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One-Dimension Wave

One-Dimension Wave. 虞台文. Contents. The Wave Equation of Vibrating String Solution of the Wave Equation Discrete Time Traveling Wave. One-Dimension Wave. The Wave Equation of Vibrating String. u. T 2. Q. . P. . T 1. 0. l. x. x +  x. Modeling of Vibrating String. u. T 2. Q.

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One-Dimension Wave

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  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

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