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THIS LECTURE. Waves on a string. Standing waves. Dispersive and non-dispersive waves. Travelling waves. No boundaries. x. With boundaries. Standing waves. Two ends fixed. One end fixed. Standing waves. Two ends fixed. Standing waves. Two ends fixed. x. x. Travelling waves.

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PowerPoint Slideshow about ' Waves on a string' - elton-sheppard


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Presentation Transcript
slide1

THIS LECTURE

Waves on a string

  • Standing waves
  • Dispersive and non-dispersive waves
slide2

Travelling waves

No boundaries

x

With boundaries

Standing waves

Two ends fixed

One end fixed

slide3

Standing waves

Two ends fixed

slide4

Standing waves

Two ends fixed

slide5

x

x

Travelling waves

Standing waves

Boundaries

No boundaries

2

2

Each section of the string vibrates with same frequency w

Each section of the string vibrates

with same amplitude A

Each section of the string vibrates

with different phase f = kx

slide6

x

x

Travelling waves

Standing waves

Boundaries

No boundaries

2

2

Each section of the string vibrates with same frequency w

Each section of the string vibrates with same frequency w

Each section of the string vibrates

with same amplitude A

Each section of the string vibrates

with different amplitude 2Asin(knx)

Each section of the string vibrates

with different phase f = kx

Each section of the string vibrates

with phase 0 or out of phase by p

slide7

Standing waves

One end fixed

slide11

Dispersive wave: it changes shape

t = 0

t > 0

Dispersive and non-dispersive waves

Non-dispersive wave: it does not change shape

t = 0

t > 0

slide12

Two velocities to describe the wave

Group velocity, Vg

Velocity at which the envelope

of wave peaks moves

Phase velocity, Vp

Velocity at which successive

peaks move

For non-dispersive waves Vg =Vp

For dispersive waves VgVp

http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-further-dispersive.htm

slide13

Group velocity

If VpVg  dispersive wave

If Vp=Vg  non-dispersive wave

Group and phase velocity

Phase velocity

Relation between Vg and Vp

slide14

Superposition

Wave-packet

Superposition of sinusoidal waves

Sinusoidal waves

w1, k1

w2, k2

w3, k3

slide15

Dispersive wave

Sinusoidal waves have different speed

w1/ k1= c1

w2/ k2= c2

w3/ k3= c3

Non-dispersive wave

Sinusoidal waves have the same speed

Wave propagates

with speed c

maintaining its shape

w1/ k1= c

w2/ k2= c

t = 0

w3/ k3= c

t > 0

Wave

changes its shape

t = 0

t > 0

slide16

Dispersion relation

c= slope

Real string (e.g. a piano string)

c2

c1

Waves on a string

Ideal string

Vp=w/k=c does not

depend on k

Non-dispersive wave

Vp=w/k=c

depends on k

Dispersive wave

slide17

Dispersion relation

Group velocity

Phase velocity

Real string

Ideal string

slide18

Problem

Determine phase and group velocity for waves whose dispersion relation is described by :

slide19

The resulting wave is given by

Group velocity

2p/Dk

Phase velocity

2p/k

Superposition of sinusoidal waves

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