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

4. Modeling of Vibrating StringP Q T1 T2   x
x+x l u

5. Modeling of Vibrating StringP Q T1 T2   x
x+x l u

6. Modeling of Vibrating StringP Q T1 T2   x
x+x l u

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

8. Solution of the Wave Equation
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: 
Case 1:
G(t)  0
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
Three Cases:
= 0
< 0

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

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

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

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

17. k = p2 (<0)

18. Solution of Vibrating Strings

19. Initial Conditions

20. Initial Conditions l
f(x)

21. Initial Conditions

22. The Solution

23. Special Case: g(x)=0 

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

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

26. Special Case: g(x)=0

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

28. Example l l l l

29. Discrete-Time Traveling Wave
One-Dimension Wave
Discrete-Time Traveling Wave

30. Discrete-Time Simulation1 2 1 2 4 1 2