Research ArticleSound Transmission in a Duct with Sudden Area ExpansionExtended Inlet and Lined Walls in Overlapping Region

Ahmet Demir

Engineering Faculty Department of Mechatronics Karabuk University 78100 Karabuk Turkey

Correspondence should be addressed to Ahmet Demir ademirkarabukedutr

Received 19 July 2016 Accepted 26 October 2016

Academic Editor Toru Otsuru

Copyright copy 2016 Ahmet DemirThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The transmission of sound in a duct with sudden area expansion and extended inlet is investigated in the case where the walls ofthe duct lie in the finite overlapping region lined with acoustically absorbent materials By using the series expansion in the overlapregion and using the Fourier transform technique elsewhere we obtain a Wiener-Hopf equation whose solution involves a set ofinfinitely many unknown expansion coefficients satisfying a system of linear algebraic equations Numerical solution of this systemis obtained for various values of the problem parameters whereby the effects of these parameters on the sound transmission arestudied

1 Introduction

One can reduce the unwanted noise propagating along aduct by using a reactive or a dissipative silencer In reactivesilencers sudden area changes in cross-sectional area helpto reduce the energy in the transmitted wave via internalreflections Having a sudden area expansion together with asudden area contraction simple expansion chambers worksin accordance with this principle and is widely investigatedin literature [1ndash4] In further investigations it has been shownthat the extension of inlet and outlet tubes into the expansionchamber increased the acoustic attenuation performance [5ndash7]

On the other hand it has been proved that the treatmentof the duct walls with an acoustically absorbent lining isanother effective method in reducing unwanted noise [8]Application of locally reacting linings or expansion chambersin ducts are efficient methods for noise reduction Thesetwo methods were combined in [9] to discover transmissionproperties of a combination silencer consist of an expansionchamber whose walls are treated by acoustic liners and havebeen analysed by the author previously

In this paper the transmission of sound in an extendedtube resonator whose walls are in overlapping region whereextended inlet and expanding duct walls overlap are treatedby locally reacting lining is investigated So themain objective

of this paper is to reveal the influence of the partial liningon the transmitted field and to present an alternative methodof formulation The method previously employed in [10 11]consists of expanding the field in the overlap region into aseries of complete set of orthogonal eigenfunctions and usingthe Fourier transform technique elsewhere The problem isthen reduced directly into a Wiener-Hopf equation whosesolution involves a set of infinitely many unknown expansioncoefficients satisfying an infinite systems of linear algebraicequations Numerical solution to these systems is obtainedfor various values of the parameters of the problem such asthe radii of the semi-infinite waveguides the overlap lengthand the impedance loading whereby the effects of theseparameters on the transmitted field are presented graphically

The time dependence is assumed to be exp (minus119894120596119905) with 120596being the angular frequency and suppressed throughout

2 Materials and Methods

Consider two opposite semi-infinite circular cylindricalwaveguides of different radii with common longitudinal axissay 119911 in a cylindrical polar coordinate system (120588 120601 119911) Theyoccupy the regions 120588 = 119886 and 119911 lt 119897 and 120588 = 119887 gt 119886 and119911 gt 0 respectively where 119897 represents the overlap lengthThese two waveguides are connected with a vertical wall at

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2016 Article ID 9485163 8 pageshttpdxdoiorg10115520169485163

2 Advances in Acoustics and Vibration

119886 lt 120588 lt 119887 and 119911 = 0 The parts of the surfaces 119903 = 119886 + 0and 120588 = 119887 minus 0 lying in the overlap region 0 lt 119911 lt 119897 of thewaveguides and the vertical wall are assumed to be treatedby acoustically absorbing linings which are characterized byconstant but different surface admittances say 1205781 1205782 and1205783 respectively while the remaining parts are perfectly rigid(see Figure 1) The waveguides are immersed in an inviscidand compressible stationary fluid of density 0 and soundspeed 119888 A plane sound wave is incident from the positive 119911-direction through the waveguide of radius 120588 = 119886 From thesymmetry of the geometry of the problem and the incidentfield the acoustic field everywhere will be independent of the

120601 coordinate We shall therefore introduce a scalar potential119906(120588 119911) which defines the acoustic pressure and velocity by119901 = 1198941205960119906 and k = grad 119906 respectivelyLet the incident field be given by

119906119894 = exp (119894119896119911) (1)

where 119896 = 120596119888 denotes the wave number For the sake ofanalytical convenience we will assume that the surroundingmedium is slightly lossy and 119896 has a small positive imaginarypart The lossless case can be obtained by letting Im 119896 rarr 0 atthe end of the analysis

The total field 119906119879(120588 119911) can be written as

119906119879 (120588 119911) = 1199061 (120588 119911) + 119906119894 (120588 119911) 120588 isin (0 119886) 119911 isin (minusinfininfin) 119906(1)2 (120588 119911) [H (119911) minusH (119911 minus 119897)] + 119906(2)2 (120588 119911)H (119911 minus 119897) 120588 isin (119886 119887) 119911 isin (0infin) (2)

1199061(120588 119911) and 119906(119895)2 (120588 119911) (119895 = 1 2) denote the scattered fieldswhich satisfy the Helmholtz equation

[1120588 120597120597120588 (120588 120597120597120588) + 12059721205971199112 + 1198962][1199061 (120588 119911)119906(119895)2 (120588 119911)] = 0

119895 = 1 2(3)

and are to be determined with the help of the followingboundary and continuity relations

1205971205971205881199061 (119886 119911) = 0 119911 lt 119897 (4a)

120597120597119911119906(1)2 (120588 0) = 0119886 lt 120588 lt 119887

(4b)

[1198941198961205781 + 120597120597120588] 119906(1)2 (119886 119911) = 00 lt 119911 lt 119897

(4c)

[1198941198961205782 minus 120597120597120588] 119906(1)2 (119887 119911) = 00 lt 119911 lt 119897

(4d)

[1198941198961205783 + 120597120597119911] 119906(1)2 (120588 0) = 0119886 lt 120588 lt 119887

(4e)

120597120597120588119906(2)2 (119887 119911) = 0 119911 gt 119897 (4f)

119906(1)2 (120588 119897) minus 119906(2)2 (120588 119897) = 0119886 lt 120588 lt 119887 (4g)

120597120597119911119906(1)2 (120588 119897) minus 120597120597119911119906(2)2 (120588 119897) = 0119886 lt 120588 lt 119887

(4h)

1199061 (119886 119911) + 119906119894 (119886 119911) minus 119906(2)2 (119886 119911) = 0 119911 gt 119897 (4i)

1205971205971205881199061 (119886 119911) + 120597120597120588119906119894 (119886 119911) minus 120597120597120588119906(2)2 (119886 119911) = 0 119911 gt 119897 (4j)

In addition to these boundary and continuity relations onehas to take into account the following radiation and edgeconditions to ensure the uniqueness of the mixed boundaryvalue problem stated by (3) and (4a)ndash(4j)

119906 sim 119890119894119896119903119903 119903 = radic1205882 + 1199112 997888rarr infin (5)

119906119879 (120588 119911) = 119874 (1) 119911 997888rarr 119897 (6a)

120597120597120588119906119879 (120588 119911) = 119874 ((119911 minus 119897)minus12) 119911 997888rarr 119897 (6b)

21 The Wiener-Hopf Equations Consider the Fourier trans-form of theHelmholtz equation satisfied by the scattered field1199061(120588 119911) in the region 120588 lt 119886 for 119911 isin (minusinfininfin) namely

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)] 119865 (120588 120572) = 0 (7)

where 119865(120588 120572) is the Fourier transform of the field 1199061(120588 119911)defined to be

119865 (120588 120572) = intinfinminusinfin

1199061 (120588 119911) 119890119894120572119911119889119911= 119890119894120572119897 [119865+ (120588 120572) + 119865minus (120588 120572)]

(8a)

Advances in Acoustics and Vibration 3

with

119865minus (120588 120572) = int119897minusinfin 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8b)

