1. Group 2 Bhadouria, Arjun Singh Glave, Theodore Dean Han, Zhe Chapter 5. Laplace Transform Chapter 19. Wave Equation

2. Wave Equation Chapter 19

3. Overview 19.1 – Introduction – Derivation – Examples 19.2 – Separation of Variables / Vibrating String – 19.2.1 – Solution by Separation of Variables – 19.2.2 – Travelling Wave Interpretation 19.3 – Separation of Variables/ Vibrating Membrane 19.4 – Solution of wave equation – 19.4.1 – d’Alembert’s solution – 19.4.2 – Solution by integral transforms

4. 19.1 - Introduction http://www.math.ubc.ca/~feldman/m267/separation.pdf

5. Derivation u(x, t) = vertical displacement of the string from the x axis at position x and time t θ(x, t) = angle between the string and a horizontal line at position x and time t T(x, t) = tension in the string at position x and time t ρ(x) = mass density of the string at position x http://www.math.ubc.ca/~feldman/m267/separation.pdf

6. Derivation http://www.math.ubc.ca/~feldman/m267/separation.pdf

7. Derivation Vertical Component of Motion Divide by Δx and taking the limit as Δx → 0. http://www.math.ubc.ca/~feldman/m267/separation.pdf

8. Derivation http://www.math.ubc.ca/~feldman/m267/separation.pdf

9. Derivation For small vibrations: Therefore, http://www.math.ubc.ca/~feldman/m267/separation.pdf

10. Derivation Substitute into (2) into (1) http://www.math.ubc.ca/~feldman/m267/separation.pdf

11. Derivation Horizontal Component of the Motion Divide by Δx and taking the limit as Δx → 0. http://www.math.ubc.ca/~feldman/m267/separation.pdf

12. Derivation http://www.math.ubc.ca/~feldman/m267/separation.pdf

13. Solution http://www.math.ubc.ca/~feldman/m267/separation.pdf

14. Separation of Variables; Vibrating String 19.2.1 - Solution by Separation of Variables

15. Scenario http://logosfoundation.org/kursus/wave.pdf

16. Procedure http://logosfoundation.org/kursus/wave.pdf

17. Step 1 – Finding Factorized Solutions http://logosfoundation.org/kursus/wave.pdf

18. Step 1 – Finding Factorized Solutions http://logosfoundation.org/kursus/wave.pdf

19. Step 1 – Finding Factorized Solutions http://logosfoundation.org/kursus/wave.pdf

20. Step 1 – Finding Factorized Solutions http://logosfoundation.org/kursus/wave.pdf

21. Step 1 – Finding Factorized Solutions http://logosfoundation.org/kursus/wave.pdf

22. Step 2 – Imposition of Boundaries http://logosfoundation.org/kursus/wave.pdf

23. Step 2 – Imposition of Boundaries http://logosfoundation.org/kursus/wave.pdf

24. Step 2 – Imposition of Boundaries http://logosfoundation.org/kursus/wave.pdf

25. Step 2 – Imposition of Boundaries http://logosfoundation.org/kursus/wave.pdf

26. Step 3 – Imposition of the Initial Condition http://logosfoundation.org/kursus/wave.pdf

27. Step 3 – Imposition of the Initial Condition The previous expression must also satisfy the initial conditions (4) and (5): (4 ’ ) (5 ’ ) http://logosfoundation.org/kursus/wave.pdf

28. Step 3 – Imposition of the Initial Condition For any (reasonably smooth) function, h(x) defined on the interval 0<x<l, has a unique representation based on its Fourier Series: (7) Which can also be written as: http://logosfoundation.org/kursus/wave.pdf

29. Step 3 – Imposition of the Initial Condition http://logosfoundation.org/kursus/wave.pdf

30. Step 3 – Imposition of the Initial Condition Therefore, (8) Where, http://logosfoundation.org/kursus/wave.pdf

31. Step 3 – Imposition of the Initial Condition http://logosfoundation.org/kursus/wave.pdf

32. Step 3 – Imposition of the Initial Condition The first 3 modes at fixed t’s. http://logosfoundation.org/kursus/wave.pdf

