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Transverse waves

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Transverse waves on a string

T

x

y

x = y = 0

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Horizontal forces:

Vertical forces:

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Solution of the wave equation

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Boundary conditions

l

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Missing modes by arbitrary plucking

F

X

All those modes having a node at X would be missing.

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F

Arbitrary disturbance may be expressed by a Fourier series

with appropriate strengths of different modes.

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Reflection and transmission of waves

Y

X=0

TT

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Boundary conditions

X=0

Y

XNot possible

T

T

X=0

Y

XPossible

TT

X=0

Y

X

Not possible

TT

X=0

Y

X

Possible

T T

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1

1 1

i r

x xY f t f t

c c

2

2

t

xY f t

c

0 0( ) i r ti Y Y f t f t f t Boundary conditions at X=0 for all time:

120 0

( )i r t

cY Yi i f t f t f t

x x c

12

( )i r t

cf t f t f t K t

c

12

i r t

cf t f t f t

c As K=0,

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Reflection coefficient

At X = 0

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Transmission coefficient

At X = 0

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Stokes relations

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Note

But

At X=0 for all time

with and

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Setting

We also have

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Special cases

Case 1:

Change in sign of reflected pulse

Abrupt phase change of on reflection

1 2 1 2 12 0c c r

Case 2:1 2 1 2 12

0c c r

No change in sign of reflected pulse

No abrupt phase change on reflection

At the knot or joint (X = 0)

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Case 3: for both the cases

No phase change on transmission

Case 4: At a fixed knot/joint

Total reflection with an inversion of shape

Case 5: At a free end with

Total reflection without inversion of shape

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Case 6: Impedance matching at the joint

Total transmission without reflection