119865+ (120588 120572) = intinfin119897 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8c)

Owing to the analytical properties of Fourier integrals119865+(120588 120572) and 119865minus(120588 120572) are regular functions in the upper half-plane Im120572 gt Im(minus119896) and in the lower half-plane Im120572 lt Im 119896respectively The solution of (7) reads

119865 (120588 120572) = minus119860 (120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) (9)

where 119860(120572) is a spectral coefficient to be determined and119870(120572) is the square-root function119870 (120572) = radic1198962 minus 1205722 (10)

which is defined in the complex 120572-plane cut as shown inFigure 2 such that 119870(0) = 119896 Consider now the Fouriertransform of (4a) namely

minus (119886 120572) = 0 (11)

The differentiation of (9) with respect to 120588 and putting 120588 = 119886gives

119890119894120572119897+ (119886 120572) = 119860 (120572) (12)

Substituting (12) into (9) yields

119865+ (120588 120572) = minus+ (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) minus 119865minus (120588 120572) (13)

In the region 119886 lt 120588 lt 119887 the field 119906(2)2 (120588 119911) satisfies theHelmholtz equation for 119911 isin (119897infin) as denoted in (3) TheFourier transform of this equation for the region in questionis

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]119866+ (120588 120572)= 119891 (120588) minus 119894120572119892 (120588)

(14)

where

119891 (120588) = 120597120597119911119906(2)2 (120588 119897) (15a)

119892 (120588) = 119906(2)2 (120588 119897) (15b)

In (14) 119866+(120588 120572) is a regular function in the upper half of thecomplex 120572-plane which is defined as

119866+ (120588 120572) = intinfin119897 119906(2)2 (120588 119911) 119890119894120572(119911minus119897)119889119911 (16)

Particular solutions to (14) can be found easily by usingGreenrsquos function which satisfies the Helmholtz equation

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]G (120588 120572) = 0120588 = 119905 120588 119905 isin (119886 119887)

(17)

with the following conditions

G (119905 + 0 119905 120572) = G (119905 minus 0 119905 120572) (18a)

120597120597120588G (119905 + 0 119905 120572) minus 120597120597120588G (119905 minus 0 119905 120572) = 1119905 (18b)

120597120597120588G (119887 119905 120572) = 0 (18c)

120597120597120588G (119886 119905 120572) = 0 (18d)

The solution is

G (120588 119905 120572) = 11198702 (120572)119872 (120572)119876 (120588 119905 120572) (19)

with

119876 (120588 119905 120572) = 1205872[1198690 (119870120588)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119886 le 120588 le 119905[1198690 (119870120588)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870119905)] 119905 le 120588 le 119887 (20a)

119872(120572) = [1198691 (119870119886) 1198841 (119870119887) minus 1198691 (119870119887) 1198841 (119870119886)] (20b)

Together with the boundary condition (4f) the solution canbe written as

119866+ (120588 120572) = 11198702 (120572)119872 (120572) 119861 (120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]

+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(21)

Here 119861(120572) is a spectral coefficient to be determined

4 Advances in Acoustics and Vibration

a

120588

bz

1205781

1205781

1205782

1205782

1205783

1205783

eikz

l

Figure 1 Geometry of the problem

minusk

k

Im 120572

Re 120572

ℒ+

ℒminus

Figure 2 Complex 120572-plane

The continuity relation in (4j) requires

+ (119886 120572) = + (119886 120572) (22)

119861(120572) can be solved uniquely from (22) as

119861 (120572) = minus+ (119886 120572) (23)

The substitution of (23) into (21) gives

119866+ (120588 120572) = 11198702 (120572)119872 (120572) minus+ (119886 120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(24)

Although the left-hand side of (24) is regular in the half-plane Im(120572) gt Im(minus119896) the regularity of the right hand sideis violated by the presence of simple poles lying at the upperhalf-plane namely at 120572 = 120572119898 (Im(120572119898) gt Im(119896)) with

1198702 (120572119898) [1198691 (119870119898119886) 1198841 (119870119898119887) minus 1198691 (119870119898119887) 1198841 (119870119898119886)]= 0 119870119898 = 119870 (120572119898) 119898 = 0 1 2 (25)

In order to provide regularity of the right hand side of (24) inthe upper half of the 120572-plane these poles must be eliminatedby imposing that their residues are zero This gives

+ (119886 119896) = 1120587 [1198910 minus 1198941198961198920] 1198862 minus 1198872119886119887 (26a)

+ (119886 120572119898) = 1120587 [119891119898 minus 119894120572119898119892119898]11986921 (119870119898119886) minus 11986921 (119870119898119887)1198691 (119870119898119886) 1198691 (119870119898119887) (26b)

for 119898 = 0 1 2 Here 119891119898 and 119892119898rsquos are the expansioncoefficients of the functions 119891(120588) and 119892(120588) respectivelywhichmay be represented trough the following complete setsof orthogonal functions

[119891 (120588)119892 (120588)] =2120587119887 [

11989101198920]

+ infinsum119898=1

[119891119898119892119898] [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)]

(27)

Using the continuity relation (4i) together with (24) andtaking into account (13) give

+ (119886 120572)1198702 (120572) 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) +1198862119865minus (119886 120572) = 12

sdot 11198702 (120572)119872 (120572) int119887

119886[119891 (119905) minus 119894120572119892 (119905)]

sdot [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119905 119889119905 minus 1198862sdot 119890119894119896119897119894 (120572 + 119896)

+ (119886 120572)1198702 (120572) 119873 (120572) + 1198862119865minus (119886 120572) = 119887120587119886[1198910 minus 1198941205721198920]1198962 minus 1205722 + 1120587

sdot infinsum119898=1

[119891119898 minus 119894120572119892119898]1205722119898 minus 12057221198691 (119870119898119887)1198691 (119870119898119886) minus

1198862 119890119894119896119897119894 (120572 + 119896)

(28)

which is the Wiener-Hopf equation to be solved throughclassical procedures Here119873(120572) stands for

119873(120572) = 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) (29)

The final solution of the W-H equation is determined to be

+ (119886 120572)(119896 + 120572) 119873+ (120572) = minus 119886120587119887[1198910 + 1198941198961198920](119896 + 120572)119873+ (119896) +

1120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894 (120572 + 119896)119873+ (119896)

(30)

Advances in Acoustics and Vibration 5

where 119873plusmn(120572) are the split functions resulting from theWiener-Hopf factorization of119873(120572) as

119873(120572) = 119873+ (120572)119873minus (120572) (31)

Their explicit expressions are given in [12] as

119873+ (120572) = [120587sdot 1198691 (119896119886)1198691 (119896119887) [1198691 (119896119886) 1198841 (119896119887) minus 1198691 (119896119887) 1198841 (119896119886)]]

minus12

sdot 119890minus120572120594 infinprod119899=0

(1 + 120572radic1198962 minus (119895119899119887)2)(1 + 120572radic1198962 minus (119895119899119886)2) (1 + 120572120572119899)

(32)

Here 119895119899rsquos are the roots of the Bessel function of the first kind1198691 (119895119899) = 0 119899 = 0 1 (33a)

120594 = 119894120587 [119887 ln 119887 minus 119886 ln 119886 minus (119887 minus 119886) ln (119887 minus 119886)] (33b)

with

119873minus (120572) = 119873+ (minus120572) (33c)

22 Determination of the Unknown Coefficients The field inthe region 119886 lt 120588 lt 119887 119911 isin (0 119897) can be expressed is terms ofthe waveguide normal modes as

119906(1)2 (120588 119911)= infinsum119899=0119886119899 [119890119894120573119899119911 minus 119875119899119890minus119894120573119899119911] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)] (34)

with

119875119899 = 1198941198961205783 + 1198941205731198991198941198961205783 minus 119894120573119899 (35a)

119877119899 = 11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)= 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887)

(35b)

In (34) 120585119899rsquos are the roots of the following equation11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)

minus 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887) = 0(36)

while 120573119899rsquos are defined as

120573119899 = radic1198962 minus 1205852119899 (37)

