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