Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 532682, 19 pages
http://dx.doi.org/10.5402/2011/532682
Research Article

Acoustic Wave Propagation in a Trifurcated Lined Waveguide

Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

Received 11 March 2011; Accepted 30 March 2011

Academic Editors: C. Chen, P. J. García Nieto, and C. Lu

Copyright © 2011 M. Ayub et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The diffraction of sound from a semi-infinite soft duct is investigated. The soft duct is symmetrically located inside an acoustically lined but infinite duct. A closed-form solution is obtained using integral transform and Jones' method based on Wiener-Hopf technique. The graphical results are presented, which show how effectively the unwanted noise can be reduced by proper selection of different parameters. The kernel functions are factorized with different approaches. The results may be used to design acoustic barriers and noise reduction devices.

1. Introduction

The analysis of the effects of unwanted noise has been an active area of research because of its technological importance. This study is important in connection with exhaust system, steam valves, internal combustion engines of aircraft and vehicles, turbofan engines, and ducts and pipes. The analysis of wave scattering by such structures is an important area of noise reduction and relevant for many applications. Continued interest in the problem of noise reduction has attracted the attention of many scientists, physicists, and numerical simulists.

Many interesting mathematical models for the reduction of noise are discussed by several authors. In view of historical perspectives the story goes that Rawlins [1] was the first to show that the duct with a thin acoustically absorbent lining is an effective method which can be used to reduce the unwanted noise within a waveguide. As a sound attenuator, the acoustic performance of a duct can be increased significantly by lining its walls with an acoustically absorbent material [2]. Koch [3] discussed the problem of noise reduction from the engineering point of view, namely, in rectangular chambers, circular and annular geometries in the absence of mean flow situation. In another paper [4], Koch discussed the analytical solution of the problem of sound radiation from the open end of a semi-infinite two-dimensional duct whose inner side walls are lined with a locally reacting sound absorbing material of finite length. The problem was solved analytically with the help of Wiener-Hopf technique. The obtained analytical results were also discussed numerically for several parameters of interest.

Jones [5] discussed the problem of scattering of plane waves from three parallel soft semi-infinite and equidistant plates and calculated the far field and the field within the waveguide. Later on, Asghar et al. [6] extended Jones analysis [5] for the case of line source and point source scattering in still air and as the medium is convective. Afterwards Hassan and Rawlins [7] analyzed an acoustic diffraction problem considering a semi-infinite hard-soft duct and obtained exact, closed-form solution valid for all plate spacings. Asghar and Hayat [8] obtained an exact solution for the problem of scattering of sound within absorbing parallel plates using Wiener-Hopf technique. In [9] the analytical solution of the sound field of a semi-infinite acoustically soft cylindrical duct, accounting for diffraction at the outlet, has been obtained applying Wiener-Hopf technique. In a similar context, Büyükaksoy and Polat [10] studied diffraction phenomenon in a bifurcated waveguide by considering a dominant mode wave incident on a soft-hard half-plane centered inside an infinite parallel plate waveguide with hard boundaries.

Related work regarding the diffraction of dominant acoustic wave modes from the trifurcated waveguide having the same geometric design but with different combinations of the boundary conditions (soft, hard, mixed (Robin type)) in the case of still air and for convective flow may be found in [1114]. Keeping in view of the importance of the above-mentioned configuration, in this paper we have attempted to solve the problem of diffraction of a dominant acoustic mode propagation out of the mouth of a semi-infinite soft duct which is symmetrically located within an infinite lined duct on which general third-type mixed boundary conditions of the Robin type are satisfied. The boundary value problem is solved analytically with the help of standard Wiener-Hopf procedure based on Jones' method, more mechanical and straightforward, rather than by using the cumbersome integral equation apparatus. This method has applications in almost all modern branches of science, engineering, and technology. For more details one is referred to [15, 16]. The geometry of the trifurcated waveguide problem under consideration is shown in Figure 1, and the paper is organized as follows.

532682.fig.001
Figure 1: Schematic diagram of the trifurcated waveguide.

The problem statement is presented in Section 2. The Wiener-Hopf (WH) equation is formed in Section 3. The problem is further solved in Section 4. The solution procedure involves the complex contour integrals. These integrals are evaluated in Section 5 by an application of Cauchy residue theorem [17]. The explicit factorization of the Wiener-Hopf kernel function is accomplished in the appendix. Numerical and graphical results are also presented.