Taking into account (27) and (34) the continuity relations(4g) and (4h) can be written in the following form

2120587119887 [1198910 + 1198941205721198920] +infinsum119898=1

[119891119898 + 119894120572119892119898]sdot [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)] = 119894infinsum

119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)]

(38)

Multiplying (38) by 2120588120587119887 and [1198690(119870119898120588)1198701198981198841(119870119898119887) minus1198701198981198691(119870119898119887)1198840(119870119898120588)]120588 respectively and then integratingover 120588 from 119886 to 119887 read

[1198910 + 1198941205721198920]= 1198941198780infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]0119899 (39a)

[119891119898 + 119894120572119892119898]= 119894119878119898

infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]119898119899 (39b)

with

0119899 = 2120587120585119899 119886119887 [1198691 (120585119899119886) minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887)

minus 1198771198991198841 (120585119899119887)] (40a)

119898119899 = 2120585119899120587 (1205852119899 minus 1198702119898) 1198691 (119870119898119887)1198691 (119870119898119886) [1198691 (120585119899119886)

minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887) minus 1198771198991198841 (120585119899119887)] (40b)

where 1198780 and 119878119898 stand for

1198780 = 21205872 1198862 minus 11988721198872 (41a)

119878119898 = 2120587211986921 (119870119898119886) minus 11986921 (119870119898119887)11986921 (119870119898119886) (41b)

6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

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2 Advances in Acoustics and Vibration

119886 lt 120588 lt 119887 and 119911 = 0 The parts of the surfaces 119903 = 119886 + 0and 120588 = 119887 minus 0 lying in the overlap region 0 lt 119911 lt 119897 of thewaveguides and the vertical wall are assumed to be treatedby acoustically absorbing linings which are characterized byconstant but different surface admittances say 1205781 1205782 and1205783 respectively while the remaining parts are perfectly rigid(see Figure 1) The waveguides are immersed in an inviscidand compressible stationary fluid of density 0 and soundspeed 119888 A plane sound wave is incident from the positive 119911-direction through the waveguide of radius 120588 = 119886 From thesymmetry of the geometry of the problem and the incidentfield the acoustic field everywhere will be independent of the

120601 coordinate We shall therefore introduce a scalar potential119906(120588 119911) which defines the acoustic pressure and velocity by119901 = 1198941205960119906 and k = grad 119906 respectivelyLet the incident field be given by

119906119894 = exp (119894119896119911) (1)

where 119896 = 120596119888 denotes the wave number For the sake ofanalytical convenience we will assume that the surroundingmedium is slightly lossy and 119896 has a small positive imaginarypart The lossless case can be obtained by letting Im 119896 rarr 0 atthe end of the analysis

The total field 119906119879(120588 119911) can be written as

119906119879 (120588 119911) = 1199061 (120588 119911) + 119906119894 (120588 119911) 120588 isin (0 119886) 119911 isin (minusinfininfin) 119906(1)2 (120588 119911) [H (119911) minusH (119911 minus 119897)] + 119906(2)2 (120588 119911)H (119911 minus 119897) 120588 isin (119886 119887) 119911 isin (0infin) (2)

1199061(120588 119911) and 119906(119895)2 (120588 119911) (119895 = 1 2) denote the scattered fieldswhich satisfy the Helmholtz equation

[1120588 120597120597120588 (120588 120597120597120588) + 12059721205971199112 + 1198962][1199061 (120588 119911)119906(119895)2 (120588 119911)] = 0

119895 = 1 2(3)

and are to be determined with the help of the followingboundary and continuity relations

1205971205971205881199061 (119886 119911) = 0 119911 lt 119897 (4a)

120597120597119911119906(1)2 (120588 0) = 0119886 lt 120588 lt 119887

(4b)

[1198941198961205781 + 120597120597120588] 119906(1)2 (119886 119911) = 00 lt 119911 lt 119897

(4c)

[1198941198961205782 minus 120597120597120588] 119906(1)2 (119887 119911) = 00 lt 119911 lt 119897

(4d)

[1198941198961205783 + 120597120597119911] 119906(1)2 (120588 0) = 0119886 lt 120588 lt 119887

(4e)

120597120597120588119906(2)2 (119887 119911) = 0 119911 gt 119897 (4f)

119906(1)2 (120588 119897) minus 119906(2)2 (120588 119897) = 0119886 lt 120588 lt 119887 (4g)

120597120597119911119906(1)2 (120588 119897) minus 120597120597119911119906(2)2 (120588 119897) = 0119886 lt 120588 lt 119887

(4h)

1199061 (119886 119911) + 119906119894 (119886 119911) minus 119906(2)2 (119886 119911) = 0 119911 gt 119897 (4i)

1205971205971205881199061 (119886 119911) + 120597120597120588119906119894 (119886 119911) minus 120597120597120588119906(2)2 (119886 119911) = 0 119911 gt 119897 (4j)

In addition to these boundary and continuity relations onehas to take into account the following radiation and edgeconditions to ensure the uniqueness of the mixed boundaryvalue problem stated by (3) and (4a)ndash(4j)

119906 sim 119890119894119896119903119903 119903 = radic1205882 + 1199112 997888rarr infin (5)

119906119879 (120588 119911) = 119874 (1) 119911 997888rarr 119897 (6a)

120597120597120588119906119879 (120588 119911) = 119874 ((119911 minus 119897)minus12) 119911 997888rarr 119897 (6b)

21 The Wiener-Hopf Equations Consider the Fourier trans-form of theHelmholtz equation satisfied by the scattered field1199061(120588 119911) in the region 120588 lt 119886 for 119911 isin (minusinfininfin) namely

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)] 119865 (120588 120572) = 0 (7)

where 119865(120588 120572) is the Fourier transform of the field 1199061(120588 119911)defined to be

119865 (120588 120572) = intinfinminusinfin

1199061 (120588 119911) 119890119894120572119911119889119911= 119890119894120572119897 [119865+ (120588 120572) + 119865minus (120588 120572)]

(8a)

Advances in Acoustics and Vibration 3

with

119865minus (120588 120572) = int119897minusinfin 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8b)

119865+ (120588 120572) = intinfin119897 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8c)

Owing to the analytical properties of Fourier integrals119865+(120588 120572) and 119865minus(120588 120572) are regular functions in the upper half-plane Im120572 gt Im(minus119896) and in the lower half-plane Im120572 lt Im 119896respectively The solution of (7) reads

119865 (120588 120572) = minus119860 (120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) (9)

where 119860(120572) is a spectral coefficient to be determined and119870(120572) is the square-root function119870 (120572) = radic1198962 minus 1205722 (10)

which is defined in the complex 120572-plane cut as shown inFigure 2 such that 119870(0) = 119896 Consider now the Fouriertransform of (4a) namely

minus (119886 120572) = 0 (11)

The differentiation of (9) with respect to 120588 and putting 120588 = 119886gives

119890119894120572119897+ (119886 120572) = 119860 (120572) (12)

Substituting (12) into (9) yields

119865+ (120588 120572) = minus+ (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) minus 119865minus (120588 120572) (13)

In the region 119886 lt 120588 lt 119887 the field 119906(2)2 (120588 119911) satisfies theHelmholtz equation for 119911 isin (119897infin) as denoted in (3) TheFourier transform of this equation for the region in questionis

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]119866+ (120588 120572)= 119891 (120588) minus 119894120572119892 (120588)

(14)

where

119891 (120588) = 120597120597119911119906(2)2 (120588 119897) (15a)

119892 (120588) = 119906(2)2 (120588 119897) (15b)

In (14) 119866+(120588 120572) is a regular function in the upper half of thecomplex 120572-plane which is defined as

119866+ (120588 120572) = intinfin119897 119906(2)2 (120588 119911) 119890119894120572(119911minus119897)119889119911 (16)

Particular solutions to (14) can be found easily by usingGreenrsquos function which satisfies the Helmholtz equation

