reflection and transmission of plane acoustic waves in an infinite annular duct with a finite gap on...

Research Article

Received 23 September 2009 Published online 24 August 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1351MOS subject classification: 74 J 20; 42 B 10; 35 J 25

Reflection and transmission of plane acousticwaves in an infinite annular duct with a finitegap on the inner wall

Gökhan Çınara∗†, Hülya Öztürkb and Özge Yanaz Çınarb

Communicated by W. Sprößig

The diffraction of acoustic waves by an infinitely long annular duct having a finite gap on the inner wall is investigatedrigorously. The related boundary-value problem is formulated into a modified Wiener–Hopf equation, which is thenreduced to a pair of simultaneous Fredholm integral equations of the second kind. At the end of the analysis, numericalresults illustrating the effects of the width of the coaxial cylindrical waveguide and the gap length on the diffractionphenomenon are presented. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: wave equation; diffraction; scattering; acoustics; annular duct; finite gap; Wiener–Hopf technique

1. Introduction

The propagation of plane acoustic waves along coaxial waveguides is an important topic in diffraction theory and relevant to manyapplications including reduction of noise in exhaust systems, jet fans, room ventilators, sonar projectors and large dryers in thermalengineering. That is why the need for accurate techniques to understand the propagation of scalar waves along coaxial ducts hasbeen essential for the last half century.

The reduction of noise in duct systems is generally achieved by silencers, which can be classified as ‘dissipative’ and ‘reactive’.Dissipative silencers are special materials that can dissipate the acoustic energy into heat energy and reduce the transmitted acousticfield outside the duct. On the other hand, reactive silencers are formed of sudden changes on the geometry or the impedance ofwalls, such as expansion chambers. The design of silencers involve other engineering problems than noise reduction. For instance,in exhaust systems of internal combustion engines, improved sound dissipation and/or absorption generally do so at the expenseof reduced performance, i.e. lower horsepower (this also means higher power consumption).

When a sound wave propagates along a duct, rapid changes in the cross-sectional area of the duct can cause significant reflection,which helps reducing the energy in the transmitted wave. This, together with the cavity resonance mechanism, is the method bywhich silencer box helps to reduce noise in the car exhaust system [1]. Modelling of simple silencer geometries has been doneand classified depending on the assumption of locally- or bulk-reacting lining. The effect of a type of silencer can be analyzedrigorously by applying integral transform approach. Then, for the systems involving more than one type of silencers, the analysisis most conveniently carried out using the transfer matrix approach by considering the known behaviors of silencer elements [2].The detailed design procedures for silencers are available in [3].

Considering modifications only on the geometry of an infinitely long annular duct, in order to provide noise reduction, anexpansion chamber on the outer wall or a finite gap on the inner wall can act as a reactive silencer. Expansion chambers havebeen widely analyzed in the literature (e.g. [4--9]). In this study, the effect of the existence of a finite gap on the inner wall of anannular duct to the sound propagation is investigated rigorously and some numerical results are presented illustrating the effectsof the waveguide width and gap length on the reflected and transmitted fields (hence, the pressure drop and noise reduction). Byapplying direct Fourier transform, the problem is reduced into the solution of a modified Wiener–Hopf equation. This equation issolved via a pair of simultaneous Fredholm integral equations of the second type and the reflection and transmission coefficientsare determined explicitly in terms of infinite number of unknown coefficients, which are solved numerically.

The time dependence of the waves are assumed and suppressed as e−i�t throughout the paper.

aElectronics Engineering Department, Gebze Institute of Technology, Cayirova Campus, 41400, Gebze, Kocaeli, TurkeybDepartment of Mathematics, Gebze Institute of Technology, Cayirova Campus, 41400, Gebze, Kocaeli, Turkey∗Correspondence to: Gökhan Çınar, Electronics Engineering Department, Gebze Institute of Technology, Cayirova Campus, 41400, Gebze, Kocaeli, Turkey.†E-mail: [email protected]

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Copyright © 2010 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 220–230

G. ÇINAR, H. ÖZTÜRK AND Ö. Y. ÇINAR

Figure 1. Geometry of the problem.