2. Problem Statement

The physical situation considered is that of the diffraction of first mode of the inside waveguide (which is the only propagative mode) as incident mode that propagates out of the end of a semi-infinite soft duct. The wave mode is propagating in the positive 𝑥-direction parallel to 𝑥-axis. The semi-infinite soft duct is placed inside the absorbingly lined duct. If 𝜙(𝑥,𝑦,𝑡) is a scalar potential, then the velocity 𝐮 and acoustic pressure 𝑝 can be written as𝐮=grad𝜙,𝑝=𝜌𝑜𝜕𝜙,𝜕𝑡(2.1) respectively, where grad is the gradient operator, 𝜌𝑜 is the density, and 𝑡 is the time. Writing 𝜙(𝑥,𝑦,𝑡)=𝑒𝑖𝜔𝑡Φ(𝑥,𝑦)(2.2) and omitting 𝑒𝑖𝜔𝑡 throughout, we have to solve the following Helmholtz equation:𝜕2Φ𝜕𝑥2+𝜕2Φ𝜕𝑦2+𝑘2Φ=0,(2.3) where 𝑘=𝜔/𝑐 (𝜔 is the angular frequency and 𝑐 is the speed of sound) is the wave number.

The boundary conditions and continuity conditions associated with the problem are of the formΦ+𝑖𝜁𝑘𝜕Φ𝜕𝑦=0,𝑦=𝑏,<𝑥<,(2.4)Φ=0,𝑦=𝑎,<𝑥<0,(2.5)Φ=0,𝑦=𝑎,<𝑥<0,(2.6)Φ𝑖𝜁𝑘𝜕Φ𝜕𝜕𝑦=0,𝑦=𝑏,<𝑥<,(2.7)Φ𝜕𝑦𝑥,𝑎+=𝜕𝜕𝑦Φ(𝑥,𝑎𝜕),𝑥>0,(2.8)Φ𝜕𝑦𝑥,𝑎+=𝜕Φ𝜕𝑦(𝑥,𝑎),𝑥>0,(2.9) where it is assumed that 𝑏>𝑎. In (2.4) and (2.7), 𝜁 represents the specific impedance of the infinite duct lining and it is necessary that for an absorbent surface Re𝜁>0.

Besides the conditions prescribed in (2.4)–(2.9), we require those conditions at infinity which are relevant to the nature of the lowest propagating modes in various duct regions. The radiation conditions at infinity suggest the following [14].

For the region (𝑎𝑦𝑎,𝑥<0), one may writeΦ(𝑥,𝑦)=𝑒𝑖𝜒1𝑥sin𝜋(𝑦𝑎)+2𝑎𝑛=1𝑅𝑛𝑒𝑖𝜒𝑛𝑥sin𝑛𝜋(𝑦𝑎)2𝑎,(2.10) where 𝜒𝑛=(𝑘2𝛼2𝑛)1/2, (𝑛=1,2,3,,) and 𝛼𝑛 satisfy𝛼sin2𝑛𝑎=0,(2.11)𝛼𝑛=𝑛𝜋/2𝑎 with 0<Im𝜒1<Im𝜒2<Im𝜒3. The lowest-order plane wave mode can propagate only when 𝜋/2<𝑘𝑎<𝜋.

The value of Φ(𝑥,𝑦) for (𝑏𝑦𝑏,𝑥>0) isΦ(𝑥,𝑦)=𝑛=1𝑇𝑛𝑒𝑖𝜎𝑛𝑥sin𝛽𝑛(𝑦𝑏)+𝑖𝜁𝑘𝛽𝑛cos𝛽𝑛(𝑦𝑏),(2.12) where 𝜎𝑛=(𝑘2𝛽2𝑛)1/2(𝑛=1,2,3,) and 𝛽𝑛 satisfy the equationsin2𝑏𝛽𝑛+2𝑖𝛽𝑛𝜁𝑘cos2𝑏𝛽𝑛+𝛽2𝑛𝜁2𝑘2sin2𝑏𝛽𝑛=0,(2.13) with 0<Im𝜎1<Im𝜎2<Im𝜎3.

For (𝑎𝑦𝑏,𝑥<0) one hasΦ(𝑥,𝑦)=𝑛=1𝑇𝑛𝑒𝑖𝛼𝑛𝑥sin𝛿𝑛(𝑦𝑏)+𝑖𝜁𝑘𝛿𝑛cos𝛿𝑛(𝑦𝑏),(2.14) where 𝛼𝑛=(𝑘2𝛿2𝑛)1/2(𝑛=1,2,3,) and 𝛿𝑛 represent the roots of the equationsin𝛿𝑛(𝑏𝑎)+𝑖𝜍𝑘𝛿𝑛cos𝛿𝑛(𝑏𝑎)=0,(2.15) with 0<Im𝛼1<Im𝛼2<Im𝛼3.

