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, Bรผyรผkaksoy 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 [11โ€“14]. 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.

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 ๐‘ฅ-direction parallel to ๐‘ฅ-axis. The semi-infinite soft duct is placed inside the absorbingly lined duct. If ๐œ™(๐‘ฅ,๐‘ฆ,๐‘ก) is a scalar potential, then the velocity ๐ฎ and acoustic pressure ๐‘ can be written as๐ฎ=grad๐œ™,๐‘=โˆ’๐œŒ๐‘œ๐œ•๐œ™,๐œ•๐‘ก(2.1) respectively, where grad is the gradient operator, ๐œŒ๐‘œ is the density, and ๐‘ก is the time. Writing ๐œ™(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘’โˆ’๐‘–๐œ”๐‘กฮฆ(๐‘ฅ,๐‘ฆ)(2.2) and omitting ๐‘’โˆ’๐‘–๐œ”๐‘ก throughout, we have to solve the following Helmholtz equation:๐œ•2ฮฆ๐œ•๐‘ฅ2+๐œ•2ฮฆ๐œ•๐‘ฆ2+๐‘˜2ฮฆ=0,(2.3) where ๐‘˜=๐œ”/๐‘ (๐œ” is the angular frequency and ๐‘ is the speed of sound) is the wave number.

The boundary conditions and continuity conditions associated with the problem are of the form๎‚ตฮฆ+๐‘–๐œ๐‘˜๎‚ถ๐œ•ฮฆ๎‚ต๐œ•๐‘ฆ=0,๐‘ฆ=๐‘,โˆ’โˆž<๐‘ฅ<โˆž,(2.4)ฮฆ=0,๐‘ฆ=๐‘Ž,โˆ’โˆž<๐‘ฅ<0,(2.5)ฮฆ=0,๐‘ฆ=โˆ’๐‘Ž,โˆ’โˆž<๐‘ฅ<0,(2.6)ฮฆโˆ’๐‘–๐œ๐‘˜๎‚ถ๐œ•ฮฆ๐œ•๐œ•๐‘ฆ=0,๐‘ฆ=โˆ’๐‘,โˆ’โˆž<๐‘ฅ<โˆž,(2.7)ฮฆ๎€ท๐œ•๐‘ฆ๐‘ฅ,โˆ’๐‘Ž+๎€ธ=๐œ•๐œ•๐‘ฆฮฆ(๐‘ฅ,โˆ’๐‘Žโˆ’๐œ•),๐‘ฅ>0,(2.8)ฮฆ๎€ท๐œ•๐‘ฆ๐‘ฅ,๐‘Ž+๎€ธ=๐œ•ฮฆ๐œ•๐‘ฆ(๐‘ฅ,๐‘Žโˆ’),๐‘ฅ>0,(2.9) where it is assumed that ๐‘>๐‘Ž. In (2.4) and (2.7), ๐œ represents the specific impedance of the infinite duct lining and it is necessary that for an absorbent surface Re๐œ>0.

Besides the conditions prescribed in (2.4)โ€“(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 (โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž,๐‘ฅ<0), one may writeฮฆ(๐‘ฅ,๐‘ฆ)=๐‘’๐‘–๐œ’1๐‘ฅ๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚น+2๐‘Žโˆž๎“๐‘›=1๐‘…๐‘›๐‘’โˆ’๐‘–๐œ’๐‘›๐‘ฅ๎‚ธsin๐‘›๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚น2๐‘Ž,(2.10) where ๐œ’๐‘›=(๐‘˜2โˆ’๐›ผ2๐‘›)1/2, (๐‘›=1,2,3,โ€ฆ,) and ๐›ผ๐‘› satisfy๎€ท๐›ผsin2๐‘›๐‘Ž๎€ธ=0,(2.11)๐›ผ๐‘›=๐‘›๐œ‹/2๐‘Ž with 0<Im๐œ’1<Im๐œ’2<Im๐œ’3โ€ฆ. The lowest-order plane wave mode can propagate only when ๐œ‹/2<๐‘˜๐‘Ž<๐œ‹.

The value of ฮฆ(๐‘ฅ,๐‘ฆ) for (โˆ’๐‘โ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ>0) isฮฆ(๐‘ฅ,๐‘ฆ)=โˆž๎“๐‘›=1๐‘‡๐‘›๐‘’๐‘–๐œŽ๐‘›๐‘ฅ๎‚ธโˆ’sin๐›ฝ๐‘›(๐‘ฆโˆ’๐‘)+๐‘–๐œ๐‘˜๐›ฝ๐‘›cos๐›ฝ๐‘›(๎‚น๐‘ฆโˆ’๐‘),(2.12) where ๐œŽ๐‘›=(๐‘˜2โˆ’๐›ฝ2๐‘›)1/2(๐‘›=1,2,3,โ€ฆ) and ๐›ฝ๐‘› satisfy the equation๎€ทsin2๐‘๐›ฝ๐‘›๎€ธ+2๐‘–๐›ฝ๐‘›๐œ๐‘˜๎€ทcos2๐‘๐›ฝ๐‘›๎€ธ+๐›ฝ2๐‘›๐œ2๐‘˜2๎€ทsin2๐‘๐›ฝ๐‘›๎€ธ=0,(2.13) with 0<Im๐œŽ1<Im๐œŽ2<Im๐œŽ3โ€ฆ.