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]G (120588 120572) = 0120588 = 119905 120588 119905 isin (119886 119887)

(17)

with the following conditions

G (119905 + 0 119905 120572) = G (119905 minus 0 119905 120572) (18a)

120597120597120588G (119905 + 0 119905 120572) minus 120597120597120588G (119905 minus 0 119905 120572) = 1119905 (18b)

120597120597120588G (119887 119905 120572) = 0 (18c)

120597120597120588G (119886 119905 120572) = 0 (18d)

The solution is

G (120588 119905 120572) = 11198702 (120572)119872 (120572)119876 (120588 119905 120572) (19)

with

119876 (120588 119905 120572) = 1205872[1198690 (119870120588)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119886 le 120588 le 119905[1198690 (119870120588)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870119905)] 119905 le 120588 le 119887 (20a)

119872(120572) = [1198691 (119870119886) 1198841 (119870119887) minus 1198691 (119870119887) 1198841 (119870119886)] (20b)

Together with the boundary condition (4f) the solution canbe written as

119866+ (120588 120572) = 11198702 (120572)119872 (120572) 119861 (120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]

+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(21)

Here 119861(120572) is a spectral coefficient to be determined

4 Advances in Acoustics and Vibration

a

120588

bz

1205781

1205781

1205782

1205782

1205783

1205783

eikz

l

Figure 1 Geometry of the problem

minusk

k

Im 120572

Re 120572

ℒ+

ℒminus

Figure 2 Complex 120572-plane

The continuity relation in (4j) requires

+ (119886 120572) = + (119886 120572) (22)

119861(120572) can be solved uniquely from (22) as

119861 (120572) = minus+ (119886 120572) (23)

The substitution of (23) into (21) gives

119866+ (120588 120572) = 11198702 (120572)119872 (120572) minus+ (119886 120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(24)

Although the left-hand side of (24) is regular in the half-plane Im(120572) gt Im(minus119896) the regularity of the right hand sideis violated by the presence of simple poles lying at the upperhalf-plane namely at 120572 = 120572119898 (Im(120572119898) gt Im(119896)) with

1198702 (120572119898) [1198691 (119870119898119886) 1198841 (119870119898119887) minus 1198691 (119870119898119887) 1198841 (119870119898119886)]= 0 119870119898 = 119870 (120572119898) 119898 = 0 1 2 (25)

In order to provide regularity of the right hand side of (24) inthe upper half of the 120572-plane these poles must be eliminatedby imposing that their residues are zero This gives

+ (119886 119896) = 1120587 [1198910 minus 1198941198961198920] 1198862 minus 1198872119886119887 (26a)

+ (119886 120572119898) = 1120587 [119891119898 minus 119894120572119898119892119898]11986921 (119870119898119886) minus 11986921 (119870119898119887)1198691 (119870119898119886) 1198691 (119870119898119887) (26b)

for 119898 = 0 1 2 Here 119891119898 and 119892119898rsquos are the expansioncoefficients of the functions 119891(120588) and 119892(120588) respectivelywhichmay be represented trough the following complete setsof orthogonal functions

[119891 (120588)119892 (120588)] =2120587119887 [

11989101198920]

+ infinsum119898=1

[119891119898119892119898] [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)]

(27)

Using the continuity relation (4i) together with (24) andtaking into account (13) give

+ (119886 120572)1198702 (120572) 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) +1198862119865minus (119886 120572) = 12

sdot 11198702 (120572)119872 (120572) int119887

119886[119891 (119905) minus 119894120572119892 (119905)]

sdot [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119905 119889119905 minus 1198862sdot 119890119894119896119897119894 (120572 + 119896)

+ (119886 120572)1198702 (120572) 119873 (120572) + 1198862119865minus (119886 120572) = 119887120587119886[1198910 minus 1198941205721198920]1198962 minus 1205722 + 1120587

sdot infinsum119898=1

[119891119898 minus 119894120572119892119898]1205722119898 minus 12057221198691 (119870119898119887)1198691 (119870119898119886) minus

1198862 119890119894119896119897119894 (120572 + 119896)

(28)

which is the Wiener-Hopf equation to be solved throughclassical procedures Here119873(120572) stands for

119873(120572) = 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) (29)

The final solution of the W-H equation is determined to be

+ (119886 120572)(119896 + 120572) 119873+ (120572) = minus 119886120587119887[1198910 + 1198941198961198920](119896 + 120572)119873+ (119896) +

1120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894 (120572 + 119896)119873+ (119896)

(30)

Advances in Acoustics and Vibration 5

where 119873plusmn(120572) are the split functions resulting from theWiener-Hopf factorization of119873(120572) as

119873(120572) = 119873+ (120572)119873minus (120572) (31)

Their explicit expressions are given in [12] as

119873+ (120572) = [120587sdot 1198691 (119896119886)1198691 (119896119887) [1198691 (119896119886) 1198841 (119896119887) minus 1198691 (119896119887) 1198841 (119896119886)]]

minus12

sdot 119890minus120572120594 infinprod119899=0

(1 + 120572radic1198962 minus (119895119899119887)2)(1 + 120572radic1198962 minus (119895119899119886)2) (1 + 120572120572119899)

(32)

Here 119895119899rsquos are the roots of the Bessel function of the first kind1198691 (119895119899) = 0 119899 = 0 1 (33a)

120594 = 119894120587 [119887 ln 119887 minus 119886 ln 119886 minus (119887 minus 119886) ln (119887 minus 119886)] (33b)

with

119873minus (120572) = 119873+ (minus120572) (33c)

22 Determination of the Unknown Coefficients The field inthe region 119886 lt 120588 lt 119887 119911 isin (0 119897) can be expressed is terms ofthe waveguide normal modes as

119906(1)2 (120588 119911)= infinsum119899=0119886119899 [119890119894120573119899119911 minus 119875119899119890minus119894120573119899119911] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)] (34)

with

119875119899 = 1198941198961205783 + 1198941205731198991198941198961205783 minus 119894120573119899 (35a)

119877119899 = 11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)= 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887)

(35b)

In (34) 120585119899rsquos are the roots of the following equation11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)

minus 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887) = 0(36)

while 120573119899rsquos are defined as

120573119899 = radic1198962 minus 1205852119899 (37)

Taking into account (27) and (34) the continuity relations(4g) and (4h) can be written in the following form

2120587119887 [1198910 + 1198941205721198920] +infinsum119898=1

[119891119898 + 119894120572119892119898]sdot [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)] = 119894infinsum

119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)]

(38)

Multiplying (38) by 2120588120587119887 and [1198690(119870119898120588)1198701198981198841(119870119898119887) minus1198701198981198691(119870119898119887)1198840(119870119898120588)]120588 respectively and then integratingover 120588 from 119886 to 119887 read

[1198910 + 1198941205721198920]= 1198941198780infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]0119899 (39a)

[119891119898 + 119894120572119892119898]= 119894119878119898

infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]119898119899 (39b)

with

0119899 = 2120587120585119899 119886119887 [1198691 (120585119899119886) minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887)

minus 1198771198991198841 (120585119899119887)] (40a)

119898119899 = 2120585119899120587 (1205852119899 minus 1198702119898) 1198691 (119870119898119887)1198691 (119870119898119886) [1198691 (120585119899119886)

minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887) minus 1198771198991198841 (120585119899119887)] (40b)

where 1198780 and 119878119898 stand for

1198780 = 21205872 1198862 minus 11988721198872 (41a)

119878119898 = 2120587211986921 (119870119898119886) minus 11986921 (119870119898119887)11986921 (119870119898119886) (41b)

6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

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Advances in Acoustics and Vibration 3

with

119865minus (120588 120572) = int119897minusinfin 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8b)

119865+ (120588 120572) = intinfin119897 1199061 (120588 119911) 119890119894120572(119911minus119897)119889119911 (8c)

