1. This lecture A wide variety of waves
Dispersive and non-dispersive waves
A wide variety of waves
Waves on a string
Acoustic waves
Electromagnetic waves
Water waves

2. Dispersive and non-dispersive waves

3. 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

5. Superposition of sinusoidal waves
The resulting wave is given by
Group velocity
Phase velocity

6. Group and phase velocity
Relation between Vg and Vp
If  Vp = Vg  non-dispersive wave
If  Vp  Vg  dispersive wave

7. Waves on a string Ideal string Real string Dispersion relation
Group velocity
Ideal string
Dispersion relation
Phase velocity
Non dispersive waves
Real string
Dispersive waves

8. Acoustic waves What is the disturbance?
Y: pressure variation in the medium
Transverse, Longitudinal or T+L Wave ?
Longitudinal wave
What is the speed of the wave ?
c = (1/kr)1/2
k = compressibility: elastic property
r = density of mass: inertial property

9. Acoustic waves Acoustic waves are non-dispersive waves Vg=Vp
Dispersion relation
Slope=c
c = (1/kr)1/2
k = compressibility
r = density of mass
Group velocity
Acoustic waves are
non-dispersive waves
Vg=Vp
Phase velocity

10. Electromagnetic waves
What is the disturbance?
Y: electric or magnetic field
Transverse, Longitudinal or T+L Wave ?
Transverse wave
What is the speed of the wave ?
c = (1/me)1/2
e = electrical permittivity
m = magnetic permeability

11. Electromagnetic waves in vacuum
Dispersion relation
Group velocity
Electromagnetic waves
in vacuum are non-dispersive
Vg= Vp
Phase velocity

12. Electromagnetic waves in the ionosphere
An altitude of 100 km is usually thought of as the boundary where our Earth ends and the emptiness of outer space begins. In fact the atmosphere extends much farther than that, into a region called the ‘ionosphere’. A vanishingly thin scattering of oxygen atoms reaches to a few hundred kilometres, blending into an even thinner scattering of hydrogen atoms yet further beyond. Exposed to the full ultraviolet glare of the Sun, the electrons are stripped away from these atmospheric gases to produce plasma – an electrically conductive 'soup' of positively and negatively charged particles (ions and free-flying electrons). These insubstantial layers of gas and plasma are the interface between the Sun’s electromagnetic energies and our planet’s environment. The beguiling ‘Northern Lights’ (and their Southern counterparts) are the beautiful side effects of the electromagnetic link to our Sun.
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.

13. Electromagnetic waves in the ionosphere
Plasma: System of positive ions and free- electrons + + + + + + + + + + + + + + + + + + + + + + + +
Electromagnetic wave
Electromagnetic waves with w < wp are reflected by the plasma

14. Electromagnetic waves in the ionosphere
Plasma: System of positive ions and free- electrons + + + + + + + + + + + + + + + + + + + + + + + +
Electromagnetic
wave
Electromagnetic waves with w > wp propagate through the plasma

15. Electromagnetic waves in the ionosphere
Plasma: System of positive ions and free- electrons + + + + + + + + + + + + + + + + + + + + + + + +
Dispersion relation for waves with frequency w > wp
Plot the dispersion relation, i.e. w versus k
Calculate the group and phase velocity
State with reasons if these waves are dispersive or non-dispersive

16. Electromagnetic waves
Electromagnetic waves in the ionosphere
Group velocity
Electromagnetic waves
are dispersive waves
Vg Vp
Phase velocity

17. Water Waves Disturbance y: displacement of water from equilibrium
Direction
of propagation
Disturbance y: displacement of water from equilibrium
Water flows in a circle
 T+L waves
g = acceleration of gravity
h = equilibrium depth of water
Dispersion relation

18. Water Waves l << h hk >> 1 Deep water waves
Wind-generated wave
l << h
Deep water waves
hk >> 1
Typical speeds
Vg  Vp  Deep water waves are dispersive waves

19. Water Waves l >> h hk << 1 Shallow water waves
Direction
of propagation
Tsunami
Typical speeds of Tsunami
Vg = Vp  Shallow water waves are non-dispersive waves

20. Tsunami High-speed wave (as fast as a jet plane!) Long wavelengths
l ~ km

21. Tsunami The height of the wave increases as it approaches the coast
The wave looses speed as it approaches land.
Since the energy is conserved and E ~ A2Vg
(A is the amplitude)
A increases as the wave approaches land

22. Plot the dependence of Vg and Vp versus k for different types of waves
Waves on a non-ideal string
Electr. waves in the ionosphere
Water waves

23. Dispersive and non-dispersive waves
Non-dispersive wave: it does not change shape
t = 0
t > 0
Dispersive wave: it changes shape
t = 0
t > 0