2. Formulation of the problem

Consider a time-harmonic incident field propagating along a coaxial cylindrical waveguide with a finite gap on the inner wall, asshown in Figure 1, given by

ui(�, z)=eikz. (1)

The total field can be defined as

uT (�, z)={

u2(�, z), �<a

ui(�, z)+u1(�, z), a<�<b(2)

where u1(�, z) and u2(�, z) denote the scattered fields in their relevant regions.The field terms satisfy the below boundary conditions and continuity relations:

��

u1(b, z) = 0, z ∈ (−∞,∞) (3a)

��

u1(a, z) = 0, z ∈ (−∞, 0)∪(l,∞) (3b)

��z

u2(�, 0) = 0, �∈ (0, a) (3c)

��z

u2(�, l) = 0, �∈ (0, a) (3d)

��

u1(a, z) = ��

u2(a, z), z ∈ (0, l) (3e)

and

u1(a, z)+ui(a, z)=u2(a, z), z ∈ (0, l). (3f)

To ensure the uniqueness of the solution, the radiation

�u1

�r− iku1 =O(r−1/2), r =

√�2 +z2 →∞ (3g)

and the edge conditions

uT (a, z)=O(1), z →0, l (3h)

and

��

uT (a, z)=O(z−1/3), z →0, l (3i)

should be taken into account [10]. Notice that dos Santos and Teixeira have shown that the edge conditions allow to specifyingthe space of solutions to the Wiener–Hopf equation as a Sobolev space [11]. The existence of a unique solution and continuousdependence on the known data may be discussed by formulating the boundary value problem in a Sobolev space setting by lookingfor solutions in the finite energy norm space and using the operator-theoretic methods. But this fairly difficult task is beyond thescope of this paper and for practical purposes we have chosen the classical formulation here.


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2.1. Derivation of the Wiener–Hopf equation

In the region a<�<b, −∞<z<∞, the scattered field u1(�, z) satisfies the Helmholtz equation[1

�

��

(�

��

)+ �2

�z2+k2

]u1(�, z)=0 (4)

whose Fourier transform with respect to z yields[1

�

��

(�

��

)+K2(�)

]�(�,�)=0 (5)

where �(�,�) stands for the Fourier transform of u1(�, z). Here, K(�)=√

k2 −�2 is the square-root function defined in the complex�-plane, cut along �=k to �=k+ i∞ and �=−k to �=−k− i∞, such that K(0)=k. It can be expressed as

�(�,�)=�−(�,�)+�1(�,�)+ei�l�+(�,�) (6)

with

�−(�,�) =∫ 0

−∞u1(�, z)ei�z dz, (7a)

�+(�,�) =∫ ∞

lu1(�, z)ei�(z−l) dz (7b)

and

�1(�,�)=∫ l

0u1(�, z)ei�z dz. (7c)

By taking into account the following asymptotic behavior (as described in [12]) of u1(�, z)

u1(�, z)=O(e±ikz), z →±∞, (8)

one can show that �−(�,�) and �+(�,�) are regular functions of � in the half-planes Im�>Im(−k) and Im�<Im(k), respectively, while�1(�,�) is an entire function of �. The general solution of (5) yields

�(�,�)=−A(�)Jo(K�)

KJ1(Kb)−B(�)

Yo(K�)

KY1(Kb)(9)

where A(�) and B(�) are unknown spectral coefficients to be found and Jn(z) and Yn(z)(n=0, 1) are the well-known Bessel andNeumann functions. The Fourier transform of the boundary condition (3a) gives

�′(b,�)=0 (10)

which reads

B(�)=−A(�). (11)

In (10), the prime denotes the first-order derivative with respect to �. On the other hand, the Fourier transform of the boundarycondition (3b) yields

�′−(a,�)+ei�l�′+(a,�)=0. (12)

Taking into account this relation, together with (6), (9) and (11), one can write