Owing to the analytical properties of Fourier integrals119865+(120588 120572) and 119865minus(120588 120572) are regular functions in the upper half-plane Im120572 gt Im(minus119896) and in the lower half-plane Im120572 lt Im 119896respectively The solution of (7) reads

119865 (120588 120572) = minus119860 (120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) (9)

where 119860(120572) is a spectral coefficient to be determined and119870(120572) is the square-root function119870 (120572) = radic1198962 minus 1205722 (10)

which is defined in the complex 120572-plane cut as shown inFigure 2 such that 119870(0) = 119896 Consider now the Fouriertransform of (4a) namely

minus (119886 120572) = 0 (11)

The differentiation of (9) with respect to 120588 and putting 120588 = 119886gives

119890119894120572119897+ (119886 120572) = 119860 (120572) (12)

Substituting (12) into (9) yields

119865+ (120588 120572) = minus+ (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886) minus 119865minus (120588 120572) (13)

In the region 119886 lt 120588 lt 119887 the field 119906(2)2 (120588 119911) satisfies theHelmholtz equation for 119911 isin (119897infin) as denoted in (3) TheFourier transform of this equation for the region in questionis

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]119866+ (120588 120572)= 119891 (120588) minus 119894120572119892 (120588)

(14)

where

119891 (120588) = 120597120597119911119906(2)2 (120588 119897) (15a)

119892 (120588) = 119906(2)2 (120588 119897) (15b)

In (14) 119866+(120588 120572) is a regular function in the upper half of thecomplex 120572-plane which is defined as

119866+ (120588 120572) = intinfin119897 119906(2)2 (120588 119911) 119890119894120572(119911minus119897)119889119911 (16)

Particular solutions to (14) can be found easily by usingGreenrsquos function which satisfies the Helmholtz equation

[1120588 120597120597120588 (120588 120597120597120588) + 1198702 (120572)]G (120588 120572) = 0120588 = 119905 120588 119905 isin (119886 119887)

(17)

with the following conditions

G (119905 + 0 119905 120572) = G (119905 minus 0 119905 120572) (18a)

120597120597120588G (119905 + 0 119905 120572) minus 120597120597120588G (119905 minus 0 119905 120572) = 1119905 (18b)

120597120597120588G (119887 119905 120572) = 0 (18c)

120597120597120588G (119886 119905 120572) = 0 (18d)

The solution is

G (120588 119905 120572) = 11198702 (120572)119872 (120572)119876 (120588 119905 120572) (19)

with

119876 (120588 119905 120572) = 1205872[1198690 (119870120588)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119886 le 120588 le 119905[1198690 (119870120588)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870120588)] [1198690 (119870119905)1198701198841 (119870119886) minus 1198701198691 (119870119886) 1198840 (119870119905)] 119905 le 120588 le 119887 (20a)

119872(120572) = [1198691 (119870119886) 1198841 (119870119887) minus 1198691 (119870119887) 1198841 (119870119886)] (20b)

Together with the boundary condition (4f) the solution canbe written as

119866+ (120588 120572) = 11198702 (120572)119872 (120572) 119861 (120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]

+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(21)

Here 119861(120572) is a spectral coefficient to be determined

4 Advances in Acoustics and Vibration

a

120588

bz

1205781

1205781

1205782

1205782

1205783

1205783

eikz

l

Figure 1 Geometry of the problem

minusk

k

Im 120572

Re 120572

ℒ+

ℒminus

Figure 2 Complex 120572-plane

The continuity relation in (4j) requires

+ (119886 120572) = + (119886 120572) (22)

119861(120572) can be solved uniquely from (22) as

119861 (120572) = minus+ (119886 120572) (23)

The substitution of (23) into (21) gives

119866+ (120588 120572) = 11198702 (120572)119872 (120572) minus+ (119886 120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(24)

Although the left-hand side of (24) is regular in the half-plane Im(120572) gt Im(minus119896) the regularity of the right hand sideis violated by the presence of simple poles lying at the upperhalf-plane namely at 120572 = 120572119898 (Im(120572119898) gt Im(119896)) with

1198702 (120572119898) [1198691 (119870119898119886) 1198841 (119870119898119887) minus 1198691 (119870119898119887) 1198841 (119870119898119886)]= 0 119870119898 = 119870 (120572119898) 119898 = 0 1 2 (25)

In order to provide regularity of the right hand side of (24) inthe upper half of the 120572-plane these poles must be eliminatedby imposing that their residues are zero This gives

+ (119886 119896) = 1120587 [1198910 minus 1198941198961198920] 1198862 minus 1198872119886119887 (26a)

+ (119886 120572119898) = 1120587 [119891119898 minus 119894120572119898119892119898]11986921 (119870119898119886) minus 11986921 (119870119898119887)1198691 (119870119898119886) 1198691 (119870119898119887) (26b)

for 119898 = 0 1 2 Here 119891119898 and 119892119898rsquos are the expansioncoefficients of the functions 119891(120588) and 119892(120588) respectivelywhichmay be represented trough the following complete setsof orthogonal functions

[119891 (120588)119892 (120588)] =2120587119887 [

11989101198920]

+ infinsum119898=1

[119891119898119892119898] [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)]

(27)

Using the continuity relation (4i) together with (24) andtaking into account (13) give

+ (119886 120572)1198702 (120572) 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) +1198862119865minus (119886 120572) = 12

sdot 11198702 (120572)119872 (120572) int119887

119886[119891 (119905) minus 119894120572119892 (119905)]

sdot [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119905 119889119905 minus 1198862sdot 119890119894119896119897119894 (120572 + 119896)

+ (119886 120572)1198702 (120572) 119873 (120572) + 1198862119865minus (119886 120572) = 119887120587119886[1198910 minus 1198941205721198920]1198962 minus 1205722 + 1120587

sdot infinsum119898=1

[119891119898 minus 119894120572119892119898]1205722119898 minus 12057221198691 (119870119898119887)1198691 (119870119898119886) minus

1198862 119890119894119896119897119894 (120572 + 119896)

(28)

which is the Wiener-Hopf equation to be solved throughclassical procedures Here119873(120572) stands for

119873(120572) = 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) (29)

The final solution of the W-H equation is determined to be

+ (119886 120572)(119896 + 120572) 119873+ (120572) = minus 119886120587119887[1198910 + 1198941198961198920](119896 + 120572)119873+ (119896) +

1120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894 (120572 + 119896)119873+ (119896)

(30)

Advances in Acoustics and Vibration 5

where 119873plusmn(120572) are the split functions resulting from theWiener-Hopf factorization of119873(120572) as

119873(120572) = 119873+ (120572)119873minus (120572) (31)

Their explicit expressions are given in [12] as

119873+ (120572) = [120587sdot 1198691 (119896119886)1198691 (119896119887) [1198691 (119896119886) 1198841 (119896119887) minus 1198691 (119896119887) 1198841 (119896119886)]]

minus12

sdot 119890minus120572120594 infinprod119899=0

(1 + 120572radic1198962 minus (119895119899119887)2)(1 + 120572radic1198962 minus (119895119899119886)2) (1 + 120572120572119899)

(32)

Here 119895119899rsquos are the roots of the Bessel function of the first kind1198691 (119895119899) = 0 119899 = 0 1 (33a)

120594 = 119894120587 [119887 ln 119887 minus 119886 ln 119886 minus (119887 minus 119886) ln (119887 minus 119886)] (33b)

with

119873minus (120572) = 119873+ (minus120572) (33c)

22 Determination of the Unknown Coefficients The field inthe region 119886 lt 120588 lt 119887 119911 isin (0 119897) can be expressed is terms ofthe waveguide normal modes as

119906(1)2 (120588 119911)= infinsum119899=0119886119899 [119890119894120573119899119911 minus 119875119899119890minus119894120573119899119911] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)] (34)

with

119875119899 = 1198941198961205783 + 1198941205731198991198941198961205783 minus 119894120573119899 (35a)

119877119899 = 11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)= 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887)

(35b)

