1. Acoustic diffraction by an Oscillating strip

2. This problem is basically solved by a technique called Wiener Hopf technique

3. Scheme of Presentation This presentation includes: 1.Introduction of Originators of this Technique 2.A general tour of Wiener Hopf technique 4.Applications of Wiener-Hopf technique 5.Acoustic diffraction by an Oscillating strip a)Geometry b)Formation of the Problem c)Solution of the problem d)Graphical / Numerical Computation

4. Introduction of Originators of this Technique

7. A general tour of Wiener Hopf technique

11. Application of Wiener-Hopf technique

13. Acoustic diffraction by an Oscillating strip

15. What is an acoustic Plane Wave?

16. A wave coming from a source located at an infinite distance with constant frequency and amplitude.

17. How it looks like?

19. Geometry

20. Strip Geometry

22. Formulation of the Problem

23.  We consider the scattering of plane acoustic wave from an oscillating strip occupying the space at and is oscillating in a direction perpendicular to the screen with velocity, where is a periodic function of time whose generalized Fourier series is given by

24. Here, the Fourier coefficients are given by with non zero fundamental frequency

25. Assume the continuity of the velocity across the boundary as given in P. A Cannel paper where the total velocity potential satisfies the wave equation For convenience, we write

26. where φ is the diffracted field and is the incident field given by Where is the frequency and is the speed of the sound.

27. We need to solve the following boundary value problem: value problem:

28. Scheme of Solution 1.Fourier temporal transform of the Problem is taken to convert it from time domain to frequency domain 2.Then Fourier Integral transform is taken to switch the problem to Complex domain from Spatial domain 3.Problem is now solved using Wiener-Hopf technique in Complex domain 4.Inverse Fourier transform is taken bring back the problem in Spatial domain 5.Then Inverse Temporal transform is taken to bring it back to time domain

29. Solution of the Problem

30. Define the temporal Fourier transform pair as

31. We need to solve the following boundary value problem: value problem:

32. Taking the temporal Fourier transform, the above take the form as follows: where

33. Define Spatial Fourier transform over the variable as follows: with

34. Transforming above equations

35. The Solution of after using the continuity of Ψ′ across y=0 is given by Now using above and following equation, we have and

36.  Adding and subtracting above two equations we have First Wiener Hopf equation: where

37. and Now, solving again and

38. We arrive at the Second Wiener Hopf equation where equations mentioned above in red boxes and after solving We get

39.  Solution of above W.E equation, as laid down in B. Noble’s book, we arrive at the following value of the

41. where

43. Far Field Approximation Far Field Approximation The far field approximation can found asymptotically. We arrive at the following results

47. Graphical Discussion

48. Case 1 We consider the case where the angle of incidence to the oscillating strip (with constant frequency) is different say 30 o, 45 o and 90 o

53. Amplitude of the separated field for different values of angle of incidence

54. Case 2 We consider the case where the frequency of the incidence ray is different and the strip is oscillating with constant frequency.

59. Amplitude of the separated field for different values of wave frequency

60. Case 3 We consider the case where the frequency of the incidence ray is constant and the strip is oscillating with different frequencies.

65. Amplitude of the separated field for different values of strip frequency

66. Respected Gentlemen I take pride to introduce a New idea in Wiener Hopf technique in the Strip Geometry

67. To understand this, we need to have the idea of the half plane and the strip geometries. Lets have a trip to these. To understand this, we need to have the idea of the half plane and the strip geometries. Lets have a trip to these.

68. Half Plane Geometry

69. Strip Geometry

70. D iscussion If we gradually increase the value of the strip length l. Please Imagine what will happen?

71. Discussion The Strip geometry supposed to be converted into Half plane geometry.

72. This phenomenon must be implemented to check the validity of the results obtained for the Strip geometry both Mathematically and Graphically.

73. Mathematical Verification

76. Graphical Verification

77. Amplitude of the separated field for different values of strip frequency for

78. Amplitude of the separated field for different values of strip frequency for Amplitude of the separated field for different values of strip frequency for

79. Amplitude of the separated field for different values of strip frequency for

80. Bashir’s Half Plane result

81. Amplitude of the diffracted field for different values of half plane frequency

82. Amplitude of the separated field for different values of Oscillating Strip frequency for Amplitude of the diffracted field for different values of Half Plane frequency

84. Question/Answer Session Session