When (𝑏𝑦𝑎,𝑥<0), we haveΦ(𝑥,𝑦)=𝑛=1𝑇𝑛𝑒𝑖𝛼𝑛𝑥sin𝛿𝑛(𝑦+𝑏)+𝑖𝜁𝑘𝛿𝑛cos𝛿𝑛(𝑦+𝑏).(2.16) In region (𝑎𝑦𝑎,𝑥<0) the acoustic wave shows incident and reflected behavior while in the regions (𝑏𝑦𝑎,𝑥<0) and (𝑎𝑦𝑏,𝑥<0) transmission behavior is observed. In the text 𝑅𝑛 and 𝑇𝑛 represent reflection and transmission coefficients, respectively. To arrive at a unique solution, we also require the “edge conditions” [18]Φ(𝑥,±𝑎)=𝑂(1),Φ𝑦𝑥(𝑥,±𝑎)=𝑂1/2as𝑥0.(2.17)

3. The Wiener-Hopf (WH) Equations

For analytic convenience, we will assume that 𝑘=𝑘1+𝑖𝑘2  (𝑘1>0, 𝑘20) since the time dependence is taken to be of the form 𝑒𝑖𝜔𝑡 [15]. Let us define Fourier transform and its inverse byΦ(𝛼,𝑦)=Φ(𝑥,𝑦)𝑒𝑖𝛼𝑥Φ𝑑𝑥=+Φ(𝛼,𝑦)+1(𝛼,𝑦),(3.1)Φ(𝑥,𝑦)=2𝜋Φ(𝛼,𝑦)𝑒𝑖𝛼𝑥𝑑𝛼.(3.2) In (3.1),Φ+(𝛼,𝑦)=0Φ(𝑥,𝑦)𝑒𝑖𝛼𝑥Φ𝑑𝑥,(𝛼,𝑦)=0Φ(𝑥,𝑦)𝑒𝑖𝛼𝑥𝑑𝑥,(3.3) where 𝛼 is a complex variable with𝛼=𝜎+𝑖𝜏.(3.4)