In (34) 120585119899rsquos are the roots of the following equation11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)

minus 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887) = 0(36)

while 120573119899rsquos are defined as

120573119899 = radic1198962 minus 1205852119899 (37)

Taking into account (27) and (34) the continuity relations(4g) and (4h) can be written in the following form

2120587119887 [1198910 + 1198941205721198920] +infinsum119898=1

[119891119898 + 119894120572119892119898]sdot [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)] = 119894infinsum

119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)]

(38)

Multiplying (38) by 2120588120587119887 and [1198690(119870119898120588)1198701198981198841(119870119898119887) minus1198701198981198691(119870119898119887)1198840(119870119898120588)]120588 respectively and then integratingover 120588 from 119886 to 119887 read

[1198910 + 1198941205721198920]= 1198941198780infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]0119899 (39a)

[119891119898 + 119894120572119892119898]= 119894119878119898

infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]119898119899 (39b)

with

0119899 = 2120587120585119899 119886119887 [1198691 (120585119899119886) minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887)

minus 1198771198991198841 (120585119899119887)] (40a)

119898119899 = 2120585119899120587 (1205852119899 minus 1198702119898) 1198691 (119870119898119887)1198691 (119870119898119886) [1198691 (120585119899119886)

minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887) minus 1198771198991198841 (120585119899119887)] (40b)

where 1198780 and 119878119898 stand for

1198780 = 21205872 1198862 minus 11988721198872 (41a)

119878119898 = 2120587211986921 (119870119898119886) minus 11986921 (119870119898119887)11986921 (119870119898119886) (41b)

6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

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4 Advances in Acoustics and Vibration

a

120588

bz

1205781

1205781

1205782

1205782

1205783

1205783

eikz

l

Figure 1 Geometry of the problem

minusk

k

Im 120572

Re 120572

ℒ+

ℒminus

Figure 2 Complex 120572-plane

The continuity relation in (4j) requires

+ (119886 120572) = + (119886 120572) (22)

119861(120572) can be solved uniquely from (22) as

119861 (120572) = minus+ (119886 120572) (23)

The substitution of (23) into (21) gives

119866+ (120588 120572) = 11198702 (120572)119872 (120572) minus+ (119886 120572)sdot [1198690 (119870120588)1198701198841 (119870119887) minus 1198840 (119870120588)1198701198691 (119870119887)]+ int119887119886[119891 (119905) minus 119894120572119892 (119905)] 119876 (119905 120588 120572) 119905 119889119905

(24)

Although the left-hand side of (24) is regular in the half-plane Im(120572) gt Im(minus119896) the regularity of the right hand sideis violated by the presence of simple poles lying at the upperhalf-plane namely at 120572 = 120572119898 (Im(120572119898) gt Im(119896)) with

1198702 (120572119898) [1198691 (119870119898119886) 1198841 (119870119898119887) minus 1198691 (119870119898119887) 1198841 (119870119898119886)]= 0 119870119898 = 119870 (120572119898) 119898 = 0 1 2 (25)

In order to provide regularity of the right hand side of (24) inthe upper half of the 120572-plane these poles must be eliminatedby imposing that their residues are zero This gives

+ (119886 119896) = 1120587 [1198910 minus 1198941198961198920] 1198862 minus 1198872119886119887 (26a)

+ (119886 120572119898) = 1120587 [119891119898 minus 119894120572119898119892119898]11986921 (119870119898119886) minus 11986921 (119870119898119887)1198691 (119870119898119886) 1198691 (119870119898119887) (26b)

for 119898 = 0 1 2 Here 119891119898 and 119892119898rsquos are the expansioncoefficients of the functions 119891(120588) and 119892(120588) respectivelywhichmay be represented trough the following complete setsof orthogonal functions

[119891 (120588)119892 (120588)] =2120587119887 [

11989101198920]

+ infinsum119898=1

[119891119898119892119898] [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)]

(27)

Using the continuity relation (4i) together with (24) andtaking into account (13) give

+ (119886 120572)1198702 (120572) 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) +1198862119865minus (119886 120572) = 12

sdot 11198702 (120572)119872 (120572) int119887

119886[119891 (119905) minus 119894120572119892 (119905)]

sdot [1198690 (119870119905)1198701198841 (119870119887) minus 1198701198691 (119870119887) 1198840 (119870119905)] 119905 119889119905 minus 1198862sdot 119890119894119896119897119894 (120572 + 119896)

+ (119886 120572)1198702 (120572) 119873 (120572) + 1198862119865minus (119886 120572) = 119887120587119886[1198910 minus 1198941205721198920]1198962 minus 1205722 + 1120587

sdot infinsum119898=1

[119891119898 minus 119894120572119892119898]1205722119898 minus 12057221198691 (119870119898119887)1198691 (119870119898119886) minus

1198862 119890119894119896119897119894 (120572 + 119896)

(28)

which is the Wiener-Hopf equation to be solved throughclassical procedures Here119873(120572) stands for

119873(120572) = 1198691 (119870119887)120587119872 (120572) 1198691 (119870119886) (29)

The final solution of the W-H equation is determined to be

+ (119886 120572)(119896 + 120572) 119873+ (120572) = minus 119886120587119887[1198910 + 1198941198961198920](119896 + 120572)119873+ (119896) +

1120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894 (120572 + 119896)119873+ (119896)

(30)

Advances in Acoustics and Vibration 5

where 119873plusmn(120572) are the split functions resulting from theWiener-Hopf factorization of119873(120572) as

119873(120572) = 119873+ (120572)119873minus (120572) (31)

Their explicit expressions are given in [12] as

119873+ (120572) = [120587sdot 1198691 (119896119886)1198691 (119896119887) [1198691 (119896119886) 1198841 (119896119887) minus 1198691 (119896119887) 1198841 (119896119886)]]

minus12

sdot 119890minus120572120594 infinprod119899=0

(1 + 120572radic1198962 minus (119895119899119887)2)(1 + 120572radic1198962 minus (119895119899119886)2) (1 + 120572120572119899)

(32)

Here 119895119899rsquos are the roots of the Bessel function of the first kind1198691 (119895119899) = 0 119899 = 0 1 (33a)

120594 = 119894120587 [119887 ln 119887 minus 119886 ln 119886 minus (119887 minus 119886) ln (119887 minus 119886)] (33b)

with

119873minus (120572) = 119873+ (minus120572) (33c)

22 Determination of the Unknown Coefficients The field inthe region 119886 lt 120588 lt 119887 119911 isin (0 119897) can be expressed is terms ofthe waveguide normal modes as

119906(1)2 (120588 119911)= infinsum119899=0119886119899 [119890119894120573119899119911 minus 119875119899119890minus119894120573119899119911] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)] (34)

with

119875119899 = 1198941198961205783 + 1198941205731198991198941198961205783 minus 119894120573119899 (35a)

119877119899 = 11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)= 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887)

(35b)

In (34) 120585119899rsquos are the roots of the following equation11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)

minus 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887) = 0(36)

while 120573119899rsquos are defined as

120573119899 = radic1198962 minus 1205852119899 (37)

Taking into account (27) and (34) the continuity relations(4g) and (4h) can be written in the following form

2120587119887 [1198910 + 1198941205721198920] +infinsum119898=1

[119891119898 + 119894120572119892119898]sdot [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)] = 119894infinsum

119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)]

(38)

Multiplying (38) by 2120588120587119887 and [1198690(119870119898120588)1198701198981198841(119870119898119887) minus1198701198981198691(119870119898119887)1198840(119870119898120588)]120588 respectively and then integratingover 120588 from 119886 to 119887 read

[1198910 + 1198941205721198920]= 1198941198780infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]0119899 (39a)

[119891119898 + 119894120572119892119898]= 119894119878119898

infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]119898119899 (39b)

with

0119899 = 2120587120585119899 119886119887 [1198691 (120585119899119886) minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887)

minus 1198771198991198841 (120585119899119887)] (40a)