For (๐‘Žโ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ<0) one hasฮฆ(๐‘ฅ,๐‘ฆ)=โˆž๎“๐‘›=1๎‚๐‘‡๐‘›๐‘’โˆ’๐‘–๎‚๐›ผ๐‘›๐‘ฅ๎‚ธโˆ’sin๐›ฟ๐‘›(๐‘ฆโˆ’๐‘)+๐‘–๐œ๐‘˜๐›ฟ๐‘›cos๐›ฟ๐‘›(๎‚น๐‘ฆโˆ’๐‘),(2.14) where ๎‚๐›ผ๐‘›=(๐‘˜2โˆ’๐›ฟ2๐‘›)1/2(๐‘›=1,2,3,โ€ฆ) and ๐›ฟ๐‘› represent the roots of the equationsin๐›ฟ๐‘›(๐‘โˆ’๐‘Ž)+๐‘–๐œ๐‘˜๐›ฟ๐‘›cos๐›ฟ๐‘›(๐‘โˆ’๐‘Ž)=0,(2.15) with 0<Im๎‚๐›ผ1<Im๎‚๐›ผ2<Im๎‚๐›ผ3โ€ฆ.

When (โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž,๐‘ฅ<0), we haveฮฆ(๐‘ฅ,๐‘ฆ)=โˆž๎“๐‘›=1๎๐‘‡๐‘›๐‘’โˆ’๐‘–๎๐›ผ๐‘›๐‘ฅ๎‚ธsin๐›ฟ๐‘›(๐‘ฆ+๐‘)+๐‘–๐œ๐‘˜๐›ฟ๐‘›cos๐›ฟ๐‘›(๎‚น๐‘ฆ+๐‘).(2.16) In region (โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž,๐‘ฅ<0) the acoustic wave shows incident and reflected behavior while in the regions (โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž,๐‘ฅ<0) and (๐‘Žโ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ<0) transmission behavior is observed. In the text ๐‘…๐‘› and ๐‘‡๐‘› represent reflection and transmission coefficients, respectively. To arrive at a unique solution, we also require the โ€œedge conditionsโ€ [18]ฮฆ(๐‘ฅ,ยฑ๐‘Ž)=๐‘‚(1),ฮฆ๐‘ฆ๎€ท๐‘ฅ(๐‘ฅ,ยฑ๐‘Ž)=๐‘‚โˆ’1/2๎€ธas๐‘ฅโŸถ0.(2.17)

3. The Wiener-Hopf (WH) Equations

For analytic convenience, we will assume that ๐‘˜=๐‘˜1+๐‘–๐‘˜2โ€‰โ€‰(๐‘˜1>0, ๐‘˜2โ‰ฅ0) since the time dependence is taken to be of the form ๐‘’โˆ’๐‘–๐œ”๐‘ก [15]. Let us define Fourier transform and its inverse by๎๎€œฮฆ(๐›ผ,๐‘ฆ)=โˆžโˆ’โˆžฮฆ(๐‘ฅ,๐‘ฆ)๐‘’๐‘–๐›ผ๐‘ฅ๎ฮฆ๐‘‘๐‘ฅ=+๎ฮฆ(๐›ผ,๐‘ฆ)+โˆ’1(๐›ผ,๐‘ฆ),(3.1)ฮฆ(๐‘ฅ,๐‘ฆ)=๎€œ2๐œ‹โˆžโˆ’โˆž๎ฮฆ(๐›ผ,๐‘ฆ)๐‘’โˆ’๐‘–๐›ผ๐‘ฅ๐‘‘๐›ผ.(3.2) In (3.1),๎ฮฆ+๎€œ(๐›ผ,๐‘ฆ)=โˆž0ฮฆ(๐‘ฅ,๐‘ฆ)๐‘’๐‘–๐›ผ๐‘ฅ๎ฮฆ๐‘‘๐‘ฅ,โˆ’๎€œ(๐›ผ,๐‘ฆ)=0โˆ’โˆžฮฆ(๐‘ฅ,๐‘ฆ)๐‘’๐‘–๐›ผ๐‘ฅ๐‘‘๐‘ฅ,(3.3) where ๐›ผ is a complex variable with๐›ผ=๐œŽ+๐‘–๐œ.(3.4)