Use of (3.1) to (2.3) gives𝑑2Φ𝑑𝑦2+𝜅2Φ=0,(3.5) where𝜅(𝛼)=𝑘2𝛼2.(3.6) The suitable solutions of (3.5) in the trifurcated regions areΦ(𝛼,𝑦)=𝐴1(𝛼)cos𝜅𝑦+𝐵1(𝛼)sin𝜅𝑦(𝑏𝑦𝑎),(3.7)Φ(𝛼,𝑦)=𝑖𝛼+𝜒1sin𝜋(𝑦𝑎)2𝑎+𝐴2(𝛼)cos𝜅𝑦+𝐵2(𝛼)sin𝜅𝑦(𝑎𝑦𝑎),(3.8)Φ(𝛼,𝑦)=𝐴3(𝛼)cos𝜅𝑦+𝐵3(𝛼)sin𝜅𝑦(𝑎𝑦𝑏),(3.9) where the first term on the right-hand side of (3.8) comes from the incident field. On taking Fourier transform, (2.4)–(2.9) become Φ(𝛼,𝑏)+𝑖𝜁𝑘ΦΦ(𝛼,𝑏)=0,(3.10)Φ(𝛼,𝑎)=0,(3.11)(𝛼,𝑎)=0,(3.12)Φ(𝛼,𝑏)𝑖𝜁𝑘ΦΦ(𝛼,𝑏)=0,(3.13)+𝛼,𝑎+=Φ+(𝛼,𝑎Φ),(3.14)+𝛼,𝑎+=Φ+(𝛼,𝑎),(3.15) where prime denotes the differentiation with respect to 𝑦. In order to determine the unknowns 𝐴𝑗(𝛼) and 𝐵𝑗(𝛼)  (𝑗=1,2 and 3), we proceed to satisfy boundary conditions (3.10)–(3.13). Thus, by invoking (3.12) and (3.13) in (3.7), we may write 𝐴1(𝛼)cos𝜅𝑎𝐵1(𝛼)sin𝜅𝑎=Φ+1𝐴(𝛼),1(𝛼)cos𝜅𝑏𝑖𝜁𝑘𝜅sin𝜅𝑏𝐵1(𝛼)sin𝜅𝑏+𝑖𝜁𝑘𝜅cos𝜅𝑏=0,(3.16) where Φ+1Φ(𝛼)=+(𝛼,𝑎) is analytic in Im𝛼>Im𝑘. Solving, above equations for 𝐴1(𝛼) and 𝐵1(𝛼) we obtain 𝐴1(𝛼)=sin𝜅𝑏+(𝑖𝜁/𝑘)𝜅cos𝜅𝑏Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+1𝐵(𝛼),1(𝛼)=cos𝜅𝑏(𝑖𝜁/𝑘)𝜅sin𝜅𝑏Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+1(𝛼).(3.17) Again, using (3.11) and (3.12) in (3.8), we may write𝐴2(𝛼)cos𝜅𝑎+𝐵2(𝛼)sin𝜅𝑎=Φ+2𝐴(𝛼),2(𝛼)cos𝜅𝑎𝐵2(𝛼)sin𝜅𝑎=Φ+1(𝛼),(3.18) where Φ+2Φ(𝛼)=+(𝛼,𝑎) is analytic in Im𝛼>Im𝑘. Solving (3.18) for 𝐴2(𝛼) and 𝐵2(𝛼), we obtain𝐴2Φ(𝛼)=sin𝜅𝑎+1(𝛼)+Φ+2(𝛼),𝐵sin2𝜅𝑎2Φ(𝛼)=cos𝜅𝑎+2(𝛼)Φ+1(𝛼).sin2𝜅𝑎(3.19) With the use of (3.10) and (3.11) in (3.9), we may write𝐴3(𝛼)cos𝜅𝑏𝑖𝜁𝑘𝜅sin𝜅𝑏+𝐵3(𝛼)sin𝜅𝑏+𝑖𝜁𝑘𝐴𝜅cos𝜅𝑏=0,3(𝛼)cos𝜅𝑎+𝐵3(𝛼)sin𝜅𝑎=Φ+2(𝛼).(3.20) Solving for 𝐴3(𝛼) and 𝐵3(𝛼), we obtain 𝐴3(𝛼)=sin𝜅𝑏+(𝑖𝜁/𝑘)𝜅cos𝜅𝑏Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+2𝐵(𝛼),3(𝛼)=cos𝜅𝑏(𝑖𝜁/𝑘)𝜅sin𝜅𝑏Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+2(𝛼).(3.21) By substituting the above values of 𝐴𝑗(𝛼) and 𝐵𝑗(𝛼) (𝑗=1,2 and 3) in (3.7)–(3.9) we getΦ(𝛼,𝑦)=sin𝜅(𝑦+𝑏)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦+𝑏)Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+1(𝛼)(𝑏𝑦𝑎),Φ(𝛼,𝑦)=𝑖𝛼+𝜒1sin𝜋(𝑦𝑎)+12𝑎Φsin2𝜅𝑎+2(𝛼)sin𝜅(𝑦+𝑎)Φ+1(𝛼)sin𝜅(𝑦𝑎)(𝑎𝑦𝑎),Φ(𝛼,𝑦)=sin𝜅(𝑏𝑦)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦𝑏)Φsin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)+2(𝛼)(𝑎𝑦𝑏).(3.22) Invoking (3.22) in (3.14) and (3.15), we arrive at 𝜅(sin𝜅(𝑏+𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏+𝑎))Φ+1(𝛼)+sin2𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎))𝜅Φ+2(𝛼)𝜆sin2𝜅𝑎1𝛼+𝜒1=Φ1(𝛼),𝜅(sin𝜅(𝑏+𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏+𝑎))Φ+2(𝛼)sin2𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎))𝜅Φ+1(𝛼)+𝜆sin2𝜅𝑎1𝛼+𝜒1=Φ2(𝛼),(3.23) where𝜆1=𝑖𝜋,Φ2𝑎1(Φ𝛼)=(𝛼,𝑎)Φ𝛼,𝑎+,Φ2Φ(𝛼)=(𝛼,𝑎Φ)𝛼,𝑎+.(3.24) The functions Φ1(𝛼) and Φ2(𝛼) are analytic in the region Im𝛼<Im𝑘. Addition and subtraction of (3.23) will give𝑊(𝛼)𝐷+(𝛼)=𝑆(𝛼),𝐾(𝛼)𝑆+(𝛼)+2𝜆1𝛼+𝜒1=𝐷(𝛼),(3.25) whereΦ±2(𝛼)Φ±1(𝛼)=𝐷±Φ(𝛼),(3.26)±2(𝛼)+Φ±1(𝛼)=𝑆±(𝛼),(3.27)𝐾(𝛼)=𝜅(cos𝜅𝑏(𝑖𝜁/𝑘)𝜅sin𝜅𝑏)𝜅cos𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)),(3.28)𝑊(𝛼)=(sin𝜅𝑏+(𝑖𝜁/𝑘)𝜅cos𝜅𝑏).sin𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎))(3.29)