119898119899 = 2120585119899120587 (1205852119899 minus 1198702119898) 1198691 (119870119898119887)1198691 (119870119898119886) [1198691 (120585119899119886)

minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887) minus 1198771198991198841 (120585119899119887)] (40b)

where 1198780 and 119878119898 stand for

1198780 = 21205872 1198862 minus 11988721198872 (41a)

119878119898 = 2120587211986921 (119870119898119886) minus 11986921 (119870119898119887)11986921 (119870119898119886) (41b)

6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

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Advances in Acoustics and Vibration 5

where 119873plusmn(120572) are the split functions resulting from theWiener-Hopf factorization of119873(120572) as

119873(120572) = 119873+ (120572)119873minus (120572) (31)

Their explicit expressions are given in [12] as

119873+ (120572) = [120587sdot 1198691 (119896119886)1198691 (119896119887) [1198691 (119896119886) 1198841 (119896119887) minus 1198691 (119896119887) 1198841 (119896119886)]]

minus12

sdot 119890minus120572120594 infinprod119899=0

(1 + 120572radic1198962 minus (119895119899119887)2)(1 + 120572radic1198962 minus (119895119899119886)2) (1 + 120572120572119899)

(32)

Here 119895119899rsquos are the roots of the Bessel function of the first kind1198691 (119895119899) = 0 119899 = 0 1 (33a)

120594 = 119894120587 [119887 ln 119887 minus 119886 ln 119886 minus (119887 minus 119886) ln (119887 minus 119886)] (33b)

with

119873minus (120572) = 119873+ (minus120572) (33c)

22 Determination of the Unknown Coefficients The field inthe region 119886 lt 120588 lt 119887 119911 isin (0 119897) can be expressed is terms ofthe waveguide normal modes as

119906(1)2 (120588 119911)= infinsum119899=0119886119899 [119890119894120573119899119911 minus 119875119899119890minus119894120573119899119911] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)] (34)

with

119875119899 = 1198941198961205783 + 1198941205731198991198941198961205783 minus 119894120573119899 (35a)

119877119899 = 11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)= 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887)

(35b)

In (34) 120585119899rsquos are the roots of the following equation11989411989612057811198690 (120585119899119886) minus 1205851198991198691 (120585119899119886)11989411989612057811198840 (120585119899119886) minus 1205851198991198841 (120585119899119886)

minus 11989411989612057821198690 (120585119899119887) + 1205851198991198691 (120585119899119887)11989411989612057821198840 (120585119899119887) + 1205851198991198841 (120585119899119887) = 0(36)

while 120573119899rsquos are defined as

120573119899 = radic1198962 minus 1205852119899 (37)

Taking into account (27) and (34) the continuity relations(4g) and (4h) can be written in the following form

2120587119887 [1198910 + 1198941205721198920] +infinsum119898=1

[119891119898 + 119894120572119892119898]sdot [1198690 (119870119898120588)1198701198981198841 (119870119898119887)minus 1198701198981198691 (119870119898119887) 1198840 (119870119898120588)] = 119894infinsum

119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897] [1198690 (120585119899120588) minus 1198771198991198840 (120585119899120588)]

(38)

Multiplying (38) by 2120588120587119887 and [1198690(119870119898120588)1198701198981198841(119870119898119887) minus1198701198981198691(119870119898119887)1198840(119870119898120588)]120588 respectively and then integratingover 120588 from 119886 to 119887 read

[1198910 + 1198941205721198920]= 1198941198780infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]0119899 (39a)

[119891119898 + 119894120572119892119898]= 119894119878119898

infinsum119899=0119886119899 [(120572 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572 minus 120573119899) 119890minus119894120573119899119897]119898119899 (39b)

with

0119899 = 2120587120585119899 119886119887 [1198691 (120585119899119886) minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887)

minus 1198771198991198841 (120585119899119887)] (40a)

119898119899 = 2120585119899120587 (1205852119899 minus 1198702119898) 1198691 (119870119898119887)1198691 (119870119898119886) [1198691 (120585119899119886)

minus 1198771198991198841 (120585119899119886)] minus [1198691 (120585119899119887) minus 1198771198991198841 (120585119899119887)] (40b)

where 1198780 and 119878119898 stand for

1198780 = 21205872 1198862 minus 11988721198872 (41a)

119878119898 = 2120587211986921 (119870119898119886) minus 11986921 (119870119898119887)11986921 (119870119898119886) (41b)

6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

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6 Advances in Acoustics and Vibration

Using the W-H solution (30) together with (26a) and(26b) we obtain a set of linear algebraic equations in termsof the unknown coefficients 119891119898 and 1198921198981205872 119887119886 [1198910 minus 1198941198961198920] 1198780119873+ (119896) = 119887120587119886

[1198910 + 1198941198961198920]119873+ (119896) + 119896120587sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886) minus

119896119886119890119894119896119897119894119873+ (119896) (42a)

1205872 [119891119903 minus 119894120572119903119892119903]1198691 (119870119903119886)1198691 (119870119903119887) 119878119903119873+ (120572119903) =

119887120587119886[1198910 + 1198941198961198920]119873+ (119896)

+ 1120587 (119896 + 120572119903)sdot infinsum119898=1

[119891119898 + 119894120572119898119892119898]2120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(42b)

and taking into account (39a) and (39b) we obtain now a setof equations to determine the unknown expansion coefficient119886119899 as

minus 1198941205872 119887119886119873+ (119896)infinsum119899=0119886119899 [(119896 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 + 120573119899) 119890minus119894120573119899119897]0119899

= 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897 minus 119875119899 (119896 minus 120573119899)

sdot 119890minus119894120573119899119897]0119899 + 119894 119896120587infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897

minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897] 119898119899120572119898119873+ (120572119898)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

minus 119896119886119890119894119896119897119894119873+ (119896)

(43a)

minus 11989412058721198691 (119870119903119886)1198691 (119870119903119887)119873+ (120572119903)

infinsum119899=0119886119899 [(120572119903 minus 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119903

+ 120573119899) 119890minus119894120573119899119897]119903119899 = 119887120587119886 1119873+ (119896)1198941198780infinsum119899=0119886119899 [(119896 + 120573119899) 119890119894120573119899119897

minus 119875119899 (119896 minus 120573119899) 119890minus119894120573119899119897]0119899 + 119894120587sdot infinsum119899=0

infinsum119898=1

119886119899 [(120572119898 + 120573119899) 119890119894120573119899119897 minus 119875119899 (120572119898 minus 120573119899) 119890minus119894120573119899119897]

sdot (119896 + 120572119903)1198981198992120572119898 (120572119898 + 120572119903)1198691 (119870119898119887)1198781198981198691 (119870119898119886)

119896 + 120572119898119873+ (120572119898)minus 119896119886119890119894119896119897119894119873+ (119896)

(43b)

23 Reflected and Transmitted Fields According to (8a) thescattered field in the region 0 lt 120588 lt 119886 that is 1199061(120588 119911) can be

obtained by taking the inverse Fourier transform of 119865(120588 120572)By considering (13) we write

1199061 (120588 119911)= minus 12120587 intL + (119886 120572) 1198690 (119870120588)119870 (120572) 1198691 (119870119886)119890

minus119894120572(119911minus119897)119889120572 (44)

The evaluation of this integral for 119911 lt 119897 and 119911 gt 119897 will give usthe reflectedwave propagating backward in the inner cylinderand the transmitted wave respectively

For 119911 lt 119897 the integral is calculated by closing thecontour in the upper half-plane and evaluating the residuescontributions from the simple poles occurring at the zerosof 1198691(119870119886) lying in the upper 120572-half-plane namely at 119870119886 =119895119899 The reflection coefficient R of the fundamental mode isdefined as the complex coefficient multiplying the travellingwave term exp(minus119894119896119911) and is computed from the contributionof the first pole at 120572 = 119896The result is

R = minus 1198901198942119896119897[119873+ (119896)]2 minus

119894120587[1198910 + 1198941198961198920]119896119887 [119873+ (119896)]2 119890

