1. RANDOM AND COHERENT SOURCES
If you have N harmonic waves of identical frequency, their sum is a harmonic wave of the same frequency, of
amplitude:
… (9.1)
phase:
… (9.2)
Can distinguish two important cases :
N randomly phased sources of equal amplitude and frequency (N being a large number)
N coherent sources of the same type (they are in phase)

2. N randomly phased sources of equal amplitude and frequency (N being a large number)
as phases are random, phase differences (i  j) are also random
-then as N increases, because cos (i  j)
take fractions which are equally divided between negative and positive values ranging from 1 to +1
therefore, resultant amplitude is given by:
… (9.3a)
(as the amplitudes of N sources are equal) 
… (9.3b)
Conclusion: The resultant irradiance of N identical but randomly phased sources is the sum of the individual irradiances

3.   N coherent sources of the same type (they are in phase)
all i are equal, so 
… (9.4a) 
… (9.4b)
Conclusion: The resultant irradiance of N identical coherent sources, radiating in phase with each other, is N2 times the irradiance of the individual sources.
Note: N need not be a large number

4. STANDING WAVES
Superposition of two equal waves in the same medium but in opposite direction of propagation results in standing waves
Assuming an ideal situation in which no energy is lost on reflection and no energy is absorbed by the transmitting medium
- forward wave:
… (9.5a)
… (9.5b)
- reverse wave:
Standing wave created when a forward wave and its reflection exist along the same medium. (Though not shown, a phase shift usually occurs on reflection)

5. Resultant wave (in the same medium), by principle of superposition, is
… (9.6)
Defining:
and
and using trigonometric identity of:
Eqn.(9.6) simplifies to:
… (9.7)
Space-dependent amplitude
 a standing wave equation
Resultant displacement of standing wave at various instants. (Solid line  max displacement; displacement at nodes (N) is always zero)

6. At any point x along the medium, oscillations are given by:
where
values of x whereby A(x) = 0 occur when or or
… (9.8)
at these locations,
for all t;
-they are called NODES, and are separated by ½ wavelength
at different times, the standing wave appears as a sine wave of different amplitude but always passes through zero at fixed nodal points

7. When ,
has maximum value at all points; or
when
therefore, outermost envelope of standing wave (solid line in Fig. (b)) happens at times given by:
, T  period
Standing wave is everywhere zero (horizontal line on x-axis of Fig. (b)) when
or at periodic times of:

8. Standing waves transmit NO energy – all energy used to sustain the oscillations between nodes
In practice, e.g. for mirrors which are not perfect reflectors, the transmitting medium absorbs some of the wave energy, and wave amplitude attenuates with x
unless the source continue to replace the lost energy, amplitude also decreases with time
thus, two waves do not cancel completely at nodes and do not add to maximum of 2E0 at antinodes (halfway between nodes)
resultant wave will have a traveling wave component that carries energy to the mirror and back
If a relative phase between the waves of Eqns. 9.5a & 9.5b exists (as expected on reflection), the cos and sin factors of Eqn. (9.7) will have a phase angle component. Nodes are then displaced though their separation remains ½. Times at which wave is everywhere zero or everywhere maximum are also different. But principal features of standing wave are unaffected.

9. A Standing Wave
The points where the amplitude is always zero are called “nodes.” The points where the amplitude oscillates maximally are called “anti-nodes.”

10. A Standing Wave Again…

11. A Standing Wave: Experiment
3.9 GHz microwaves
Mirror
Input beam
The same effect occurs in lasers.
Note the node at the reflector at left (there’s a phase shift on reflection).

12. Interfering spherical waves also yield a standing wave
Antinodes

13. Two Point Sources
Different separations. Note the different node patterns.

14. PHASE AND GROUP VELOCITIES
Superposition also applies to waves of same or comparable amplitude but different frequency (implying different wavelength and velocity).
Two waves of different frequency and wave number are given by:
… (9.9a)
… (9.9b)
Superposition of the waves results in:
using trigonometric identity:
and defining:
and

15. we have:
… (9.10)
let ,
… (9.11a)
… (9.11b)
and
Eqn. (9.10) simplifies to:
… (9.12)
 a product of two cosine waves:
1st cosine function: p = average of frequencies of component waves;
and kp = average of propagation constants of component waves
2nd cosine function: g  difference in frequencies of component waves; and kg  difference in propagation constants of component waves

16. Separate plots of cosine factors of eqn. (9
Separate plots of cosine factors of eqn. (9.12) at x = x0 with p >> g
may consider cosine function that varies slowly as a fraction in the range 1 to +1 that limits the displacement of the fast varying function
resulting in the low-frequency wave serving as an envelope modulating the high-frequency wave, exhibiting beats.