4. Solution of the Problem

Writing (see the appendix) 𝑊(𝛼)=𝑊+(𝛼)𝑊(𝛼),𝐾(𝛼)=𝐾+(𝛼)𝐾(𝛼),(4.1) where (+) is the subscript assigned to the function regular in the upper half plane Im𝛼>Im𝑘 and the subscript (−) represents the function regular in the lower half plane Im𝛼<Im𝑘. Now from (3.25), we have𝑊+(𝛼)𝐷+𝑆(𝛼)=(𝛼)𝑊(,𝐾𝛼)+(𝛼)𝑆+(𝛼)+2𝜆1𝛼+𝜒1𝐾𝜒1=𝐷(𝛼)𝐾(𝛼)2𝜆1𝛼+𝜒11𝐾1(𝛼)𝐾𝜒1.(4.2) Note that the left-hand side of both equations is analytic in Im𝛼>Im𝑘 and the right-hand side is analytic in Im𝛼<Im𝑘. Also, when |𝛼|,𝐾±(𝛼)=𝑂|𝛼|1/2,𝑊±(𝛼)=𝑂|𝛼|1/2,(4.3) and Fourier transform of edge conditions (2.17) helps us to determine the asymptotic behavior of 𝐷±(𝛼) and 𝑆±(𝛼). For |𝛼|,𝐷(𝛼)=𝑂|𝛼|1,𝑆𝛼(𝛼)=𝑂1𝐷forIm𝛼<Im𝑘,+(𝛼)=𝑂|𝛼|1,𝑆+𝛼(𝛼)=𝑂1/2forIm𝛼>Im𝑘.(4.4) Now the use of (4.3)–(4.4) and standard Wiener-Hopf procedure [15] on (4.2) give𝐷+𝐾(𝛼)=0,+(𝛼)𝑆+(𝛼)+2𝜆1𝛼+𝜒1𝐾+𝜒1=0,(4.5) where 𝐾(𝜒1)=𝐾+(𝜒1). Using (3.26) and (3.27) in (4.5), we getΦ+1𝜆(𝛼)=1𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1,Φ+2(𝜆𝛼)=1𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1.(4.6) From (3.22) and (4.6), we obtain the field representations for the different regions as follows.

When (𝑏𝑦𝑎),𝜆Φ(𝛼,𝑦)=1sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)sin𝜅(𝑦+𝑏)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦+𝑏)𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1.(4.7) For (𝑎𝑦𝑎), one obtainsΦ(𝛼,𝑦)=𝑖𝛼+𝜒1sin𝜋(𝑦𝑎)𝜆2𝑎1cos𝜅𝑎cos𝜅𝑦𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1.(4.8) For (𝑎𝑦𝑏), one has𝜆Φ(𝛼,𝑦)=1sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)sin𝜅(𝑏𝑦)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦𝑏)𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1.(4.9) Taking inverse Fourier transform of (4.7)–(4.9), we obtain the following.

For the region (𝑏𝑦𝑎,𝑥<0), one may write𝜆Φ(𝑥,𝑦)=12𝜋+𝑖𝜏+𝑖𝜏𝑒𝑖𝛼𝑥×sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)sin𝜅(𝑦+𝑏)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦+𝑏)𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1𝑑𝛼.(4.10) When (𝑎𝑦𝑎,𝑥<0), we haveΦ(𝑥,𝑦)=𝑒𝑖𝜒1𝑥sin𝜋(𝑦𝑎)𝜆2𝑎12𝜋+𝑖𝜏+𝑖𝜏𝑒𝑖𝛼𝑥cos𝜅𝑎cos𝜅𝑦𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1𝑑𝛼.(4.11) For (𝑎𝑦𝑏,𝑥<0), we arrive at𝜆Φ(𝑥,𝑦)=12𝜋+𝑖𝜏+𝑖𝜏𝑒𝑖𝛼𝑥×sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)sin𝜅(𝑦𝑏)(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦𝑏)𝛼+𝜒1𝐾+(𝛼)𝐾+𝜒1𝑑𝛼.(4.12) For (𝑏𝑦𝑏,𝑥>0), one may write𝜆Φ(𝑥,𝑦)=12𝜋+𝑖𝜏+𝑖𝜏𝑒𝑖𝛼𝑥×(cos𝜅𝑏(𝑖𝜁/𝑘)𝜅sin𝜅𝑏sin𝜅(𝑦𝑏)(𝑖𝜁/𝑘)𝜅cos𝜅(𝑦𝑏))cos𝜅𝑎𝐾(𝛼)𝜅𝛼+𝜒1𝐾+𝜒1𝑑𝛼.(4.13) In (4.10)–(4.13), 𝜏 is the imaginary part of 𝛼, 𝜅(𝛼)=𝑘2𝛼2 and branch cuts are taken to be from 𝑘 to 𝑖 and 𝑘 to 𝑖, and 0arg𝜅𝜋 (Figure 2). Note that the integrands have no singularities which lie on the contour of integration. To evaluate the integrals in (4.10)–(4.13), it is noted that the contour of integration in these equations lies in the strip Im𝑘<Im𝛼<Im𝑘.

532682.fig.002
Figure 2: Strip of analyticity and branch cuts in the complex 𝛼-plane.

