Waves on a stringWaves on a string

THIS LECTURE

• Standing wavesStanding waves

• Dispersive and non-dispersive Dispersive and non-dispersive waveswaves

Travelling waves

x

Standing waves

No boundaries

With boundaries

Two ends fixed

One end fixed

Standing wavesStanding waves

Two ends fixed

txkAtx nnn sin)sin(2,

Lnkn

...3,2,1n

n

Ln

2

L

ncn

L

ncn 2

Standing wavesStanding wavesTwo ends fixed

Travelling wavesTravelling waves

tkxAtx cos,

Each section of the string vibrates with same frequency

Each section of the string vibrateswith different phase = kx

Each section of the string vibrateswith same amplitude A

No boundaries

tkxAtx cos, x

x

Standing wavesStanding waves

tfxA

txkAtx

nn

nn

2sin)2

sin(

sin)sin(,

Boundaries

2

2

Travelling wavesTravelling waves

tkxAtx cos,

Each section of the string vibrates with same frequency

Each section of the string vibrateswith different phase = kx

Each section of the string vibrateswith same amplitude A

No boundaries

tkxAtx cos, x

x

Standing wavesStanding waves

tfxA

txkAtx

nn

nn

2sin)2

sin(

sin)sin(,

Boundaries

Each section of the string vibrateswith phase 0 or out of phase by

Each section of the string vibrateswith different amplitude 2Asin(knx)

Each section of the string vibrates with same frequency

2

2

One end fixedStanding wavesStanding waves

Superposition of standing wavesSuperposition of standing waves

n

nnn txkAtx sin)sin(,

Relative intensities of the harmonics Relative intensities of the harmonics for different instrumentsfor different instruments

Playing different instrumentsPlaying different instruments

n

nnn txkAtx sin)sin(,

tx, tx,

x x

Dispersive and non-dispersive wavesNon-dispersive waveNon-dispersive wave: it does not change shape

t = 0

t > 0

Dispersive waveDispersive wave: it changes shape

t = 0

t > 0

x

Two velocities to describe the wave

Group velocity, Vg

Velocity at which the envelopeof 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

Group velocitydk

d

kkkVg

~

21

21

Phase velocitykkk

Vp

21

21

Group and phase velocity

dk

kVd

dk

dV p

g

)(

Relation between Vg and Vp

If Vp Vg dispersive wavedispersive wave0dk

dVp

If Vp = Vg non-dispersive wavedispersive wave0dk

dVp

dk

dVkV

dk

kVd

dk

dV p

pp

g )(

N

iiii txkAtx

1

cos,

Superposition of sinusoidal waves

Sinusoidal waves

1, k1

2, k2

3, k3

Superposition Wave-packet

Wave propagates with speed c

maintaining its shape

t = 0

t > 0

Wavechanges its shape

t = 0

t > 0

Sinusoidal waves have the same speed

1/ k1= c

2/ k2= c

3/ k3= c

Non-dispersive wave

0dk

dVpck

Vp

0dk

dVpconstk

Vp

Sinusoidal waves have different speed

1/ k1= c1

2/ k2= c2

3/ k3= c3

Dispersive wave

Ideal stringIdeal string

T

kc

Real string Real string (e.g. a piano string)(e.g. a piano string)

2kT

kc

Vp=/k=c does not depend on k

Vp=/k=c depends on k

c= slope

Dispersion relation

k

k

c1

c2

Non-dispersive wave

Dispersive wave

Waves on a stringWaves on a string

kT

ck

2kT

k

Ideal stringIdeal string

Tk

Dispersion relation

k

k

Real stringReal string

2kT

k

Group velocity

T

dk

dVg

Phase velocity

T

kVp

2

22

kT

kT

dk

dVg

2kT

kVp

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

222 kcp

Group velocity

kVg

Phase velocity

kVp

tkxtkxA 21

21 coscos2The resulting wave is given by

2121

2121 , kkk 2121 , kkk

x

txkAtxkA 222111 coscos

Superposition of sinusoidal waves

1

11 k

c

2

22 k

c

k

k