17. beat frequency is 2X frequency of modulating cosine wave, i.e.
envelope
Modulated wave representing eqn. (9.12) at x = x0 with p >> g
as the radiant flux density  (wave displacement)2, energy delivered by traveling sequence of pulses is pulsating at beat frequency, b.
beat frequency is 2X frequency of modulating cosine wave, i.e.
… (9.13)

18. When two waves of different frequency interfere, they produce "beats."
Indiv-
idual
waves
Sum
Envel-
ope
Irrad-
iance:

19. When two light waves of different frequency interfere, they also produce beats.

20. When light propagates through a refractive medium, dispersion occurs because light components of different wavelengths travel with different speeds
- in practice, monochromatic light has a narrow spread of wavelengths
- any two wavelength components of a light beam, moving through a dispersive medium can then be represented by:
and
producing a resultant wave like that shown as the enveloped modulated wave
Velocity of the higher-frequency wave and lower-frequency envelope can be deduced through the relation:

21. velocity of higher-frequency wave  phase velocity:
… (9.14)
employing the approximation
and
for close-by frequency and wavelength components
velocity of the envelope  group velocity:
… (9.15)
differences between frequencies and propagation constants are assumed to be small.

22. Comparing vp with vg :
- if vp > vg, high-frequency waves appear to move to right relative to envelope
- if vp < vg, relative motion is reversed
- if vp = vg, high-frequency waves and envelope move together at the same rate, with no relative motion
Relation between vp and vg derived as follows:
substituting 9.14 into 9.15; 
… (9.16)
when velocity of a wave is independent on wavelength, i.e. in non-dispersive medium such as in vacuum,
and vg = vp = c

23. in dispersive media,
n is the refractive index which is a function of  or k, then n = n(k) and
Incorporating Eqn.9.16, another relation between phase and group velocities is obtained:
… (9.17)
Using
and
, it is rewritten as:
… (9.18)

24. if harmonic waves with a wide range of frequencies constitute the waveform
narrow pulses separated by relatively longer intervals of low field amplitude.
Snapshot of a waveform that is the sum of 10 equal-amplitude harmonic waves with frequency spacing about 1/50 of the average frequency of the constituent harmonic waves.

25. The group velocity is less than the phase velocity in non-absorbing regions.
vg = c0 / (n + w dn/dw)
In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies.

26. in regions of normal dispersion, and
dn/d < 0

27. These results holds in general for > 2 waves with a narrow range of frequencies – may be characterized both by vp (average velocity of individual waves) and by vg (velocity of modulating wave form)
As vg determines speed with which energy is transmitted, it is the directly measurable speed of the waves
- in Amplitude Modulation (AM) of radiowaves, carrier wave (vp) modulated to contain information (vg = signal velocity)
- when light pulses made up of a number of harmonic waves (extending over range of frequencies) are transmitted through dispersive medium, vg = velocity of pulses ( velocity of individual harmonic waves)
- in wave mechanics of electron, it is represented by localized wave packet (constituting a number of harmonic waves with a range of wavelengths); measured velocity of electron = v(wave packet) = vg

28. The group velocity can exceed c0 when dispersion is anomalous.
vg = c0 / (n + w dn/dw)
dn/dw is negative in regions of anomalous dispersion, that is, near a
resonance. So vg can exceed c0 for these frequencies!
One problem is that absorption is strong in these regions. Also, dn/dw is only steep when the resonance is narrow, so only a narrow range of
frequencies has vg > c0. Frequencies outside this range have vg < c0.
Pulses of light (which are broadband) therefore break up into a mess.

29. Group-velocity dispersion is undesirable in telecommunications systems.
Train of input telecom pulses
Dispersion causes short pulses to spread in time and to become long pulses.
Many km of fiber
Dispersion dictates the wavelengths at which telecom systems must operate and requires fiber to be very carefully designed to compensate for dispersion.
Train of output telecom pulses