One-Dimension Wave

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Contents

The Wave Equation of Vibrating StringSolution of the Wave EquationDiscrete Time Traveling Wave

One-Dimension Wave

The Wave Equation of Vibrating String

Modeling of Vibrating String

.coscos 21 constTTT

2

2

12 sinsint

uxTT

0 l

u

PQ

T1

T2

x x+x

Modeling of Vibrating String

.coscos 21 constTTT

2

2

12 sinsint

uxTT

0 l

u

PQ

T1

T2

x x+x

2

2

1

1

2

2

cos

sin

cos

sin

t

u

T

x

T

T

T

T

2

2

tantant

u

T

x

xx

u

tan

xxx

u

tan

Modeling of Vibrating String

0 l

u

PQ

T1

T2

x x+x

2

2

1

1

2

2

cos

sin

cos

sin

t

u

T

x

T

T

T

T

2

2

tantant

u

T

x

xx

u

tan

xxx

u

tan

2

2

1

x x x

u u u

x x x T t

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

Tc2

Tc2

2

2

x

u

1D Wave Equation

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

u(x, t) = ?

Boundary Conditions:

Initial Conditions:

ttlutu allfor 0),( ,0),0(

)()0,( xfxu )(),(

0

xgt

txu

t

0 l

u

One-Dimension Wave

Solution of the

Wave Equation

Separation of Variables

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

).()(),( tGxFtxu Assume

GFt

u

2

2

GFx

u

2

2

GFcGF 2

F

F

Gc

G

2

functionof t

functionof x

k

constantwhy?

Separation of Variables

F

F

Gc

G

2

k

0 kFF 0 kFF

02 kGcG 02 kGcG

Separation of Variables

0 kFF 0 kFF

02 kGcG 02 kGcG

).()(),( tGxFtxu

Boundary Conditions:0),( ,0),0( tlutu

0)()0(),0( tGFtu

0)()(),( tGlFtlu

Case 1:

Case 2:

G(t) 0

F(0) = 0F(l ) =0

不是我們要的不是我們要的

Separation of Variables

0 kFF 0 kFF

Boundary Conditions:

F(0) = 0, F(l) =0

F(x) = ?

Three Cases: k = 0> 0

< 0

k = 0

0 kFF 0 kFF

Boundary Conditions:

F(0) = 0, F(l) =0

F(x) = ?

0F

baxxF )(

a = 0 and b = 0

不是我們要的不是我們要的

k =2 (>0)

0 kFF 0 kFF

Boundary Conditions:

F(0) = 0, F(l) =0

F(x) = ?

02 FF

xx BeAexF )(

A = 0B = 0

不是我們要的不是我們要的

k = p2 (<0)

0 kFF 0 kFF

Boundary Conditions:

F(0) = 0, F(l) =0

F(x) = ?

02 FpFpxBpxAxF sincos)(

0)0( AF

0sin)( plBlF

0B npl

l

np

k = p2 (<0)

0 kFF 0 kFF

Boundary Conditions:

F(0) = 0, F(l) =0

F(x) = ?

pxBpxAxF sincos)( l

np

xl

nxFn

sin)(Define

Any linear combinationof Fn(x) is a solution.

k = p2 (<0)

02 kGcG 02 kGcGl

np

02 GG n 02 GG n

l

cnn

tBtBtG nnnnn sincos)( *

Solution of Vibrating Strings

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

).()(),( tGxFtxu

xl

nxFn

sin)(

tBtBtG nnnnn sincos)( *

l

cnn

,2,1 ,sin)sincos()( *

nxl

ntBtBtu nnnnn

1

*

1

sin)sincos(),(),(n

nnnnn

n xl

ntBtBtxutxu

1

*

1

sin)sincos(),(),(n

nnnnn

n xl

ntBtBtxutxu

Initial Conditions

)()0,( xfxu

)(0

xgt

u

t

)(sin1

xfxl

nB

nn

01

*

0

sin)cossin(

tnnnnnnn

t

xl

ntBtB

t

u

)(sin1

* xgxl

nB

nnn

1

*

1

sin)sincos(),(),(n

nnnnn

n xl

ntBtBtxutxu

1

*

1

sin)sincos(),(),(n

nnnnn

n xl

ntBtBtxutxu

Initial Conditions

)()0,( xfxu

)(0

xgt

u

t

)(sin1

xfxl

nB

nn

)(sin1

* xgxl

nB

nnn

l

n xdxl

nxf

lB

0sin)(

2

l

n xdxl

nxf

lB

0sin)(

2

0 l

f(x)

Initial Conditions

)()0,( xfxu

)(0

xgt

u

t

)(sin1

xfxl

nB

nn

)(sin1

* xgxl

nB

nnn

l

nn xdxl

nxg

lB

0

* sin)(2

l

n xdxl

nxg

cnB

0

* sin)(2

l

n xdxl

nxg

cnB

0

* sin)(2

l

n xdxl

nxf

lB

0sin)(

2

l

n xdxl

nxf

lB

0sin)(

2

The Solution

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

1

* sin)sincos(),(n

nnnn xl

ntBtBtxu

l

n xdxl

nxg

cnB

0

* sin)(2

l

n xdxl

nxf

lB

0sin)(

2

,2,1n

Special Case: g(x)=0

2

2

22

2 1

t

u

cx

u

2

2

22

2 1

t

u

cx

u

1

* sin)sincos(),(n

nnnn xl

ntBtBtxu

l

n xdxl

nxg

cnB

0

* sin)(2

l

n xdxl

nxf

lB

0sin)(

2

,2,1n

Special Case: g(x)=0

1

sincos),(n

nn xl

ntBtxu

l

cnn

)]sin()[sin(2

1cossin )]sin()[sin(

2

1cossin

11

)(sin2

1)(sin

2

1),(

nn

nn ctx

l

nBctx

l

nBtxu

lxxuxfxl

nB

nn

0 ),0,()(sin1

0 l

f(x)

Special Case: g(x)=0)]sin()[sin(

2

1cossin )]sin()[sin(

2

1cossin

xxfxl

nB

nn ),(*sin

1

0 l

f*(x)

1

sincos),(n

nn xl

ntBtxu

l

cnn

11

)(sin2

1)(sin

2

1),(

nn

nn ctx

l

nBctx

l

nBtxu

Special Case: g(x)=0

)](*)(*[2

1),( ctxfctxftxu )](*)(*[

2

1),( ctxfctxftxu

1

sincos),(n

nn xl

ntBtxu

l

cnn

11

)(sin2

1)(sin

2

1),(

nn

nn ctx

l

nBctx

l

nBtxu

xxfxl

nB

nn ),(*sin

1

Interpretationf*(x) f*(xct)f*(x+ct)

)](*)(*[2

1),( ctxfctxftxu )](*)(*[

2

1),( ctxfctxftxu

Example)](*)(*[

2

1),( ctxfctxftxu )](*)(*[

2

1),( ctxfctxftxu

0 l 0 l

0 l 0 l

0 l 0 l

0 l 0 l

One-Dimension Wave

Discrete-Time Traveling Wave

Discrete-Time Simulation

1 2 1

1 2 1

1 12 4 2