A(�)=�′1(a,�)

J1(Kb)Y1(Kb)

[J1(Ka)Y1(Kb)−J1(Kb)Y1(Ka)]. (13)

The scattered field u2(�, z) satisfies the Helmholtz equation in the region �<a, 0<z<l[1

�

��

(�

��

)+ �2

�z2+k2

]u2(�, z)=0 (14)

whose Fourier transform is [1

�

��

(�

��

)+K2(�)

]�1(�,�)= i�[ei�lf (�)−g(�)] (15)

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where the boundary conditions (3c) and (3d) are taken into account while �1(�,�), f (�) and g(�) stand for

�1(�,�)=∫ l

0u2(�, z)ei�z dz (16)

and

f (�)=u2(�, l), g(�)=u2(�, 0), (17)

respectively. A particular solution of (15) can be expressed in terms of the Green’s function related to this differential equation,which satisfies [

1

�

��

(�

��

)+K2(�)

]G(�, t,�)= 1

t�(�−t), �, t ∈ (0, a) (18)

with the condition that G(�, t,�) must be limited at �=0 and the followings:

G(t+0, t,�)−G(t−0, t,�) = 0 (19a)

��

G(t+0, t,�)− ��

G(t−0, t,�) = 1

t(19b)

��

G(a, t,�) = 0. (19c)

The solution of (18) under the conditions given above is

G(�, t,�)= 1

J1(Ka)Q(�, t,�) (20)

with

Q(�, t,�)= �

2

{Jo(K�)[J1(Ka)Yo(Kt)−Jo(Kt)Y1(Ka)], 0��t

Jo(Kt)[J1(Ka)Yo(K�)−Jo(K�)Y1(Ka)], t��a(21)

Hence, the general solution of (15) can be expressed as

�1(�,�)= 1

KJ1(Ka)

{C(�)Jo(K�)+ i�K

∫ a

0[ei�lf (t)−g(t)]Q(t,�,�)t dt

}. (22)

Taking into account the continuity relation (3e), one gets

C(�)=−�′1(a,�) (23)

which yields

�1(a,�)= 1

KJ1(Ka)

{−Jo(Ka)�′

1(a,�)+ i�K

∫ a

0[ei�lf (t)−g(t)]Q(t, a,�)t dt

}. (24)

Although the left-hand side of (24) is an entire function of �, the regularity of the right-hand side is violated by the presenceof simple poles occurring at the zeros of KJ1(Ka) lying in the upper and lower halves of the complex �-plane, namely, at �=±�msatisfying

KmJ1(Kma)=0, m=0, 1, 2,. . . (25a)

with

�o =k and �m =√

k2 −K2m, m=1, 2,. . . (25b)

These poles can be eliminated by imposing that their residues are zero. This gives

e±iklfo −go =∓ 2i

ka�′

1(a,±k) (26a)

and

e±i�mlfm −gm =∓ 2i

a�mJo(Kma)�′

1(a,±�m), m=1, 2,. . . (26b)

with [fo

go

]= 2

a2

∫ a

0

[f (t)

g(t)

]t dt (26c)


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and [fm

gm

]= 2

a2J2o(Kma)

∫ a

0

[f (t)

g(t)

]Jo(Kmt)t dt, m=1, 2,. . . (26d)

Owing to (26c) and (26d), f (�) and g(�) can be expanded into Dini series [13] as follows[f (�)

g(�)

]=

[fo

go

]+

∞∑m=1

[fm

gm

]Jo(Km�). (27)

Now, substituting (24) into the Fourier transform of the continuity relation (3f) and making use of (6), (9) and (13), one gets

�−(a,�)+ei�l�+(a,�)− 2

�a

�′1(a,�)

(k2 −�2)M(�)= [ei(�+k)l −1]

i(�+k)− i�

KaJ1(Ka)(ei�lfo −go)

∫ a

0Jo(Kt)t dt

− i�

KaJ1(Ka)

∞∑m=1

(ei�lfm −gm)

