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A wide variety of wavesA wide variety of wavesWaves on a stringAcoustic wavesElectromagnetic wavesWater waves
Dispersive and non-dispersive wavesDispersive and non-dispersive waves
This lectureThis lecture
Dispersive and non-dispersive waves
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 velocity
kVg
Phase velocity
kVp
kxttkxA 21
21 coscos2
The 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
Group and phase velocity
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 )(
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
Waves on a stringWaves on a string
Non dispersive waves
Dispersive waves
Acoustic waves
What is the disturbance? : pressure variation in the medium
Transverse, Longitudinal or T+L Wave ?Longitudinal wave
What is the speed of the wave ? c = (1/)1/2
= compressibility: elastic property = density of mass: inertial property
Acoustic wavesAcoustic waves
Group velocity
cdk
dVg
Phase velocity
ck
Vp
ck
c = (1/)1/2
= compressibility = density of mass
Acoustic waves are non-dispersive
wavesVg=Vp
Dispersion relation
k
Slope=c
Electromagnetic waves
What is the disturbance? : electric or magnetic field
Transverse, Longitudinal or T+L Wave ?Transverse wave
What is the speed of the wave ? c = (1/)1/2
= electrical permittivity = magnetic permeability
Electromagnetic waves in vacuumElectromagnetic waves in vacuum
ck
lightofspeedc
cdk
dVg
Group velocity
ck
Vp
Phase velocityElectromagnetic waves
in vacuum are non-dispersive Vg= Vp
Dispersion relation
k
The ionosphere is the uppermost part of the atmosphere, distinguished because it is ionized by solar radiation. It has practical importance because, among other functions, it influences the propagation of electromagnetic waves, particularly of radio waves.
Electromagnetic waves in the ionosphereElectromagnetic waves in the ionosphere
Electromagnetic waves in the ionosphereElectromagnetic waves in the ionosphere
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Plasma: System of positive ions and free- electrons
frequencynoscillatioplasmap
Electromagnetic waves with < p are reflected by the plasma
Electromagnetic wave
Electromagnetic waves in the ionosphereElectromagnetic waves in the ionosphere
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Plasma: System of positive ions and free- electrons
frequencynoscillatioplasmap
Electromagnetic waves with > p propagate through the plasma
Electromagneticwave
Electromagnetic waves in the ionosphereElectromagnetic waves in the ionosphere
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Plasma: System of positive ions and free- electrons
frequencynoscillatioplasmap
222 kcp lightofspeedc
1. Plot the dispersion relation, i.e. versus k2. Calculate the group and phase velocity3. State with reasons if these waves are dispersive or non-dispersive
Dispersion relation for waves with frequency > p
Electromagnetic waves in the ionosphereElectromagnetic waves in the ionosphere
222 kcp
frequencynoscillatioplasmap lightofspeedc
k
22
2
2
ck
c
dk
dV
p
g
Group velocity
22
2
ckk
V pp
Phase velocity
Electromagnetic waves are dispersive waves
Vg Vp
Water Waves
Water flows in a circle T+L wavesDirection
of propagation
Disturbance : displacement of water from equilibrium
k
g = acceleration of gravityh = equilibrium depth of water
Dispersion relation
)1(
)1(2
2
kh
kh
e
egk
<< h
Deep water waves
hk >> 1
gke
egk
kh
kh
~)1(
)1(2
2
22
1
2
1 g
k
g
dk
dVg
2
g
k
g
kVp
Vg Vp Deep water waves are dispersive waves
Typical speeds
m
msg
msg
k
gVg
1
8.9
1~22
1
2
1
2
1
Water WavesWater Waves Wind-generated wave
Water WavesWater Waves
>> h
Shallow water waves
hk << 1
kghe
egk
kh
kh
~)1(
)1(2
2
ghdk
dVg
ghk
Vp
Vg = Vp Shallow water waves are non-dispersive waves
Direction of propagation
Tsunami
Typical speeds of Tsunami
kmh
msg
msghVg
1
8.9
100~2
1
TsunamiHigh-speed waveHigh-speed wave(as fast as a jet plane!) (as fast as a jet plane!)
kmh
msg
msghVg
4
8.9
200~2
1
Long wavelengthsLong wavelengths ~ 10-100 km
TsunamiThe height of the wave increases as it approaches the coast
kmh
msghVg
1
100~ 1
mh
msghVg
1
3~ 1
The wave looses speed as it approaches land.
A ~ 5 m A ~ 30 m
Since the energy is conserved and E ~ A2Vg (A is the amplitude)
A increases as the wave approaches land
Plot the dependence of Vg and Vp versus k for different types of waves
)1(
)1(2
2
kh
kh
e
egk
2kT
k
222 kcp
Waves on a non-ideal string
Electr. waves in the ionosphere
Water waves
Dispersive and non-dispersive waves
Non-dispersive waveNon-dispersive wave: it does not change shape
t = 0
t > 0
Dispersive waveDispersive wave: it changes shape
t = 0
t > 0
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