In expressions (4.10)–(4.13), the pole 𝛼=𝜒1 lies below the contour of integration. One can also note that the terms in the curly brackets {} of (4.10)–(4.12) have no branch points in Im𝛼>Im𝑘 and those in (4.13) have no branch points in Im𝛼<Im𝑘. Thus the only singularities in the integrands of (4.10) and (4.12) occur at the zeros ofsin𝜅(𝑏𝑎)+𝑖𝜁𝑘𝜅cos𝜅(𝑏𝑎)=0,(4.14) that is, at𝛼=𝛼𝑛=𝑘2𝛿2𝑛1/2(𝑛=1,2,3,).(4.15) The only singularities of (4.11) occur at the zeros of cos𝜅𝑎=0, that is at𝛼=𝜒2𝑛1=𝑘2(2𝑛1)2𝜋24𝑎21/2(𝑛=1,2,3,).(4.16) The only singularities in the integrands in (4.13) occur at the zeros ofcos𝜅𝑏𝑖𝜁𝑘𝜅sin𝜅𝑏=0,(4.17) that is at 𝛼=𝜎𝑛𝑘=2𝛽2𝑛1/2(𝑛=1,2,3,).(4.18)

5. Modal Field Representation

Invoking Cauchy residue theorem [17] to the integrals in (4.10)–(4.13), we obtain the following.

When (𝑏𝑦𝑎,𝑥<0),Φ(𝑥,𝑦)=𝑛=1𝑒𝑖𝛼𝑛𝑥𝑞1𝛼𝑛sin𝛿𝑛(𝑦+𝑏)+(𝑖𝜁/𝑘)𝛿𝑛cos𝛿𝑛𝜋(𝑦+𝑏)2𝑎𝛼𝑛+𝜒1𝐾+𝛼𝑛𝐾+𝜒1,(5.1) where𝑞1(𝛼)=sin𝜅(𝑏𝑎)+𝑖𝜁𝑘𝜅cos𝜅(𝑏𝑎),𝛼𝑛=𝑘2𝛿2𝑛1/2(𝑛=1,2,3,).(5.2) For (𝑎𝑦𝑎,𝑥<0), we haveΦ(𝑥,𝑦)=𝑒𝑖𝜒1𝑥sin𝜋(𝑦𝑎)2𝑎𝑛=1𝜋𝛼2𝑛1(1)𝑛+1𝑒𝑖𝜒2𝑛1𝑥𝛼cos2𝑛1𝑦2𝑎2𝜒2𝑛1𝜒2𝑛1+𝜒1𝐾+𝜒2𝑛1𝐾+𝜒1.(5.3) For (𝑎𝑦𝑏,𝑥<0), we arrive atΦ(𝑥,𝑦)=𝑛=1𝑒𝑖𝛼𝑛𝑥𝑞1𝛼𝑛sin𝛿𝑛(𝑦𝑏)(𝑖𝜁/𝑘)𝛿𝑛cos𝛿𝑛𝜋(𝑦𝑏)2𝑎𝛼𝑛+𝜒1𝐾+𝛼𝑛𝐾+𝜒1.(5.4) When (𝑏𝑦𝑏,𝑥>0), we haveΦ(𝑥,𝑦)=𝑛=1𝑒𝑖𝜎𝑛𝑥sin𝛽𝑛(𝑦𝑏)(𝑖𝜁/𝑘)𝛽𝑛cos𝛽𝑛(𝑏𝑦)𝜋cos𝛽𝑛𝑎2𝑎𝑚𝜎𝑛𝐾+𝜒1𝜒1𝜎𝑛𝛽𝑛𝐾𝜎𝑛,(5.5) where𝑚(𝛼)=cos𝜅𝑏𝑖𝜁𝑘𝜅sin𝜅𝑏,𝛼=𝜎𝑛𝑘=2𝛽2𝑛1/2(𝑛=1,2,3,).(5.6)

6. Reflection Coefficient

Inside the waveguide field intensity is superposition of reflected and transmitted waves. Hence, it is relevant to deal with reflection or transmission coefficients which are related to relative energy. We will consider reflection coefficient for the first mode 𝑛=1 given by (5.3) as𝑅1=𝜋28𝑎3𝜒21𝐾+𝜒12.(6.1)

7. Numerical and Graphical Results

The expression of field intensity involves infinite sums/products for which we have used numerical technique and obtained the results using truncation approach [19]. We have computed the results of variation of field intensity for different boundary conditions of the semi-infinite duct at 𝑛 from 10 to 150 by step 10 given in Table 1. From this table, it is evident that the presence of soft boundary condition induces a good noise reduction as compared with hard boundary conditions.

tab1
Table 1: Field intensity |Φ| versus the truncation number 𝑛 for different boundary conditions of the semi-infinite duct.

The values of specific impedance 𝜁=𝜉+𝑖𝜂(=𝑧/𝜌0𝑐) for an absorbing sheet which seem to have practical importance are [13]fibrous sheet: 𝜉=0.5, 1<𝜂<3,perforated sheet: 0<𝜉<3, 1<𝜂<3.

The convergence of the field can be checked through the relative error for which the suitable definition would be𝐸𝑛=max𝑥,𝑦||Φ𝑛(𝑥,𝑦)Φ𝑁||(𝑥,𝑦)||Φ𝑁||(𝑥,𝑦),(7.1) where Φ𝑛(𝑥,𝑦) is the solution obtained using the truncation number 𝑛 and 𝑁 is suitably large truncation number (𝑁=150).