∫ a

0Jo(Kmt)Jo(Kt)t dt (28)

with

M(�)= J1(Ka)

J1(Kb)[J1(Ka)Y1(Kb)−J1(Kb)Y1(Ka)]. (29)

The evaluation of the integrals at the right-hand side of (28) yields

�−(a,�)+ei�l�+(a,�)− 2

�a

�′1(a,�)

(k2 −�2)M(�)= [ei(�+k)l −1]

i(�+k)− i�(ei�lfo −go)

(k2 −�2)+ i�

∞∑m=1

(ei�lfm −gm)

(�2 −�2m)

Jo(Kma) (30)

which is nothing but the modified Wiener–Hopf equation (MWHE) with which we can solve the unknown functions �±(a,�) and�1(a,�).

2.2. Solution of the Wiener–Hopf equation

By following a similar procedure described in [14], the MWHE in (30) can be reduced into the following pair of simultaneousFredholm integral equations of the second kind

M−(�)(k−�)L(�)= 1

2�i

∫L−

ei�lM−(�)(k−�)U(�)

(�−�)d�−2ik

M+(k)

(k+�)+ ik

M+(k)

(k+�)go + i

2

∞∑m=1

gmJo(Kma)

(�+�m)M+(�m)(k+�m) (31)

and

M+(�)(k+�)U(�)=− 1

2�i

∫L+

e−i�lM+(�)(k+�)L(�)

(�−�)d�+ ik

M+(k)

(k−�)fo − i

2

∞∑m=1

fmJo(Kma)

(�−�m)M+(�m)(k+�m) (32)

where L(�) and U(�) stand for

L(�)=�−(a,�)+ 1

i(k+�)− i�

(k2 −�2)go + i�

∞∑m=1

gmJo(Kma)

(�2 −�2m)

(33)

and

U(�)=�+(a,�)− eikl

i(k+�)+ i�

(k2 −�2)fo − i�

∞∑m=1

fmJo(Kma)

(�2 −�2m)

, (34)

respectively. The paths of integration L+ and L− are depicted in Figure 2.In (31) and (32), M+(�) and M−(�)=M+(−�) are the split functions regular and free of zeros in the upper and lower halves of the

complex �-plane, respectively, resulting from the Wiener–Hopf factorization of M(�) in (29) as

M+(�)=√

M(0)e�T∞∏

m=1

(1+ �

zm

)(

1+ �

pm

) (35a)

with

T = i� [b log b−a log a−(b−a) log(b−a)]. (35b)

By standard asymptotics, it can be shown that M(�)=O(|�|−1) as |�|→∞ and since M(−�)=M(�), then by the Theorem 3.2 of [15],

M±(�)=O(±�−1/2) as |�|→∞ (36)

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Figure 2. Complex �-plane.

in their respective regions of regularity. In (35a), zm’s and pm’s are the zeros and poles of the meromorphic function M(�), respectively.The zeros of this function involve �m’s defined in (25b) and �m’s which satisfy

�m =√

k2 −�2m, m=1, 2,. . . (37a)

with

J1(�ma)Y1(�mb)−J1(�mb)Y1(�ma)=0, m=1, 2,. . . (37b)

Besides, the poles of M(�) are m’s satisfying

m =√

k2 −2m, m=1, 2,. . . (38a)

with

J1(mb)=0, m=1, 2,. . . (38b)

The integrals at the right-hand sides of (31) and (32), namely

I1(�)= 1

2�i

∫L−

ei�lM−(�)(k−�)U(�)

(�−�)d� (39)

and

I2(�)=− 1

2�i

∫L+

e−i�lM+(�)(k+�)L(�)

(�−�)d� (40)

can be evaluated easily by taking into account (33) and (34). Consider first the evaluation of I1(�). By virtue of Jordan’s lemma andaccording to Cauchy’s theorem, I1(�) is equal to the sum of the residues occurring at the simple poles of M(�) lying in the upperhalf-plane, namely, at �=m’s defined by (38a). The result is determined to be