Use of (3.1) to (2.3) gives๐‘‘2๎ฮฆ๐‘‘๐‘ฆ2+๐œ…2๎ฮฆ=0,(3.5) whereโˆš๐œ…(๐›ผ)=๐‘˜2โˆ’๐›ผ2.(3.6) The suitable solutions of (3.5) in the trifurcated regions are๎ฮฆ(๐›ผ,๐‘ฆ)=๐ด1(๐›ผ)cos๐œ…๐‘ฆ+๐ต1(๐›ผ)sin๐œ…๐‘ฆ(โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž),(3.7)๎ฮฆ(๐›ผ,๐‘ฆ)=โˆ’๐‘–๐›ผ+๐œ’1๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚น2๐‘Ž+๐ด2(๐›ผ)cos๐œ…๐‘ฆ+๐ต2(๐›ผ)sin๐œ…๐‘ฆ(โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž),(3.8)๎ฮฆ(๐›ผ,๐‘ฆ)=๐ด3(๐›ผ)cos๐œ…๐‘ฆ+๐ต3(๐›ผ)sin๐œ…๐‘ฆ(๐‘Žโ‰ค๐‘ฆโ‰ค๐‘),(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)โ€“(2.9) become ๎๎‚ตฮฆ(๐›ผ,๐‘)+๐‘–๐œ๐‘˜๎‚ถ๎ฮฆ๎…ž๎ฮฆ(๐›ผ,๐‘)=0,(3.10)โˆ’๎ฮฆ(๐›ผ,๐‘Ž)=0,(3.11)โˆ’๎๎‚ต(๐›ผ,โˆ’๐‘Ž)=0,(3.12)ฮฆ(๐›ผ,โˆ’๐‘)โˆ’๐‘–๐œ๐‘˜๎‚ถ๎ฮฆ๎…ž๎ฮฆ(๐›ผ,โˆ’๐‘)=0,(3.13)๎…ž+๎€ท๐›ผ,โˆ’๐‘Ž+๎€ธ=๎ฮฆ๎…ž+(๐›ผ,โˆ’๐‘Žโˆ’๎ฮฆ),(3.14)๎…ž+๎€ท๐›ผ,๐‘Ž+๎€ธ=๎ฮฆ๎…ž+(๐›ผ,๐‘Žโˆ’),(3.15) where prime denotes the differentiation with respect to ๐‘ฆ. In order to determine the unknowns ๐ด๐‘—(๐›ผ) and ๐ต๐‘—(๐›ผ)โ€‰โ€‰(๐‘—=1,2 and 3), we proceed to satisfy boundary conditions (3.10)โ€“(3.13). Thus, by invoking (3.12) and (3.13) in (3.7), we may write ๐ด1(๐›ผ)cos๐œ…๐‘Žโˆ’๐ต1(๐›ผ)sin๐œ…๐‘Ž=ฮฆ+1๐ด(๐›ผ),1๎‚ต๎‚ต(๐›ผ)cos๐œ…๐‘โˆ’๐‘–๐œ๐‘˜๎‚ถ๎‚ถ๐œ…sin๐œ…๐‘โˆ’๐ต1๎‚ต๎‚ต(๐›ผ)sin๐œ…๐‘+๐‘–๐œ๐‘˜๎‚ถ๎‚ถ๐œ…cos๐œ…๐‘=0,(3.16) where ฮฆ+1๎ฮฆ(๐›ผ)=+(๐›ผ,โˆ’๐‘Ž) is analytic in Im๐›ผ>โˆ’Im๐‘˜. Solving, above equations for ๐ด1(๐›ผ) and ๐ต1(๐›ผ) we obtain ๐ด1(๐›ผ)=sin๐œ…๐‘+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…๐‘ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+1๐ต(๐›ผ),1(๐›ผ)=cos๐œ…๐‘โˆ’(๐‘–๐œ/๐‘˜)๐œ…sin๐œ…๐‘ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+1(๐›ผ).(3.17) Again, using (3.11) and (3.12) in (3.8), we may write๐ด2(๐›ผ)cos๐œ…๐‘Ž+๐ต2(๐›ผ)sin๐œ…๐‘Ž=ฮฆ+2๐ด(๐›ผ),2(๐›ผ)cos๐œ…๐‘Žโˆ’๐ต2(๐›ผ)sin๐œ…๐‘Ž=ฮฆ+1(๐›ผ),(3.18) where ฮฆ+2๎ฮฆ(๐›ผ)=+(๐›ผ,๐‘Ž) is analytic in Im๐›ผ>โˆ’Im๐‘˜. Solving (3.18) for ๐ด2(๐›ผ) and ๐ต2(๐›ผ), we obtain๐ด2๎€ทฮฆ(๐›ผ)=sin๐œ…๐‘Ž+1(๐›ผ)+ฮฆ+2๎€ธ(๐›ผ),๐ตsin2๐œ…๐‘Ž2๎€ทฮฆ(๐›ผ)=cos๐œ…๐‘Ž+2(๐›ผ)โˆ’ฮฆ+1๎€ธ(๐›ผ).sin2๐œ…๐‘Ž(3.19) With the use of (3.10) and (3.11) in (3.9), we may write๐ด3๎‚ต๎‚ต(๐›ผ)cos๐œ…๐‘โˆ’๐‘–๐œ๐‘˜๎‚ถ๎‚ถ๐œ…sin๐œ…๐‘+๐ต3๎‚ต๎‚ต(๐›ผ)sin๐œ…๐‘+๐‘–๐œ๐‘˜๎‚ถ๎‚ถ๐ด๐œ…cos๐œ…๐‘=0,3(๐›ผ)cos๐œ…๐‘Ž+๐ต3(๐›ผ)sin๐œ…๐‘Ž=ฮฆ+2(๐›ผ).(3.20) Solving for ๐ด3(๐›ผ) and ๐ต3(๐›ผ), we obtain ๐ด3(๐›ผ)=sin๐œ…๐‘+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…๐‘ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+2๐ต(๐›ผ),3(๐›ผ)=โˆ’cos๐œ…๐‘โˆ’(๐‘–๐œ/๐‘˜)๐œ…sin๐œ…๐‘ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+2(๐›ผ).(3.21) By substituting the above values of ๐ด๐‘—(๐›ผ) and ๐ต๐‘—(๐›ผ) (๐‘—=1,2 and 3) in (3.7)โ€“(3.9) we get๎ฮฆ(๐›ผ,๐‘ฆ)=sin๐œ…(๐‘ฆ+๐‘)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆ+๐‘)ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+1๎(๐›ผ)(โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž),ฮฆ(๐›ผ,๐‘ฆ)=โˆ’๐‘–๐›ผ+๐œ’1๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚น+12๐‘Ž๎€ทฮฆsin2๐œ…๐‘Ž+2(๐›ผ)sin๐œ…(๐‘ฆ+๐‘Ž)โˆ’ฮฆ+1๎€ธ๎(๐›ผ)sin๐œ…(๐‘ฆโˆ’๐‘Ž)(โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž),ฮฆ(๐›ผ,๐‘ฆ)=sin๐œ…(๐‘โˆ’๐‘ฆ)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆโˆ’๐‘)ฮฆsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)+2(๐›ผ)(๐‘Žโ‰ค๐‘ฆโ‰ค๐‘).(3.22) Invoking (3.22) in (3.14) and (3.15), we arrive at โˆ’๐œ…(sin๐œ…(๐‘+๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘+๐‘Ž))ฮฆ+1(๐›ผ)+sin2๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž))๐œ…ฮฆ+2(๐›ผ)โˆ’๐œ†sin2๐œ…๐‘Ž1๐›ผ+๐œ’1=ฮฆโˆ’1(๐›ผ),๐œ…(sin๐œ…(๐‘+๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘+๐‘Ž))ฮฆ+2(๐›ผ)โˆ’sin2๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž))๐œ…ฮฆ+1(๐›ผ)+๐œ†sin2๐œ…๐‘Ž1๐›ผ+๐œ’1=ฮฆโˆ’2(๐›ผ),(3.23) where๐œ†1=โˆ’๐‘–๐œ‹,ฮฆ2๐‘Žโˆ’1(๎ฮฆ๐›ผ)=๎…žโˆ’(๐›ผ,โˆ’๐‘Žโˆ’)โˆ’๎ฮฆ๎…žโˆ’๎€ท๐›ผ,โˆ’๐‘Ž+๎€ธ,ฮฆโˆ’2๎ฮฆ(๐›ผ)=๎…žโˆ’(๐›ผ,๐‘Žโˆ’๎ฮฆ)โˆ’๎…žโˆ’๎€ท๐›ผ,๐‘Ž+๎€ธ.(3.24) The functions ฮฆโˆ’1(๐›ผ) and ฮฆโˆ’2(๐›ผ) are analytic in the region Im๐›ผ<Im๐‘˜. Addition and subtraction of (3.23) will give๐‘Š(๐›ผ)๐ท+(๐›ผ)=๐‘†โˆ’(๐›ผ),๐พ(๐›ผ)๐‘†+(๐›ผ)+2๐œ†1๐›ผ+๐œ’1=๐ทโˆ’(๐›ผ),(3.25) whereฮฆยฑ2(๐›ผ)โˆ’ฮฆยฑ1(๐›ผ)=๐ทยฑฮฆ(๐›ผ),(3.26)ยฑ2(๐›ผ)+ฮฆยฑ1(๐›ผ)=๐‘†ยฑ(๐›ผ),(3.27)๐พ(๐›ผ)=๐œ…(cos๐œ…๐‘โˆ’(๐‘–๐œ/๐‘˜)๐œ…sin๐œ…๐‘)๐œ…cos๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)),(3.28)๐‘Š(๐›ผ)=(sin๐œ…๐‘+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…๐‘).sin๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž))(3.29)