33. Step 3 – Imposition of the Initial Condition http://logosfoundation.org/kursus/wave.pdf

34. Example Problem:

35. Example

37. Separation of Variables; Vibrating String 19.2.2 - Travelling Wave Interpretation

38. Travelling Wave Start with the Transport Equation: where, u(t, x) – function c – non-zero constant (wave speed) x – spatial variable Initial Conditions http://www.math.umn.edu/~olver/pd_/pdw.pdf

39. Travelling Wave Let x represents the position of an object in a fixed coordinate frame. The characteristic equation: Represents the object’s position relative to an observer who is uniformly moving with velocity c. Next, replace the stationary space-time coordinates (t, x) by the moving coordinates (t, ξ). http://www.math.umn.edu/~olver/pd_/pdw.pdf

40. Travelling Wave Re-express the Transport Equation: Express the derivatives of u in terms of those of v: http://www.math.umn.edu/~olver/pd_/pdw.pdf

41. Travelling Wave Using this coordinate system allows the conversion of a wave moving with velocity c to a stationary wave. That is, http://www.math.umn.edu/~olver/pd_/pdw.pdf

42. Travelling Wave For simplicity, we assume that v(t, ξ) has an appropriate domain of definition, such that, Therefore, the transport equation must be a function of the characteristic variable only. http://www.math.umn.edu/~olver/pd_/pdw.pdf

43. The Travelling Wave Interpretation http://www.math.umn.edu/~olver/pd_/pdw.pdf

44. Travelling Wave Revisiting the transport equation, Also recall that: http://www.math.umn.edu/~olver/pd_/pdw.pdf

45. Travelling Wave At t = 0, the wave has the initial profile When c > 0, the wave translates to the right. When c < 0, the wave translates to the left. While c = 0 corresponds to a stationary wave form that remains fixed at its original location. http://www.math.umn.edu/~olver/pd_/pdw.pdf

46. Travelling Wave As it only depends on the characteristic variable, every solution to the transport equation is constant on the characteristic lines of slope c, that is: where k is an arbitrary constant. At any given time t, the value of the solution at position x only depends on its original value on the characteristic line passing through (t, x). http://www.math.umn.edu/~olver/pd_/pdw.pdf

47. Travelling Wave http://www.math.umn.edu/~olver/pd_/pdw.pdf

48. 19.3 Separation of Variables Vibrating Membranes Let us consider the motion of a stretched membrane This is the two dimensional analog of the vibrating string problem To solve this problem we have to make some assumptions

49. Physical Assumptions 1.The mass of the membrane per unit area is constant. The membrane is perfectly flexible and offers no resistance to bending 2.The membrane is stretched and then fixed along its entire boundary in the xy plane. The tension per unit length T is the same at all points and does not change 3.The deflection u(x,y,t) of the membrane during the motion is small compared to the size of the membrane

50. Vibrating Membrane Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

51. Derivation of differential equation We consider the forces acting on the membrane Tension T is force per unit length For a small portion ∆x, ∆y forces are approximately T∆x and T∆y Neglecting horizontal motion we have vertical components on right and left side as T ∆y sin β and -T ∆y sin α Hence resultant is T∆y(sin β – sin α) As angles are small sin can be replaced with tangents F res = T∆y(tan β – tan α)

52. F res = TΔy[u x (x+ Δx,y 1 )-u x (x,y 2 )] Similarly F res on other two sides is given by F res = TΔx[u y (x 1, y+ Δy)-u y (x 2,y)] Using Newtons Second Law we get Which gives us the wave equation: …..(1)

53. Vibrating Membrane: Use of double Fourier series The two-dimensional wave equation satisfies the boundary condition (2) u = 0 for all t ≥ 0 (on the boundary of membrane) And the two initial conditions (3) u(x,y,0) = f(x,y) (given initial displacement f(x,y) And (4)