I1(�)= 1

b

∞∑n=1

U(n)

(�−n)Dn (41a)

with

Dn = n(k−n)J21(na)Y1(nb)

nM+(n)Jo(nb)einl. (41b)

Substituting (41a) in (31) yields

M−(�)(k−�)L(�)= 1

b

∞∑n=1

U(n)

(�−n)Dn −2ik

M+(k)

(k+�)+ ik

M+(k)

(k+�)go + i

2

∞∑m=1

gmJo(Kma)

(�+�m)M+(�m)(k+�m). (42)

Similarly, by virtue of Jordan’s lemma and according to Cauchy’s theorem, I2(�) is equal to the sum of the residues occurring atthe simple poles of M(�) lying in the lower half-plane, namely, at �=−m’s defined by (38a). This gives

I2(�)= 1

b

∞∑n=1

L(−n)

(�+n)Dn (43)


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yielding

M+(�)(k+�)U(�)= 1

b

∞∑n=1

L(−n)

(�+n)Dn + ik

M+(k)

(k−�)fo − i

2

∞∑m=1

fmJo(Kma)

(�−�m)M+(�m)(k+�m). (44)

Hence, the solution of the modified Wiener–Hopf equation reads

�′1(a,�) = −�a

2b(k+�)M+(�)

∞∑n=1

U(n)

(�−n)Dn − �a

2b(k−�)M−(�)ei�l

∞∑n=1

L(−n)

(�+n)Dn + i�ka

2M+(k)[M+(�)(2−go)−M−(�)ei�lfo]

− i�a

4(k+�)M+(�)

∞∑m=1

gmJo(Kma)

(�+�m)M+(�m)(k+�m)+ i�a

4(k−�)M−(�)ei�l

∞∑m=1

fmJo(Kma)

(�−�m)M+(�m)(k+�m). (45)

The solution is determined in terms of some unknown coefficients, namely, U(n)’s, L(−n)’s, fm’s and gm’s. Considering theEquations (26a, b), (42), (44) and (45) together, these coefficients can be solved numerically.

3. Analysis of the fields

The scattered field in the region a<�<b, −∞<z<∞, namely, u1(�, z) can be solved by the below inverse Fourier transform integral

u1(�, z)= 1

2�

∫L

�′1(a,�)

K[J1(Kb)Yo(K�)−Jo(K�)Y1(Kb)]

K2[J1(Ka)Y1(Kb)−J1(Kb)Y1(Ka)]e−i�z d� (46)

by considering Equations (9), (11) and (13) together. In order to determine the transmitted field, the above integral must be evaluatedfor z>l. Taking into account (36), (45) and the standard asymptotics related to the Bessel and Neumann functions, one can showthat the integrand in (46) tends to zero for |�|→∞. This allows the application of the Jordan’s lemma and by virtue of Jordan’slemma and the application of the law of residues, the above integral becomes equal to the sum of the residues related to the polesoccurring at the simple zeros of K2[J1(Ka)Y1(Kb)−J1(Kb)Y1(Ka)] lying in the lower half-plane, namely, at �=−k and �=−�m’s. Thetransmission coefficient of the fundamental mode is determined by the residue contribution of the pole at �=−k and it is foundto be

T0 =− ia

k(b2 −a2)�′

1(a,−k). (47)

Similarly, the reflected field can be determined by evaluating the integral in (46) for z<0. The integral becomes equal to the sumof the residues related to the poles occurring at the simple zeros of K2[J1(Ka)Y1(Kb)−J1(Kb)Y1(Ka)] lying in the upper half-plane,�=k and �=�m’s. The reflection coefficient of the fundamental mode is the residue contribution of the pole at �=k and it is asfollows:

R0 =− ia

k(b2 −a2)�′

1(a, k). (48)

Considering all the residue contributions, u1(�, z) can be expressed in the most general way as

u1(�, z)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

R0e−ikz +∞∑

m=1Rme−i�mz, z<0

T0eikz +∞∑

m=1Tmei�mz, z>0

(49)

involving all possible propagating modes. The reflection and transmission coefficients of the higher-order modes can be determinedby the residue contributions of �m’s and −�m’s as mentioned above. Note that, R0 and T0 are related to the fundamental modeonly, which is cut-on (propagating) for all frequencies. Higher-order modes can only propagate when the frequency of the waveexceeds their relevant cut-off frequency.