4. Solution of the Problem

Writing (see the appendix) ๐‘Š(๐›ผ)=๐‘Š+(๐›ผ)๐‘Šโˆ’(๐›ผ),๐พ(๐›ผ)=๐พ+(๐›ผ)๐พโˆ’(๐›ผ),(4.1) where (+) is the subscript assigned to the function regular in the upper half plane Im๐›ผ>โˆ’Im๐‘˜ and the subscript (โˆ’) represents the function regular in the lower half plane Im๐›ผ<Im๐‘˜. Now from (3.25), we have๐‘Š+(๐›ผ)๐ท+๐‘†(๐›ผ)=โˆ’(๐›ผ)๐‘Šโˆ’(,๐พ๐›ผ)+(๐›ผ)๐‘†+(๐›ผ)+2๐œ†1๎€ท๐›ผ+๐œ’1๎€ธ๐พโˆ’๎€ทโˆ’๐œ’1๎€ธ=๐ทโˆ’(๐›ผ)๐พโˆ’โˆ’(๐›ผ)2๐œ†1๐›ผ+๐œ’1๎ƒฌ1๐พโˆ’โˆ’1(๐›ผ)๐พโˆ’๎€ทโˆ’๐œ’1๎€ธ๎ƒญ.(4.2) Note that the left-hand side of both equations is analytic in Im๐›ผ>โˆ’Im๐‘˜ and the right-hand side is analytic in Im๐›ผ<Im๐‘˜. Also, when |๐›ผ|โ†’โˆž,๐พยฑ๎€ท(๐›ผ)=๐‘‚|๐›ผ|1/2๎€ธ,๐‘Šยฑ๎€ท(๐›ผ)=๐‘‚|๐›ผ|1/2๎€ธ,(4.3) and Fourier transform of edge conditions (2.17) helps us to determine the asymptotic behavior of ๐ทยฑ(๐›ผ) and ๐‘†ยฑ(๐›ผ). For |๐›ผ|โ†’โˆž,๐ทโˆ’๎€ท(๐›ผ)=๐‘‚|๐›ผ|โˆ’1๎€ธ,๐‘†โˆ’๎€ท๐›ผ(๐›ผ)=๐‘‚โˆ’1๎€ธ๐ทforIm๐›ผ<Im๐‘˜,+๎€ท(๐›ผ)=๐‘‚|๐›ผ|โˆ’1๎€ธ,๐‘†+๎€ท๐›ผ(๐›ผ)=๐‘‚โˆ’1/2๎€ธforIm๐›ผ>โˆ’Im๐‘˜.(4.4) Now the use of (4.3)โ€“(4.4) and standard Wiener-Hopf procedure [15] on (4.2) give๐ท+๐พ(๐›ผ)=0,+(๐›ผ)๐‘†+(๐›ผ)+2๐œ†1๎€ท๐›ผ+๐œ’1๎€ธ๐พ+๎€ท๐œ’1๎€ธ=0,(4.5) where ๐พโˆ’(โˆ’๐œ’1)=๐พ+(๐œ’1). Using (3.26) and (3.27) in (4.5), we getฮฆ+1๐œ†(๐›ผ)=โˆ’1๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ,ฮฆ+2(๐œ†๐›ผ)=โˆ’1๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ.(4.6) From (3.22) and (4.6), we obtain the field representations for the different regions as follows.

