truncate it after the firstfew terms. The truncation number is chosen as N = 5.The walls of the expansion chamber are assumed to be coated by a thin fibrous sheet for which

the surface impedance is [2]:

f‘ ¼ 0:5þ i,; �1:0 < , < 3:0: ð53Þ
The speed of the sound is taken in all the following graphical examples as 343.1 m/s.
1

itude of the transmission coefficient versus the truncation number for different values of the lining impedance

ka = 1, kb = 2, kl = 10, M = 0.1

jTj for f‘ = 0.5 + i jTj for f‘ = 0.5 + 2i jTj for f‘ = 0.5 + 3i

0.00246785266963341 0.0431074021973868 0.140471183078957

0.00276834331567489 0.0398005275161205 0.133033956097576

0.00276751773006485 0.0394138455670168 0.132116781648527

0.00276676384379854 0.0392992531127508 0.13184186741659

0.00276635479553718 0.0392508350703066 0.131725141195155

0.00276612199897657 0.0392260483393498 0.131665197155722

0.00276598009947649 0.0392117231742280 0.131630468647719

0.00276588835218191 0.0392027202285949 0.131608597846045

0.00276582611037655 0.0391967032784308 0.131593954521838

0.00276578219655870 0.0391924881226018 0.131583679635072

0.00276575019475196 0.0391894232414322 0.131576197765598

0.00276572623385736 0.0391871264809950 0.131570583552799

Fig. 3. (a) The effect of lining impedance f‘ on the transmission loss in the case where M = 0. (b) The effect of lining

impedance f‘ on the transmission loss in the case where M = 0.05. (c) The effect of lining impedance f‘ on thetransmission loss in the case where M = 0.1.


In Fig. 3a–c the influence of the specific impedance f‘ of the acoustical lining on the transmis-sion loss

TL ¼ �20 log10jTj ð54Þ
are displayed in the cases whereM = 0,M = 0.05 andM = 0.1, respectively. ForM = 0, Eq. (29a)and for M 5 0 Eq. (29b) are used to account for the perforated tube surface impedance fp. Theperforate hole diameter dh, the wall thickness of perforated tube tw and the porosity r are assumedto be dh = 2.49 · 10�3 m, tw = 0.9 · 10�3 m, r = 0.08 for the case of Fig. 3a, and dh = 2.49 · 10�3m, tw = 0.81 · 10�3 m, r = 0.037, for the cases of Fig. 3b and Figs. 4–6.In the case of Fig. 3a the mean flow in the central tube is absent. We see that an expansion
chamber with an absorbing lining have generally significantly higher acoustic attenuation thanthe one with no lining. When the surface impedance of the lining increases, the repeating domescorresponding to rigid expansion chamber are transformed to a single broad peak. The graphscorresponding to vanishing surface admittance (1/f‘ = 0) in Fig. 3a are exactly the same as theones previously obtained in [19, Fig. 6] and [20, Fig. 8], respectively. Similarly, the result related

Fig. 4. The effect of the mean flow velocity M on the transmission loss.

Fig. 5. The effect of the radius b on the transmission loss.


to the case 1/f‘ = 0, fp = 0 andM = 0, which is displayed in Fig. 3c, coincide with the result givenin [22, Fig. 4].From Fig. 4, it is seen that the mean flow velocity has an important effect on the performance of

the silencer. The magnitude of the TL peak is reduced when the mean flow velocity increases.Figs. 5 and 6 depict the transmission loss versus the frequency for different values of the expan-

sion chamber radius b and length l, respectively. It is observed that beyond 1200 Hz approxi-mately, the transmission loss decreases with increasing values of b. As to the effect of theexpansion chamber length, it is seen that the magnitude of the noise reduction peak is increasedwith the increasing values of l.Finally, Fig. 7 shows the effect of the central perforated tube on the acoustic performance of the

silencer. In the presence of mean flow, the porosity parameter r in Eq. (29b) is changed while the

Fig. 6. The effect of the length l of the expansion chamber on the transmission loss.

Fig. 7. The effect of the impedance (porosity r) of the central perforated tube on the transmission loss.


other quantities are kept as in the previous cases. It is observed that the magnitude and the cor-responding frequency of the main TL peak are increased as the porosity is increased.

4. Concluding remarks

A rigorous Wiener–Hopf approach is adopted to analyze the acoustic performance of a cylin-drical silencer whose walls are treated with a locally reacting absorbant liner. The expansionchamber is separated from the central airway which contain a uniform main gas flow by a perfo-rated cylindrical screen. The effect of the perforated center tube is modelled as a partially trans-missing surface characterized by the boundary conditions in (28a,b). The final solution involvesfour systems of linear algebraic equations involving four sets of infinitely many unknown


coefficients. Numerical solution to these systems is obtained for various values of the silencerparameters. In the case where the mean flow and the central perforated tube are absent, the resultsreduce exactly to those previously obtained in [21]. Furthermore, the graphs corresponding tovanishing surface admittance (1/f‘ = 0) in Fig. 3a and b are exactly the same as the ones previouslyobtained in [19, Fig. 6] and [20, Fig. 8], respectively. Similarly, the result related to the case 1/f‘ = 0, fp = 0 and M = 0, which is displayed in Fig. 3c, coincide with the result given in [22,Fig. 4]. These can be considered as an accurate check of the analysis carried out in this work.It is worth pointing out that this work may also serve as a benchmark solution for approximateand numerical techniques used for dealing with circular silencer problems.

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Documents

Wiener–Hopf approach for predicting the transmission loss of a circular silencer with a locally reacting lining