waves on a string
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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