119894119896119897

+ 119894120587 119890119894119896119897119886119873+ (119896)infinsum119898=1

[119891119898 + 119894120572119898119892119898]120572119898119873+ (120572119898)1198691 (119870119898119887)1198691 (119870119898119886)

(45)

The first term is the reflection coefficient related to the casewhere a semi-infinite rigid duct is inserted axially into a largerrigid tube of infinite length [12] whereas the second one is thecorrection term involving the effect of the impedances of theannular region and the overlap length

Similarly the transmission coefficient T of the funda-mental mode which is defined as the complex coefficient ofexp(119894119896119911) is obtained by evaluating the integral in (44) for119911 gt 119897This integral is now computed by closing the contourin the lower half of the complex 120572-plane The pole of interestis at 120572 = minus119896 whose contribution gives

(minus1 +T) 119890119894119896119911 + 119874(119890119894radic1198962minus(119895119899119887)2119911) (46a)

with

T = 11988621198872 + 119894119890minus119894119896119897120587119896119886 (119887119886 minus 119886119887) [1198910 + 1198941198961198920] (46b)

The first term in (46a) cancels out the incident wave in theregion 120588 lt 119886 119911 gt 119897 while the second is the transmissioncoefficient of the fundamental mode

3 Results and Discussion

In this section in order to show the effects of the parameterslike the length of the extended inlet 119897 and the surfaceadmittance 120578123 on the transmitted field some numericalresults showing variation of the transmission coefficient Twith different parameters are presented In all numericalcalculations the solution of the infinite system of algebraicequations is obtained by truncating the infinite series at119873 =5 since the transmission coefficient becomes insensitive for

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 7

0 1 2 3 4 5024

026

028

03

032

034

036

038

04

042

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = 075X1 = 05X1 = 025

ka = 1 kb = 2 kl = 1

X2

05 15 25 35 45

Figure 3 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 gt 0) for different values of 1205781 = 1198941198831

minus5 minus4 minus3 minus2 minus1 0005

015

025

035

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = minus075X1 = minus05X1 = minus025

ka = 1 kb = 2 kl = 1

X2

04

03

02

01

minus45 minus35 minus25 minus15 minus05

Figure 4 Transmission coefficientT versus the surface admittance1205782 = 1198941198832 (1198832 lt 0) for different values of 1205781 = 1198941198831

119873 gt 5 We also limit ourselves with only imaginary values ofsurface admittance 12057812 for simplicity

In Figures 3 and 4 while the admittance 1198832 gt 0 of thelateral wall of the expanding duct increases the transmittedfield is ascending until some value of 1198832 then it startsto attenuate gradually But for negative values of 1198832 theattenuation is more visible especially around minus05 lt 1198832 lt 0For different values of1198831 not much but some decrease in thetransmitted field is observed

In Figure 5 an oscillatory behaviour is seen for increasingvalues of the extended inlet length 119896119897 but this behaviour is

0 5 10 15 20 25 300

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

ka = 1 kb = 2

06

05

04

03

02

01

kl

X1 = X2 = 05

X1 = X2 = 0

X1 = X2 = minus05

Figure 5 Transmission coefficient T versus the extended inletlength 119896119897 for different values of 1205781 = 1198941198831 and 1205782 = 1198941198832

0 1 2 3 4 5

025

035

045

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

X1 = X2 = 05

04

05

02

03

05 15 25 35 45

Im(1205783) = minus05

Im(1205783) = 00

Im(1205783) = 05

Re(1205783)

ka = 1 kb = 2 kl = 1

Figure 6 Transmission coefficient T versus the real part of 1205783 fordifferent values of Im(1205783)

broken for negative values of 1198831 and 1198832 From Figure 6 itis observed that the transmission does not alter as the realpart of 1205783 increased But for small positive values of Re(1205783)imaginary part Im(1205783) becomes effectiveThemost reductionon the sound transmission is seen for the negative value ofIm(1205783)

Figure 7 shows an excellent agreement between thepresent paper (for the case of extended inlet length 119896119897 rarr 0)and the previous study [9] of the author (for the case of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Acoustics and Vibration

1 2 3 4 50

1

Mod

ulus

of t

rans

miss

ion

coeffi

cien

t

Ref [9] Present paper

ka = 1

kb

15 25 35 45

02

01

03

04

05

06

07

08

09

Figure 7 Transmission coefficientT versus the expansion chamberradius 119896119887

expansion chamber length 119896119897 rarr infin and surface admittance istaken to be zero) In this comparison transmission coefficientis calculated as though it is in a rigid-walled duct with suddenarea expansion (without extended inlet)

4 Conclusions

This paper examines the transmission of sound waves in anextended tube resonator whose walls in overlapping regionwhere extended inlet and expanding duct walls overlap aretreated by acoustically absorbing materials of finite lengthIn the present work the lined region of the inner surfaceis assumed to be finite which makes the problem morecomplicated To overcome the additional difficulty caused bythe impedance discontinuity a hybrid method of formulationconsisting of expressing the total field in terms of com-plete sets of orthogonal waveguide modes where availableand using the Fourier transform elsewhere is adopted Themixed boundary value problem is reduced to a Wiener-Hopfequation whose solution involves infinitely many expansioncoefficients satisfying an infinite system of linear algebraicequations These equations are solved numerically and theeffects of various parameters on transmitted field such asthe extended inlet length and the surface admittance of thelined section are displayed graphically As a future work asimilar problem now with an extended outlet will be studiedfollowing the same method used here

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Miles ldquoThe analysis of plane discontinuities in cylindricaltubes Part Irdquo The Journal of the Acoustical Society of Americavol 17 pp 259ndash271 1946

[2] M L Munjal Acoustics of Ducts and Mufflers Wiley-Interscience New York NY USA 1987

[3] J Kergomard and A Garcia ldquoSimple discontinuities in acousticwaveguides at low frequencies critical analysis and formulaerdquoJournal of Sound and Vibration vol 114 no 3 pp 465ndash479 1987

[4] A Selamet and P M Radavich ldquoThe effect of length onthe acoustic attenuation performance of concentric expansionchambers an analytical computational and experimental inves-tigationrdquo Journal of Sound andVibration vol 201 no 4 pp 407ndash426 1997

[5] M Abom ldquoDerivation of four-pole parameters includinghigher order mode effects for expansion chamber mufflers withextended inlet and outletrdquo Journal of Sound and Vibration vol137 no 3 pp 403ndash418 1990

[6] K S Peat ldquoThe acoustical impedance at the junction of anextended inlet or outlet ductrdquo Journal of Sound and Vibrationvol 9 pp 101ndash110 1991

[7] A Selamet and Z L Ji ldquoAcoustic attenuation performanceof circular expansion chambers with extended inletoutletrdquoJournal of Sound and Vibration vol 223 no 2 pp 197ndash212 1999

[8] A D Rawlins ldquoRadiation of sound from an unflanged rigidcylindrical ductwith an acoustically absorbing internal surfacerdquoProceedings of the Royal Society London Series AMathematicalPhysical and Engineering Sciences vol 361 no 1704 pp 65ndash911978

[9] A Demir and A Buyukaksoy ldquoTransmission of sound waves ina cylindrical duct with an acoustically lined mufflerrdquo Interna-tional Journal of Engineering Science vol 41 no 20 pp 2411ndash2427 2003

[10] A Buyukaksoy and A Demir ldquoDiffraction of sound wavesby a rigid cylindrical cavity of finite length with an internalimpedance surfacerdquoZeitschrift fur AngewandteMathematik undPhysik vol 56 no 4 pp 694ndash717 2005

[11] A Buyukaksoy G Uzgoren and F Birbir ldquoThe scattering ofa plane wave by two parallel semi-infinite overlapping screenswith dielectric loadingrdquo Wave Motion vol 34 no 4 pp 375ndash389 2001

[12] A D Rawlins ldquoA bifurcated circular waveguide problemrdquo IMAJournal of Applied Mathematics vol 54 no 1 pp 59ndash81 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of