1. One-Dimension Wave 虞台文

2. Contents • The Wave Equation of Vibrating String • Solution of the Wave Equation • Discrete Time Traveling Wave

3. One-Dimension Wave The Wave Equation of Vibrating String

4. u T2 Q  P  T1 0 l x x+x Modeling of Vibrating String

5. u T2 Q  P  T1 0 l x x+x Modeling of Vibrating String

6. u T2 Q  P  T1 0 l x x+x Modeling of Vibrating String

7. u 0 l 1D Wave Equation u(x, t) = ? Boundary Conditions: Initial Conditions:

8. One-Dimension Wave Solution of the Wave Equation

9. Separation of Variables Assume function of t function of x constant why?

10. Separation of Variables

11. Separation of Variables Boundary Conditions:  G(t)  0 Case 1: 不是我們要的 F(0) = 0 F(l ) =0 Case 2:

12. Separation of Variables F(x) = ? Boundary Conditions: F(0) = 0, F(l) =0 > 0 k = 0 Three Cases: < 0

13. F(x) = ? Boundary Conditions: F(0) = 0, F(l) =0 k = 0 a = 0 and b = 0 不是我們要的

14. F(x) = ? Boundary Conditions: F(0) = 0, F(l) =0 k =2 (>0) A = 0 B = 0 不是我們要的

15. F(x) = ? Boundary Conditions: F(0) = 0, F(l) =0 k = p2 (<0)

16. F(x) = ? Boundary Conditions: F(0) = 0, F(l) =0 k = p2 (<0) Any linear combination of Fn(x) is a solution. Define

17. k = p2 (<0)

18. Solution of Vibrating Strings

19. Initial Conditions

20. f(x) 0 l Initial Conditions

21. Initial Conditions

22. The Solution

23. Special Case: g(x)=0 

24. f(x) 0 l Special Case: g(x)=0

25. f*(x) 0 l Special Case: g(x)=0

26. Special Case: g(x)=0

27. f*(x) f*(xct) f*(x+ct) Interpretation

28. l l l l l l l 0 0 0 0 0 0 0 l 0 Example

29. One-Dimension Wave Discrete-Time Traveling Wave

30. 1 2 1 2 4 2 1 1 1 2 1 Discrete-Time Simulation