When (โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž),๎๐œ†ฮฆ(๐›ผ,๐‘ฆ)=โˆ’1๎ƒฏsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)sin๐œ…(๐‘ฆ+๐‘)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆ+๐‘)๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ.(4.7) For (โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž), one obtains๎ฮฆ(๐›ผ,๐‘ฆ)=โˆ’๐‘–๐›ผ+๐œ’1๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚นโˆ’๐œ†2๐‘Ž1๎ƒฏcos๐œ…๐‘Žcos๐œ…๐‘ฆ๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ.(4.8) For (๐‘Žโ‰ค๐‘ฆโ‰ค๐‘), one has๎๐œ†ฮฆ(๐›ผ,๐‘ฆ)=โˆ’1๎ƒฏsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)sin๐œ…(๐‘โˆ’๐‘ฆ)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆโˆ’๐‘)๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ.(4.9) Taking inverse Fourier transform of (4.7)โ€“(4.9), we obtain the following.

For the region (โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž,๐‘ฅ<0), one may write๐œ†ฮฆ(๐‘ฅ,๐‘ฆ)=โˆ’1๎€œ2๐œ‹โˆž+๐‘–๐œโˆ’โˆž+๐‘–๐œ๐‘’โˆ’๐‘–๐›ผ๐‘ฅร—๎ƒฏsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)sin๐œ…(๐‘ฆ+๐‘)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆ+๐‘)๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ๐‘‘๐›ผ.(4.10) When (โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž,๐‘ฅ<0), we haveฮฆ(๐‘ฅ,๐‘ฆ)=๐‘’๐‘–๐œ’1๐‘ฅ๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚นโˆ’๐œ†2๐‘Ž1๎€œ2๐œ‹โˆž+๐‘–๐œโˆ’โˆž+๐‘–๐œ๐‘’โˆ’๐‘–๐›ผ๐‘ฅ๎ƒฏcos๐œ…๐‘Žcos๐œ…๐‘ฆ๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ๐‘‘๐›ผ.(4.11) For (๐‘Žโ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ<0), we arrive at๐œ†ฮฆ(๐‘ฅ,๐‘ฆ)=1๎€œ2๐œ‹โˆž+๐‘–๐œโˆ’โˆž+๐‘–๐œ๐‘’โˆ’๐‘–๐›ผ๐‘ฅร—๎ƒฏsin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)sin๐œ…(๐‘ฆโˆ’๐‘)โˆ’(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆโˆ’๐‘)๎€ท๐›ผ+๐œ’1๎€ธ๐พ+(๐›ผ)๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ๐‘‘๐›ผ.(4.12) For (โˆ’๐‘โ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ>0), one may write๐œ†ฮฆ(๐‘ฅ,๐‘ฆ)=1๎€œ2๐œ‹โˆž+๐‘–๐œโˆ’โˆž+๐‘–๐œ๐‘’โˆ’๐‘–๐›ผ๐‘ฅร—๎ƒฏ(cos๐œ…๐‘โˆ’(๐‘–๐œ/๐‘˜)๐œ…sin๐œ…๐‘sin๐œ…(๐‘ฆโˆ’๐‘)โˆ’(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘ฆโˆ’๐‘))cos๐œ…๐‘Ž๐พโˆ’(๐›ผ)๐œ…๎€ท๐›ผ+๐œ’1๎€ธ๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ๐‘‘๐›ผ.(4.13) In (4.10)โ€“(4.13), ๐œ is the imaginary part of ๐›ผ, โˆš๐œ…(๐›ผ)=๐‘˜2โˆ’๐›ผ2 and branch cuts are taken to be from ๐‘˜ to ๐‘–โˆž and โˆ’๐‘˜ to โˆ’๐‘–โˆž, and 0โ‰คarg๐œ…โ‰ค๐œ‹ (Figure 2). Note that the integrands have no singularities which lie on the contour of integration. To evaluate the integrals in (4.10)โ€“(4.13), it is noted that the contour of integration in these equations lies in the strip โˆ’Im๐‘˜<Im๐›ผ<Im๐‘˜.

In expressions (4.10)โ€“(4.13), the pole ๐›ผ=โˆ’๐œ’1 lies below the contour of integration. One can also note that the terms in the curly brackets {} of (4.10)โ€“(4.12) have no branch points in Im๐›ผ>โˆ’Im๐‘˜ and those in (4.13) have no branch points in Im๐›ผ<Im๐‘˜. Thus the only singularities in the integrands of (4.10) and (4.12) occur at the zeros of๎‚ตsin๐œ…(๐‘โˆ’๐‘Ž)+๐‘–๐œ๐‘˜๎‚ถ๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)=0,(4.14) that is, at๐›ผ=๎‚๐›ผ๐‘›=๎€ท๐‘˜2โˆ’๐›ฟ2๐‘›๎€ธ1/2(๐‘›=1,2,3,โ€ฆ).(4.15) The only singularities of (4.11) occur at the zeros of cos๐œ…๐‘Ž=0, that is at๐›ผ=๎‚๐œ’2๐‘›โˆ’1=๎‚ต๐‘˜2โˆ’(2๐‘›โˆ’1)2๐œ‹24๐‘Ž2๎‚ถ1/2(๐‘›=1,2,3,โ€ฆ).(4.16) The only singularities in the integrands in (4.13) occur at the zeros of๎‚ตcos๐œ…๐‘โˆ’๐‘–๐œ๐‘˜๎‚ถ๐œ…sin๐œ…๐‘=0,(4.17) that is at ๐›ผ=โˆ’๐œŽ๐‘›๎€ท๐‘˜=โˆ’2โˆ’๐›ฝ2๐‘›๎€ธ1/2(๐‘›=1,2,3,โ€ฆ).(4.18)

