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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.

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 [

Jones [

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 [

Schematic diagram of the trifurcated waveguide.

The problem statement is presented in Section

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

The boundary conditions and continuity conditions associated with the problem are of the form

Besides the conditions prescribed in (

For the region

The value of

For

When

For analytic convenience, we will assume that

Use of (

Writing (see the appendix)

When

For the region

Strip of analyticity and branch cuts in the complex

In expressions (

Invoking Cauchy residue theorem [

When

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

The expression of field intensity involves infinite sums/products for which we have used numerical technique and obtained the results using truncation approach [

Field intensity

10 | 1.809170 | 0.824323 |

20 | 1.853300 | 0.878547 |

30 | 1.883790 | 0.899716 |

40 | 1.905980 | 0.911027 |

50 | 1.922750 | 0.918070 |

60 | 1.935870 | 0.922877 |

70 | 1.946400 | 0.926369 |

80 | 1.955050 | 0.929019 |

90 | 1.962260 | 0.931100 |

100 | 1.968380 | 0.932777 |

110 | 1.973630 | 0.934157 |

120 | 1.978180 | 0.935313 |

130 | 1.982170 | 0.936295 |

140 | 1.985700 | 0.937140 |

150 | 1.988830 | 0.937874 |

The values of specific impedance

fibrous sheet:

The convergence of the field can be checked through the relative error for which the suitable definition would be

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 (

In Figures

Variation of field intensity

Variation of field intensity

In Figure

Variation of

Variation of relative error

The main findings from the analysis are summarized in the following points.

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.

Gradually increase in the separation distance between the infinite plates yields a decrease in the value of reflected field intensity.

The absolute values of the reflection coefficient are in accordance with the conservation of energy rules.

The findings confirm that the relative error reduces by increasing the truncation number

The established results clearly show the contribution that arises because of the soft surfaces.

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 [

The main purpose of this appendix is to give the complete factorization of the kernel functions

We may note that the functions