54. Separation of Variables Let u(x,y,t) = F(x,y)G(t)…..(5) Using this in the wave equation we have Separating variables we get

55. This gives two equations: for the time function G(t) we have …..(6) And for the Amplitude function F(x,y) we have …..(7) which is known as the Helmholtz equation Separation of Helmholtz equation: F(x,y) = H(x)Q(y)…..(8) Substituting this into (7) gives

56. Separating variables Giving two ODE’s (9) And (10) where

57. Satisfying boundary conditions The general solution of (9) and (10) are H(x) = Acos(kx)+Bsin(kx) and Q(y) = Ccos(py)+Dsin(py) Using boundary condition we get H(0) = H(a) = Q(0) = Q(b) = 0 which in turn gives A = 0; k = mπ/a; C = 0; p = nπ/b m,n Ε integer

58. We thus obtain the solution H m (x) = sin (mπx/a) and Q n (y) = sin(nπy/b) Hence the functions (11)F mn (x) = H m (x)Q n (y) = sin(mπx/a)sin (nπy/b) Turning to time function As p 2 = ν 2 -k 2 and λ=cν we have λ = c(k 2 +p 2 ) 1/2 Hence λ mn = cπ(m 2 /a 2 +n 2 /b 2 ) 1/2 …..(12) Therefore …(13)

59. Solution of the Entire Problem: Double Fourier Series …..(14) Using (3)

60. Using Fourier analysis we get the generalized Euler formula And using (4) we obtain

61. Example Vibrations of a rectangular membrane Find the vibrations of a rectangular membrane of sides a = 4 ft and b = 2 ft if the Tension T is 12.5 lb/ft, the density is 2.5 slugs/ft 2, the initial velocity is zero and the initial displacement is

62. Solution

63. Which gives Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

64. 19.4 Vibrating String Solutions 19.4.1 d’Alembert’s Solution Solution for the wave equation can be obtained by transforming (1) by introducing independent variables

65. u becomes a function of v and z. The derivatives in (1) can be expressed as derivatives with respect to v and z. We transform the other derivative in (1) similarly to get

66. Inserting these two results in (1) we get which gives This is called the d’Alembert’s solution of the wave equation (1)

67. D’Alembert’s solution satisfying initial conditions Dividing (8) by c and integrating we get

68. Solving (9) with (7) gives Replacing x by x+ct for φ and x by x-ct for ψ we get the solution

69. 19.4.2 Solution by integral transforms Laplace Transform Semi Infinite string Find the displacement w(x,t) of an elastic string subject to: (i)The string is initially at rest on the x axis (ii)For time t>0 the left end of the string is moved by (iii)

70. Solution Wave equation: With f as given and using initial conditions Taking the Laplace transform with respect to t

71. We thus obtain Which gives

72. Using initial condition This implies A(s) = 0 because c>0 so e sx/c increases as x increases. So we have W(0,s) = B(s)=F(s) So W(x,s)=F(s)e -sx/c Using inverse Laplace we get

73. Travelling wave solution Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

74. References H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. 1st Edition., 2011, XIV, 600 p. 9 illus. 10.3 R. Baber. The Language of Mathematics: Utilizing Math in Practice. Appendix F Poromechanics III - Biot Centennial (1905-2005) http://www.math.ubc.ca/~feldman/m267/separa tion.pdf http://logosfoundation.org/kursus/wave.pdf http://www.math.umn.edu/~olver/pd_/pdw.pdf

75. Advanced engineering mathematics, 2 nd edition, M. D. Greenberg Advanced engineering mathematics, 8 th edition, E. Kreyszig Partial differential equations in Mechanics, 1 st edition, A.P.S. Selvadurai Partial differential equations, Graduate studies in mathematics, Volume 19, L. C. Evans Advanced engineering mathematics, 2 nd edition, A.C. Bajpai, L.R. Mustoe, D. Walker