4. Computational results

In order to show the effects of the parameters like the inner and outer radii of the annular duct and the length of the gap onthe inner wall on the pressure drop and noise, some numerical results showing variation of the reflection coefficient R0 and thetransmission coefficient T0 of the fundamental mode with different parameters are presented. On the other hand, one can find themoduli of the reflected and transmitted fields for specific values of k(b−a), kl and N (truncation number for the solution of infinitesystem of linear equations) in Figure 3. N can be chosen for desired accuracy. In Figures 4–9, the truncation number is chosen as 30.

In Figures 4 and 5, the variation of the amplitude of the reflection coefficient related to the fundamental mode with respect to klis depicted. It is observed that, the amplitude of the reflection coefficient is oscillating when kl increases. At some specific values ofkl, there occurs almost no reflection (e.g. |R0|0.0064 for kl =4, ka=1, kb=2.5 and |R0|0.0065 for kl =7.1, ka=1, kb=2.5). From

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Figure 3. Variation of the moduli of the coefficients with respect to N for f =50 Hz, ka=1.25, kb=2.5 and kl =10.

Figure 4. Variation of the reflection coefficient with respect to kl for f =50 Hz, and kb=2.5.

Figure 5. Variation of the reflection coefficient with respect to kl for f =50 Hz, and ka=1.


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Figure 6. Variation of the transmission coefficient with respect to kl for f =50 Hz, and kb=2.5.

Figure 7. Variation of the transmission coefficient with respect to kl for f =50 Hz, and ka=1.

the figures, one can realize that these specific values of kl are dependent on b and a. As for the engineering point of view, thesevalues are most suitable for minimum pressure drop where the engine will consume less power.

In Figures 6 and 7, the transmission coefficient’s modulus’ variation with respect to kl is depicted. The amplitude of the transmissioncoefficient is oscillating for increasing kl. Note that, the minimums occurring at specific values of kl mean maximum noise reduction(e.g. |T0|0.2237 for kl =6.3, ka=1, kb=2.5 and |T0|0.2237 for kl =12.6, ka=1, kb=2.5).

Finally, the variation of the amplitudes of the reflection and transmission coefficients with respect to k(b−a) are presented inFigures 8 and 9, respectively. It is observed that, the amplitudes of both coefficients decrease with increasing cross-sectional areaof the annular duct. This is because, more modes can propagate with higher cross-sectional area and some of the energy carriedby the fundamental mode is transferred to the newly propagating modes.

5. Concluding remarks

In this study, the effect of the existence of a finite gap on the inner wall of an annular duct to the sound propagation is investigatedrigorously and some numerical results are presented. The problem is formulated into a modified Wiener–Hopf equation. Thisequation is solved via a pair of Fredholm integral equations of the second type. Both the reflection and transmission coefficientsof the fundamental mode are determined explicitly in terms of infinite number of unknown coefficients. Numerical results arepresented in Section 4, and it is observed that some specific values of a, b and l can provide minimum back pressure or maximumnoise reduction.

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Figure 8. Variation of the reflection coefficient with respect to k(b−a) for f =50 Hz.

Figure 9. Variation of the transmission coefficient with respect to k(b−a) for f =50 Hz.

This analysis can also be applied to the case where a mean flow exists in the annular duct and a perforated tube is located atthe finite gap, or to the cases where different types of partial acoustical linings are applied to duct walls. In a problem involvingmean flow the instability mode must be studied carefully and the problem will become very complicated. On the other hand, aproblem involving partial acoustical linings will only bring a little complexity on the kernel function, i.e. the factorization process.These problems will be the subjects of the forthcoming papers by the authors.

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