I= SCATTERING OF WAVES BY HALF PLANES AND SLITS WITH MIXED BOUNDARY CONDITIONS
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Title of Thesis
SCATTERING OF WAVES BY HALF PLANES AND SLITS WITH MIXED BOUNDARY CONDITIONS

Author(s)
Tasawar Hayat
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University, Islamabad
Session
1999
Subject
Mathematics
Number of Pages
345
Keywords (Extracted from title, table of contents and abstract of thesis)
scattering of waves, half planes, slits, mixed boundary conditions, acoustic diffraction problems, absorbing half plane, bi-impedant half plane, wind tunnels, spherical acoustic wave

Abstract
Considerable attention has been given to the acoustic scattering from half planes. The aim of this thesis is to contribute something by studying some acoustic diffraction problems from the absorbing half plane. Although, the acoustic wave diffraction from an absorbing half plane with similar absorbing parameter has been studied in the literature, nevertheless no attempt has been made to consider the spherical wave diffraction from a bi-impedant half plane in still air. The absorption in the half plane is introduced through different absorbing parameters. The problem is formulated in terms of matrix Wiener-Hopf (W. H.) functional equation. Physically, it corresponds to a mathematical model for a noise barrier whose surface is treated with two different acoustically absorbent materials. The modified W.H. method is used to arrive at the solution.

We further calculate the diffraction of a spherical acoustic wave from a porous barrier using a simple theory of porous materials [77] in still air. Our model assumes the barrier is made from a rigid material that is riddled with small pores that are approximately normal to the plane of the barrier. We take limited account of the compressibility of the gas in the pores. However, the gas in each pore behaves primarily as an incompressible cylinder, driven back and forth by the harmonic wavefield, but opposed by the frictional force generated at the pore walls ( the flow resistance). The barrier is thin enough (with respect to wavelength) that sound is communicated from one side to the other by the motion of numerous incompressible cylinders. The approximate boundary conditions for such a situation are derived. A formal analytic solution to the complete problem is given, for the diffracted wavefield in the farfield region of the slit. The dependence on the barrier parameters of the power removed from the reflected wavefield by the diffraction at the slit is exhibited.

In the case of noise radiated by aero engines and inside wind tunnels, it is necessary to discuss acoustic diffraction in the presence of a moving fluid . Therefore, the theory of acoustic diffraction is further extended to include the case of moving fluid and the following two problems are addressed in this direction. (1) "The diffraction of spherical wave by a half plane in a moving fluid". A finite region in the vicinity of the edge of a half plane has an impedance boundary conditions; the remaining part of the half plane is taken as rigid. It is found that the field is increased in case of moving fluid when compared with still air case. The field is also independent of the direction of the flow. This model has potential application in engine noise shielding by aircraft wings. (2) "The diffraction of a spherical Gaussian pulse by an absorbing half plane in a moving fluid for trailing edge (situation)€. The trailing edge adds the complications of a trailing vortex sheet to the absorbing half plane. The motivation of this problem comes from a desire to understand the transient nature of the wavefields-since these can be expressed as linear combination of Gaussian pulses. The time dependence of the field is tackled by the use of temporal Fourier transform. It is found that field ratio of no wake to wake situation is independent of the type of acoustic sources. Also near the edge of absorbing half plane, the field of a spherical pulse caused by the Kutta-Joukowski condition is in excess of that in its absence.

Chapter 0 is devoted to the brief history of the problems of acoustic scattering. This chapter also contains the motivation for the work presented in this thesis. In chapter 1, we calculate the diffracted field by a slit in an infinite porous barrier. Chapter 2 deals with the problems of scattering by a bi-impedant half plane. Chapter 3 is devoted to the scattering of a spherical wave by a rigid screen with an absorbent edge in a moving fluid. In chapter 4, we present the scattering of a spherical Gaussian pulse near an absorbing half plane in a moving fluid.

Download Full Thesis
17716.67 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents 0
747.06 KB
2 1 Scattering Of A Spherical Acoustic Wave By A Slit In An Infinite Porous Barrier 17
1918.9 KB
  1.1 Formulation 18
  1.2 The Boundary Condition 21
  1.3 The Wiener- Hopf Problem 27
  1.4 The Solution To The Wiener- Hopf Problem 33
  1.5 The Diffracted Wavefield 37
  1.6 Far Field Asymptotic Approximations To The Diffracted Field 39
  1.7 Discussion 42
3 2 Scattering Of A Spherical Sound Wave By A Bi- Impedant Half Plane 44
2209.38 KB
  2.1 Problem Description 45
  2.2 Solution Of The Problem 47
  2.3 Asymptotic Expressions For The Far Field 57
  2.4 Concluding Remarks 63
4 3 Scattering Of A Spherical Sound Wave By A Rigid Screen With An Absorbent Edge 64
1408.78 KB
  3.1 Formulation 65
  3.2 Solution Of The Problem 67
  3.3 Approximate Solution Of Equations (3.41) And (3.42) For KsL ‰ 1 76
  3.4 Approximate Solution Of Equations (3.43), And (3.44) For KsL ‰ 1 80
  3.5 Conclusions 85
5 4 The Transient Response Of A Spherical Gaussian Pulse By An Absorbing Half Plane 86
1511.29 KB
  4.1 Formulation Of The Problem 87
  4.2 Solution Of The Problem 89
  4.3 Discussion 91
6 5 Factorization Of The Matrix K( v) 97
10164.48 KB
  5.1 Evaluation Of The Integrals I( v) And J(v) 109
  5.2 Solution Of Integrals 115
  5.3 Detail Of Calculations Of Eq . (4.11) 118
  5.4 Evaluation Of Integral In Eq . (4.11) 124
  5.5 References 129