For a comprehensive numerical study, we need a considerable number of graphs because of the number of parameters which determine the diffracted field. The computer programme “MATHEMATICA 5.2” is used for the numerical evaluation and graphical representation of the functions given by (6.1) and (7.1).

In Figures 3 and 4, the field intensity is plotted against the wave number 𝑘 for different values of noise reduction parameters, that is, 𝜉 (real part of 𝜁 ) and 𝜂 (the imaginary part of 𝜁).

532682.fig.003
Figure 3: Variation of field intensity |𝑅1| with wave number 𝑘 for several values of 𝜉 corresponding to 𝜂=0.5, 𝑎=1, and 𝑏=3𝑎.
532682.fig.004
Figure 4: Variation of field intensity |𝑅1| with wave number 𝑘 for several values of 𝜂 corresponding to 𝜉=1, 𝑎=1, and 𝑏=3𝑎.

In Figure 5, the reflected field is plotted against wave number 𝑘 for different values of 𝑏 (separation distance between the infinite plates). In Figure 6, variation of relative error 𝐸𝑛 against truncation number 𝑛 is plotted.

532682.fig.005
Figure 5: Variation of |𝑅1| with 𝑘 for several values of 𝑏 corresponding to 𝜂=1, 𝜉=2.50, and 𝑎=𝑏/2.
532682.fig.006
Figure 6: Variation of relative error 𝐸𝑛 with truncation number 𝑛 corresponding to 𝑎=0.5, 𝑏=3𝑎, 𝑥=0.1, 𝑘=0.7, 𝑦=1, 𝜉=0.5, and 𝜂=1.

The main findings from the analysis are summarized in the following points.(i)Global speaking, it is noted that the reflected field is a decreasing function of the real and imaginary parts of the absorbing parameter but with relative or local maxima and minima.(ii)Gradually increase in the separation distance between the infinite plates yields a decrease in the value of reflected field intensity.(iii)The absolute values of the reflection coefficient are in accordance with the conservation of energy rules.(iv)The findings confirm that the relative error reduces by increasing the truncation number 𝑛.(v)The established results clearly show the contribution that arises because of the soft surfaces.

8. Final Remarks

Computation of acoustic diffraction is very important in the analysis of acoustic waveguide systems. In this study the Wiener-Hopf method has been used for diffraction of acoustic waves in a trifurcated waveguide. The problem consists of absorbing and soft surfaces. A sound wave of first mode propagating out of the mouth of the semi-infinite soft duct is taken into account. The problem is formulated first and then solved analytically. For the quality of the computation, the comparison of the hard [13] with the soft boundary conditions of the semi-infinite duct is discussed in detail. To enhance the quality of the results some graphs are plotted for sundry parameters of interest using wave number versus reflection coefficient of first mode in absolute value. It is observed that soft surfaces show good noise reduction effects on the noise transmitted through the waveguide as compared with hard surfaces [13]. This is a canonical problem of mathematical interest. The reported results are shown conclusively and present a comprehensive introduction on the current state of art within the field of guide acoustics.

Appendix

The main purpose of this appendix is to give the complete factorization of the kernel functions 𝐾(𝛼) and 𝑊(𝛼) and to show their asymptotic behavior as |𝛼|. The factorization of these functions 𝐾(𝛼) and 𝑊(𝛼) is of the form𝐾(𝛼)=𝐾+(𝛼)𝐾(𝛼),𝑊(𝛼)=𝑊+(𝛼)𝑊(𝛼),(A.1) where 𝐾+(𝛼) and 𝑊+(𝛼) denote certain functions which are regular and free of zeros in upper half plane Im𝛼>Im𝑘 and 𝐾(𝛼) and 𝑊(𝛼) denote certain functions which are regular and free of zeros in lower half plane Im𝛼<Im𝑘.