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

When (โˆ’๐‘โ‰ค๐‘ฆโ‰คโˆ’๐‘Ž,๐‘ฅ<0),ฮฆ(๐‘ฅ,๐‘ฆ)=โˆ’โˆž๎“๐‘›=1๐‘’โˆ’๐‘–๎‚๐›ผ๐‘›๐‘ฅ๐‘ž๎…ž1๎€ท๎‚๐›ผ๐‘›๎€ธ๎ƒฏ๎€ทsin๐›ฟ๐‘›(๐‘ฆ+๐‘)+(๐‘–๐œ/๐‘˜)๐›ฟ๐‘›cos๐›ฟ๐‘›๎€ธ๐œ‹(๐‘ฆ+๐‘)๎€ท2๐‘Ž๎‚๐›ผ๐‘›+๐œ’1๎€ธ๐พ+๎€ท๎‚๐›ผ๐‘›๎€ธ๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ,(5.1) where๐‘ž1๎‚ต(๐›ผ)=sin๐œ…(๐‘โˆ’๐‘Ž)+๐‘–๐œ๐‘˜๎‚ถ๐œ…cos๐œ…(๐‘โˆ’๐‘Ž),๎‚๐›ผ๐‘›=๎€ท๐‘˜2โˆ’๐›ฟ2๐‘›๎€ธ1/2(๐‘›=1,2,3,โ€ฆ).(5.2) For (โˆ’๐‘Žโ‰ค๐‘ฆโ‰ค๐‘Ž,๐‘ฅ<0), we haveฮฆ(๐‘ฅ,๐‘ฆ)=๐‘’๐‘–๐œ’1๐‘ฅ๎‚ธsin๐œ‹(๐‘ฆโˆ’๐‘Ž)๎‚นโˆ’2๐‘Žโˆž๎“๐‘›=1๐œ‹๐›ผ2๐‘›โˆ’1(โˆ’1)๐‘›+1๐‘’โˆ’๐‘–๐œ’2๐‘›โˆ’1๐‘ฅ๎€ท๐›ผcos2๐‘›โˆ’1๐‘ฆ๎€ธ2๐‘Ž2๐œ’2๐‘›โˆ’1๎€ท๐œ’2๐‘›โˆ’1+๐œ’1๎€ธ๐พ+๎€ท๐œ’2๐‘›โˆ’1๎€ธ๐พ+๎€ท๐œ’1๎€ธ.(5.3) For (๐‘Žโ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ<0), we arrive atฮฆ(๐‘ฅ,๐‘ฆ)=โˆž๎“๐‘›=1๐‘’โˆ’๐‘–๎‚๐›ผ๐‘›๐‘ฅ๐‘ž๎…ž1๎€ท๎‚๐›ผ๐‘›๎€ธ๎ƒฏ๎€ทsin๐›ฟ๐‘›(๐‘ฆโˆ’๐‘)โˆ’(๐‘–๐œ/๐‘˜)๐›ฟ๐‘›cos๐›ฟ๐‘›๎€ธ๐œ‹(๐‘ฆโˆ’๐‘)๎€ท2๐‘Ž๎‚๐›ผ๐‘›+๐œ’1๎€ธ๐พ+๎€ท๎‚๐›ผ๐‘›๎€ธ๐พ+๎€ท๐œ’1๎€ธ๎ƒฐ.(5.4) When (โˆ’๐‘โ‰ค๐‘ฆโ‰ค๐‘,๐‘ฅ>0), we haveฮฆ(๐‘ฅ,๐‘ฆ)=โˆž๎“๐‘›=1๐‘’๐‘–๐œŽ๐‘›๐‘ฅ๎€ทsin๐›ฝ๐‘›(๐‘ฆโˆ’๐‘)โˆ’(๐‘–๐œ/๐‘˜)๐›ฝ๐‘›cos๐›ฝ๐‘›๎€ธ(๐‘โˆ’๐‘ฆ)๐œ‹cos๐›ฝ๐‘›๐‘Ž2๐‘Ž๐‘š๎…ž๎€ทโˆ’๐œŽ๐‘›๎€ธ๐พ+๎€ท๐œ’1๐œ’๎€ธ๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐›ฝ๐‘›๐พโˆ’๎€ทโˆ’๐œŽ๐‘›๎€ธ,(5.5) where๎‚ต๐‘š(๐›ผ)=cos๐œ…๐‘โˆ’๐‘–๐œ๐‘˜๎‚ถ๐œ…sin๐œ…๐‘,๐›ผ=โˆ’๐œŽ๐‘›๎€ท๐‘˜=โˆ’2โˆ’๐›ฝ2๐‘›๎€ธ1/2(๐‘›=1,2,3,โ€ฆ).(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 ๐‘›=1 given by (5.3) as๐‘…1=๐œ‹28๐‘Ž3๐œ’21๎€ท๐พ+๎€ท๐œ’1๎€ธ๎€ธ2.(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 ๐‘› 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.

