Group 2 Bhadouria , Arjun Singh Glave , Theodore Dean Han, Zhe

1. Group 2Bhadouria, Arjun SinghGlave, Theodore DeanHan, 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 • Wave Equation • Uses: • Electromagnetic Waves • Pulsatile blood flow • Acoustic Waves in Solids • Vibrating Strings • Vibrating Membranes 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 • Forces: • Tension pulling to the right, which has a magnitude T(x+Δx, t) and acts at an angle θ(x+Δx, t) above horizontal • Tension pulling to the left, which has magnitude T(x, t) and acts at an angle θ(x, t) below horizontal • The net magnitude of the external forces acting vertically F(x, t)Δx • Mass Distribution: 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 • For small vibrations: and Therefore, http://www.math.ubc.ca/~feldman/m267/separation.pdf

13. Solution For a constant string density ρ, independent of x The string tension T(t) is a constant, and No external forces, F 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 u(x, t) = vertical displacement of a string from the x axis at position x and time t l = string length Recall: (1) Boundry Conditions: u(0, t) = 0 for all t > 0 (2) u(l, t) = 0 for all t > 0 (3) Initial Conditions u(x, 0) = f(x) for all 0 < x <l (4) ut(x, 0) = g(x) for all 0 < x <l (5) http://logosfoundation.org/kursus/wave.pdf

16. Procedure There are three steps to consider in order to solve this problem: Step 1: • Find all solutions of (1) that are of the special form for some function that depends on x but not t and some function that depends on t but not x. Step 2: • We impose the boundary conditions (2) and (3). Step 3: • We impose the initial conditions (4) and (5). http://logosfoundation.org/kursus/wave.pdf

17. Step 1 – Finding Factorized Solutions Let: Since the left hand side is independent of t the right hand side must also be independent of t. The same goes for the right hand side being independent of x. Therefore, both sides must be constant (σ). http://logosfoundation.org/kursus/wave.pdf

18. Step 1 – Finding Factorized Solutions (6) http://logosfoundation.org/kursus/wave.pdf

19. Step 1 – Finding Factorized Solutions Solve the differential equations in (6) http://logosfoundation.org/kursus/wave.pdf

20. Step 1 – Finding Factorized Solutions If , there are two independent solutions for (6) If , http://logosfoundation.org/kursus/wave.pdf

21. Step 1 – Finding Factorized Solutions Solutions to the Wave Equation For arbitrary and arbitrary For arbitrary http://logosfoundation.org/kursus/wave.pdf

22. Step 2 – Imposition of Boundaries For Thus, this solution is discarded. http://logosfoundation.org/kursus/wave.pdf

23. Step 2 – Imposition of Boundaries For , when When Therefore, http://logosfoundation.org/kursus/wave.pdf

24. Step 2 – Imposition of Boundaries Since , in order to satisfy An integer k must be introduced such that: Therefore, http://logosfoundation.org/kursus/wave.pdf

25. Step 2 – Imposition of Boundaries Where, and are allowed to be any complex numbers and are allowed to be any complex numbers http://logosfoundation.org/kursus/wave.pdf

26. Step 3 – Imposition of the Initial Condition From the preceding: which obeys the wave equation (1) and the boundary conditions (2) and (3), for any choice of and 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 For the coefficients. We can make (7) match (4′) by choosing and . Thus . Similarly, we can make (7) match (5′) by choosing and Thus 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 The sum (8) can be very complicated, each term, called a “mode”, is quite simple. For each fixed t, the mode is just a constant times . As x runs from 0 to l, the argument of runs from 0 to , which is k half–periods of sin. Here are graphs, at fixed t, of the first three modes, called the fundamental tone, the first harmonic and the second harmonic. 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 For each fixed x, the mode is just a constant times plus a constant times . As t increases by one second, the argument, , of both and increases by , which is cycles (i.e. periods). So the fundamental oscillates at cps, the first harmonic oscillates at 2cps, the second harmonic oscillates at 3cps and so on. If the string has density ρ and tension T , then we have seen that . So to increase the frequency of oscillation of a string you increase the tension and/or decrease the density and/or shorten the string. http://logosfoundation.org/kursus/wave.pdf

34. Example Problem:

35. Example Let l = 1, therefore, It is very inefficient to use the integral formulae to evaluate and . It is easier to observe directly, just by matching coefficients.

36. 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 VariablesVibrating 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 • The mass of the membrane per unit area is constant. The membrane is perfectly flexible and offers no resistance to bending • 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 • 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, 8th Edition, Erwin Kreyszig