We may note that the functions 𝐾(𝛼) and 𝑊(𝛼) are even in Fourier transform parameter 𝛼 and more precisely their respective derivatives are zero at 𝛼=0. So these functions can be factorized by applying the infinite product expansion of an integral function with infinitely many zeros [15, 20]. For given 𝐾(𝛼) by (3.28), we have 𝐾(𝛼)=𝜅(cos𝜅𝑏(𝑖𝜁/𝑘)𝜅sin𝜅𝑏)cos𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)).(A.2) It is evident that the product factorization of 𝐾(𝛼) depends upon the factorization of 𝐿(𝛼)=cos𝜅𝑏𝑖𝜁𝑘𝜅sin𝜅𝑏,𝑁(𝛼)=sin𝜅(𝑏𝑎)𝜅+𝑖𝜁𝑘cos𝜅(𝑏𝑎),𝑃(𝛼)=cos𝜅𝑎.(A.3) By employing the procedure outlined by Mittra and Lee [21], we have 𝐿+(𝛼)=cos𝑘𝑏𝑖𝜁sin𝑘𝑏exp𝑖𝛼𝑏𝜋1𝐶ln|𝛼|𝑏𝜋+𝑖𝜋2𝑛=1𝛼1+𝜎𝑛exp𝑖𝛼𝑏,𝑁𝑛𝜋+(𝛼)=sin𝑘(𝑏𝑎)𝑘+𝑖𝜁𝑘cos𝑘(𝑏𝑎)exp𝑖𝛼(𝑏𝑎)𝜋1𝐶ln|𝛼|(𝑏𝑎)𝜋+𝑖𝜋2×𝑛=1𝛼1+𝛼𝑛exp𝑖𝛼(𝑏𝑎),𝑃𝑛𝜋+(𝛼)=cos𝑘𝑎exp𝑖2𝛼𝑎𝜋1𝐶ln|𝛼|2𝑎𝜋+𝑖𝜋2𝑛=1𝛼1+𝜒2𝑛1exp𝑖2𝛼𝑎,(2𝑛1)𝜋(A.4) with 𝐾+𝐿(𝛼)=+(𝛼)𝑁+(𝛼)𝑃+(𝛼).(A.5) Thus 𝐾(𝛼)=𝐾+(𝛼)𝐾(𝛼),(A.6) where𝐾+(𝛼)=𝑘(cos𝑘𝑏𝑖𝜁sin𝑘𝑏)×[]}cos𝑘𝑎(sin𝑘(𝑏𝑎)+𝑖𝜁cos𝑘(𝑏𝑎))exp{(𝑖𝛼𝑏/𝜋)1𝐶ln(|𝛼|𝑏/𝜋)+𝑖𝜋/2[]}×1exp{(𝑖2𝛼𝑎/𝜋)1𝐶ln(|𝛼|2𝑎/𝜋)+𝑖𝜋/2[]}×exp{(𝑖𝛼(𝑏𝑎)/𝜋)1𝐶ln(|𝛼|(𝑏𝑎)/𝜋)+𝑖𝜋/2𝑛=11+𝛼/𝜎𝑛exp(𝑖𝛼𝑏/𝑛𝜋)1+𝛼/𝜒2𝑛1exp(𝑖2𝛼𝑎/((2𝑛1)𝜋))1+𝛼/𝛼𝑛.exp(𝑖𝛼(𝑏𝑎)/𝑛𝜋)(A.7) Here 𝜎,𝑛𝑠,𝛼,𝑛𝑠 and 𝜒,2𝑛1𝑠 are the roots of the functions 𝐿(𝛼), 𝑁(𝛼), and 𝑃(𝛼), respectively,𝐿𝜎𝑛=0,𝑁𝛼𝑛=0,𝑃𝜒2𝑛1=0,𝑛=1,2,3,.(A.8) with𝐿(𝛼)=𝐿+(𝛼),𝑁(𝛼)=𝑁+(𝛼),𝑃(𝛼)=𝑃+(𝛼)(A.9) and 𝐶 being Euler's constant given by 𝐶=0.57721 and 𝐾+(𝛼)=𝐾(𝛼). In the respective region of analyticity, when |𝛼|,𝐾±(𝛼)=𝑂|𝛼|1/2.(A.10) Similarly, for 𝑊(𝛼) given by (3.29),𝑊(𝛼)=𝜅(sin𝜅𝑏+(𝑖𝜁/𝑘)𝜅cos𝜅𝑏)sin𝜅𝑎(sin𝜅(𝑏𝑎)+(𝑖𝜁/𝑘)𝜅cos𝜅(𝑏𝑎)).(A.11) By following the above procedure [21], we can write𝑊+(𝛼)=𝑘(sin𝑘𝑏+𝑖𝜁cos𝑘𝑏)×[]}sin𝑘𝑎(sin𝑘(𝑏𝑎)+𝑖𝜁cos𝑘(𝑏𝑎))exp{(𝑖𝛼𝑏/𝜋)1𝐶ln(|𝛼|𝑏/𝜋)+𝑖𝜋/2[]}×1exp{(𝑖𝛼𝑎/𝜋)1𝐶ln(|𝛼|𝑎/𝜋)+𝑖𝜋/2[]}×exp{(𝑖𝛼(𝑏𝑎)/𝜋)1𝐶ln(|𝛼|(𝑏𝑎)/𝜋)+𝑖𝜋/2𝑛=11+𝛼/𝜎𝑛exp(𝑖𝛼𝑏/𝑛𝜋)1+𝛼/𝜎𝑛exp(𝑖𝛼𝑎/𝑛𝜋)1+𝛼/𝛼𝑛,exp(𝑖𝛼(𝑏𝑎)/𝑛𝜋)(A.12) with𝑊+(𝛼)=𝑊(𝛼).(A.13) Also, when |𝛼|, in the respective region of analyticity,𝑊±(𝛼)=𝑂|𝛼|1/2.(A.14)

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