One-Dimension Wave 虞台文

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

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

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

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

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

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

Solution of the Wave Equation One-Dimension Wave Solution of the Wave Equation

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

Separation of Variables

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

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

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

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

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

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

k = p2 (<0)

Solution of Vibrating Strings

Initial Conditions

Initial Conditions l f(x)

Initial Conditions

The Solution

Special Case: g(x)=0 

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

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

Special Case: g(x)=0

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

Example l l l l

Discrete-Time Traveling Wave One-Dimension Wave Discrete-Time Traveling Wave

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