The values of specific impedance ๐œ=๐œ‰+๐‘–๐œ‚(=๐‘ง/๐œŒ0๐‘) for an absorbing sheet which seem to have practical importance are [13]fibrous sheet: ๐œ‰=0.5, โˆ’1<๐œ‚<3,perforated sheet: 0<๐œ‰<3, โˆ’1<๐œ‚<3.

The convergence of the field can be checked through the relative error for which the suitable definition would be๐ธ๐‘›=max๐‘ฅ,๐‘ฆ||ฮฆ๐‘›(๐‘ฅ,๐‘ฆ)โˆ’ฮฆ๐‘||(๐‘ฅ,๐‘ฆ)||ฮฆ๐‘||(๐‘ฅ,๐‘ฆ),(7.1) where ฮฆ๐‘›(๐‘ฅ,๐‘ฆ) is the solution obtained using the truncation number ๐‘› and ๐‘ is suitably large truncation number (๐‘=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 โ€œMATHEMATICA 5.2โ€ 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 ๐‘˜ for different values of noise reduction parameters, that is, ๐œ‰ (real part of ๐œ ) and ๐œ‚ (the imaginary part of ๐œ).

In Figure 5, the reflected field is plotted against wave number ๐‘˜ for different values of ๐‘ (separation distance between the infinite plates). In Figure 6, variation of relative error ๐ธ๐‘› against truncation number ๐‘› is plotted.

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 ๐‘›.(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 ๐พ(๐›ผ) and ๐‘Š(๐›ผ) and to show their asymptotic behavior as |๐›ผ|โ†’โˆž. The factorization of these functions ๐พ(๐›ผ) and ๐‘Š(๐›ผ) is of the form๐พ(๐›ผ)=๐พ+(๐›ผ)๐พโˆ’(๐›ผ),๐‘Š(๐›ผ)=๐‘Š+(๐›ผ)๐‘Šโˆ’(๐›ผ),(A.1) where ๐พ+(๐›ผ) and ๐‘Š+(๐›ผ) denote certain functions which are regular and free of zeros in upper half plane Im๐›ผ>โˆ’Im๐‘˜ and ๐พโˆ’(๐›ผ) and ๐‘Šโˆ’(๐›ผ) denote certain functions which are regular and free of zeros in lower half plane Im๐›ผ<Im๐‘˜.

We may note that the functions ๐พ(๐›ผ) and ๐‘Š(๐›ผ) are even in Fourier transform parameter ๐›ผ and more precisely their respective derivatives are zero at ๐›ผ=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 ๐พ(๐›ผ) by (3.28), we have ๐พ(๐›ผ)=๐œ…(cos๐œ…๐‘โˆ’(๐‘–๐œ/๐‘˜)๐œ…sin๐œ…๐‘)cos๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)).(A.2) It is evident that the product factorization of ๐พ(๐›ผ) depends upon the factorization of ๎‚ต๐ฟ(๐›ผ)=cos๐œ…๐‘โˆ’๐‘–๐œ๐‘˜๎‚ถ๐œ…sin๐œ…๐‘,๐‘(๐›ผ)=sin๐œ…(๐‘โˆ’๐‘Ž)๐œ…+๎‚ต๐‘–๐œ๐‘˜๎‚ถcos๐œ…(๐‘โˆ’๐‘Ž),๐‘ƒ(๐›ผ)=cos๐œ…๐‘Ž.(A.3) By employing the procedure outlined by Mittra and Lee [21], we have ๐ฟ+โˆš(๐›ผ)=๎‚ปcos๐‘˜๐‘โˆ’๐‘–๐œsin๐‘˜๐‘exp๐‘–๐›ผ๐‘๐œ‹๎‚ธ๎‚ต1โˆ’๐ถโˆ’ln|๐›ผ|๐‘๐œ‹๎‚ถ+๐‘–๐œ‹2๎‚น๎‚ผโˆž๎‘๐‘›=1๎‚ต๐›ผ1+๐œŽ๐‘›๎‚ถ๎‚€exp๐‘–๐›ผ๐‘๎‚,๐‘๐‘›๐œ‹+๎ƒŽ(๐›ผ)=sin๐‘˜(๐‘โˆ’๐‘Ž)๐‘˜+๎‚ต๐‘–๐œ๐‘˜๎‚ถ๎‚ปcos๐‘˜(๐‘โˆ’๐‘Ž)exp๐‘–๐›ผ(๐‘โˆ’๐‘Ž)๐œ‹๎‚ธ๎‚ต1โˆ’๐ถโˆ’ln|๐›ผ|(๐‘โˆ’๐‘Ž)๐œ‹๎‚ถ+๐‘–๐œ‹2ร—๎‚น๎‚ผโˆž๎‘๐‘›=1๎‚ต๐›ผ1+๎‚๐›ผ๐‘›๎‚ถ๎‚ตexp๐‘–๐›ผ(๐‘โˆ’๐‘Ž)๎‚ถ,๐‘ƒ๐‘›๐œ‹+โˆš(๐›ผ)=๎‚ปcos๐‘˜๐‘Žexp๐‘–2๐›ผ๐‘Ž๐œ‹๎‚ธ๎‚ต1โˆ’๐ถโˆ’ln|๐›ผ|2๐‘Ž๐œ‹๎‚ถ+๐‘–๐œ‹2๎‚น๎‚ผโˆž๎‘๐‘›=1๎‚ต๐›ผ1+๎‚๐œ’2๐‘›โˆ’1๎‚ถ๎‚ตexp๐‘–2๐›ผ๐‘Ž๎‚ถ,(2๐‘›โˆ’1)๐œ‹(A.4) with ๐พ+๐ฟ(๐›ผ)=+(๐›ผ)๐‘+(๐›ผ)๐‘ƒ+(๐›ผ).(A.5) Thus ๐พ(๐›ผ)=๐พ+(๐›ผ)๐พโˆ’(๐›ผ),(A.6) where๐พ+๎ƒŽ(๐›ผ)=๐‘˜(cos๐‘˜๐‘โˆ’๐‘–๐œsin๐‘˜๐‘)ร—[]}cos๐‘˜๐‘Ž(sin๐‘˜(๐‘โˆ’๐‘Ž)+๐‘–๐œcos๐‘˜(๐‘โˆ’๐‘Ž))exp{(๐‘–๐›ผ๐‘/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|๐‘/๐œ‹)+๐‘–๐œ‹/2[]}ร—1exp{(๐‘–2๐›ผ๐‘Ž/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|2๐‘Ž/๐œ‹)+๐‘–๐œ‹/2[]}ร—exp{(๐‘–๐›ผ(๐‘โˆ’๐‘Ž)/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|(๐‘โˆ’๐‘Ž)/๐œ‹)+๐‘–๐œ‹/2โˆž๎‘๐‘›=1๎€ท1+๐›ผ/๐œŽ๐‘›๎€ธexp(๐‘–๐›ผ๐‘/๐‘›๐œ‹)๎€ท1+๐›ผ/๎‚๐œ’2๐‘›โˆ’1๎€ธ๎€ทexp(๐‘–2๐›ผ๐‘Ž/((2๐‘›โˆ’1)๐œ‹))1+๐›ผ/๎‚๐›ผ๐‘›๎€ธ.exp(๐‘–๐›ผ(๐‘โˆ’๐‘Ž)/๐‘›๐œ‹)(A.7) Here ๐œŽ,๐‘›๐‘ ,๎‚๐›ผ,๐‘›๐‘  and ๎‚๐œ’,2๐‘›โˆ’1๐‘  are the roots of the functions ๐ฟ(๐›ผ), ๐‘(๐›ผ), and ๐‘ƒ(๐›ผ), respectively,๐ฟ๎€ท๐œŽ๐‘›๎€ธ๎€ท=0,๐‘๎‚๐›ผ๐‘›๎€ธ๎€ท=0,๐‘ƒ๎‚๐œ’2๐‘›โˆ’1๎€ธ=0,๐‘›=1,2,3,โ€ฆ.(A.8) with๐ฟโˆ’(๐›ผ)=๐ฟ+(โˆ’๐›ผ),๐‘โˆ’(๐›ผ)=๐‘+(โˆ’๐›ผ),๐‘ƒโˆ’(๐›ผ)=๐‘ƒ+(โˆ’๐›ผ)(A.9) and ๐ถ being Euler's constant given by ๐ถ=0.57721โ€ฆ and ๐พ+(๐›ผ)=๐พโˆ’(โˆ’๐›ผ). In the respective region of analyticity, when |๐›ผ|โ†’โˆž,๐พยฑ๎€ท(๐›ผ)=๐‘‚|๐›ผ|1/2๎€ธ.(A.10) Similarly, for ๐‘Š(๐›ผ) given by (3.29),๐‘Š(๐›ผ)=๐œ…(sin๐œ…๐‘+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…๐‘)sin๐œ…๐‘Ž(sin๐œ…(๐‘โˆ’๐‘Ž)+(๐‘–๐œ/๐‘˜)๐œ…cos๐œ…(๐‘โˆ’๐‘Ž)).(A.11) By following the above procedure [21], we can write๐‘Š+๎ƒŽ(๐›ผ)=๐‘˜(sin๐‘˜๐‘+๐‘–๐œcos๐‘˜๐‘)ร—[]}sin๐‘˜๐‘Ž(sin๐‘˜(๐‘โˆ’๐‘Ž)+๐‘–๐œcos๐‘˜(๐‘โˆ’๐‘Ž))exp{(๐‘–๐›ผ๐‘/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|๐‘/๐œ‹)+๐‘–๐œ‹/2[]}ร—1exp{(๐‘–๐›ผ๐‘Ž/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|๐‘Ž/๐œ‹)+๐‘–๐œ‹/2[]}ร—exp{(๐‘–๐›ผ(๐‘โˆ’๐‘Ž)/๐œ‹)1โˆ’๐ถโˆ’ln(|๐›ผ|(๐‘โˆ’๐‘Ž)/๐œ‹)+๐‘–๐œ‹/2โˆž๎‘๐‘›=1๎€ท1+๐›ผ/๎‚๐œŽ๐‘›๎€ธexp(๐‘–๐›ผ๐‘/๐‘›๐œ‹)๎€ท1+๐›ผ/๎๐œŽ๐‘›๎€ธ๎€ทexp(๐‘–๐›ผ๐‘Ž/๐‘›๐œ‹)1+๐›ผ/๎‚๐›ผ๐‘›๎€ธ,exp(๐‘–๐›ผ(๐‘โˆ’๐‘Ž)/๐‘›๐œ‹)(A.12) with๐‘Š+(๐›ผ)=๐‘Šโˆ’(โˆ’๐›ผ).(A.13) Also, when |๐›ผ|โ†’โˆž, in the respective region of analyticity,๐‘Šยฑ๎€ท(๐›ผ)=๐‘‚|๐›ผ|1/2๎€ธ.(A.14)