1. WAVE EQUATIONS 1-D Wave Eqn  A traveling wave can be expressed as:
+ to the left;  to the right ; where f can be any function
 To derive partial differential equation that represents wave equation:
- starting from y = f (x’) where x’ = x  vt
- from which
and
- using chain rule, the space derivative is:
- its 2nd derivative is:

2. similarly the time derivative and its 2nd derivative are:
Combining the two derivatives yields the 1-D differential wave equation
any wave of form y = f (x  vt) must satisfy this wave equation irrespective of the physical nature of the wave itself.

3. Harmonic Waves: Involve cosine or sine function: or
their difference is a translation of /2 radians only, i.e., sin x = cos (x – ½); A and k are constants; A  amplitude, k = 2/  propagation constant
Linear combination of sin and cos terms can represent any actual periodic wave form (square, triangular, etc.) – Fourier series A x y 
t = constant t T
x = constant
A = amplitude
 = wavelength
T = period

4. Related descriptions of wave parameters:
Symbol / relation
Units
Period T s
Wavelength  m
Propagation constant
k = 2/ or k = 2/vT
radian/m
Wave velocity
v = f
m/s
Frequency
f = 1/T Hz
Angular frequency
 = 2f
radian/s
Wave number
 = 1/
m1
Phase (argument of the sine or cosine function)
 = k(xvt)
radian

5. Common expressions for harmonic waves:; OR ;
At constant phase,  = constant; ,
Thus,
which confirms that v = wave velocity, negative when
and positive when

6. E.g., a travelling wave of: will have: (when compared to
The wave may have an arbitrary initial displacement of phase 0, and the wave equation becomes:
E.g., a travelling wave of:
will have: (when compared to )
amplitude = 0.35 m
propagation constant k = 3 rad/m
angular frequency  = 10 rad/s
wavelength  = 2/k = 2/3 m
frequency = f = /2 = 5 Hz
velocity v = f = 10/3 m/s in positive x-direction indicated by negative sign before t in phase of wave
initial phase angle 0 = /4
Displacement at x = 10 cm and t = 0; m

7. Harmonic waves expressed as complex numbers:
(more convenient to use because mathematics involving exponential functions is usually simpler to handle)
Complex number:
and
where Im Re a b 
in polar coordinates:
its magnitude (absolute value or modulus) is
where
since
and
using Euler’s formula given by:
we can write,
where
Complex conjugate: or
Useful theorem:

8. Frequently used values of ei are:0 1
90 i
180
 1
270
 i
Expression for Harmonic waves:
where
and

9. Plane waves (3-D):
Any arbitrary direction involves 3 spatial coordinates (x, y, z)
Thus, wave displacement is represented by function  such as:
 wave propagating in +x-direction
If time is fixed (e.g. t = 0), spatial extent of wave is:
If x is constant, phase
= constant  surfaces of constant phase are family of planes shown below (i.e. they make up the wavefronts of the disturbance)

10. Plane waves along x-axis.
Surfaces of constant phase are planes x = constant.
Waves penetrate planes x = a, x = b, and x = c at the points as shown.
Wave displacement given by  is the same for all points of a wavefront.

11. to be a vector quantity (its magnitude
In general,
Generalization of plane wave to an arbitrary direction
Wave direction given by vector k along x-axis
Wave direction given by vector k along an arbirary direction
wave disturbance at arbitrary point in space, defined by vector r is the same for point x along the x-axis where x = rcos
thus,
Taking
to be a vector quantity (its magnitude
), pointing in direction of propagation, we have

12. Harmonic wave becomes:
Generally,
where (kx, ky, kz) are components of propagation direction and (x, y, z) are components of point in space where displacement  is determined
The 3-D partial differential equation satisfied by the 3-D wave equations is of the form: 
where 2  Laplacian operator

13. Spherical wave:
Harmonic disturbances emitted from a point source in a homogeneous medium travel at equal rates in all directions.
Their wavefronts form spherical surfaces centered at the source.
Harmonic wave equation:
amplitude decreases as it propagates further from the source
amplitude A corresponds to when wave at r =1

14. Cylindrical Waves
ρ is the perpendicular distance from the line of symmetry to a point on the waveform.
i.e. if the z-axis is the line of symmetry, then
Note:
Wave of this form are not exact solutions to the wave equation and so do not exactly represent physical waves but rather are approximately valid for large ρ.
Still they are useful forms that approximate the wave that emerges from a slit illuminated by a plane wave.

15. Gaussian Beams
another important family of (single-frequency) approximate solutions to the wave equation -- Hermite-Gaussians (complicated but important)
excellent approximation to waveforms produced by laser systems that use spherical mirrors to form the laser cavity
beamlike i.e. beam irradiance is strongly confined in the transverse direction
w(z) –often called spot size
wo minimum spot size
also called beam waist
wavefronts are planar
beam spreads while maintaining nearly spherical wavefronts that change radius of curvature as the beam propagates
in regions close to the axis of symmetry, the beam can be described by planar waveforms

16. Electromagnetic waves:
Plane EM wave: Electric field , magnetic field , and propagation vector are everywhere mutually perpendicular.
fields variation of EM wave can be described by harmonic equations as:
both E and B wave travel with same k and frequency , thus, with same wavelength and speed.
wavefronts for a linearly polarised plane em wave.

17. From EM theory, field amplitudes are related by E0 = cB0 where c = speed of light
At any specified time and place, E = cB
and velocity c in free space is given by:
0 = permittivity of free space =  1012 (C-s)2/kg-m3
0 = permeability of free space = 4  107 kg-m/(A-s)2
c = speed of light =  108 m/s
This wave represents the transmission of energy.
Energy density associated with the E-field is:
J/m3
And energy density associated with the B-field is:
J/m3

18. The two energy densities are equal from
Thus, the total energy density is their sum: Or
The power carried by the EM wave is the rate at which energy is transported

19. In time t, energy enclosed in rectangular volume A(ct) flows through cross-sectional area A, therefore, power is
Power =
But energy density is:
Thus, power per unit area is:
In vector notation, it is called the Poynting vector:

20. In optical detection, it is more appropriate to use irradiance Ee (time average of power delivered per unit area) rather than the magnitude of Poynting vector which varies rapidly with time (VIS ranges in 1014 – 1015 Hz)
As the average of sin2 or cos2 functions over one period = ½, so or or

21. E. g. Laser beam of radius 1 mm carries power 6 kW
E.g. Laser beam of radius 1 mm carries power 6 kW. What is the average irradiance and the amplitude of its E and B fields.
Average irradiance:
W/m2
Amplitude of E:
V/m
Amplitude of B: T

22. Light Polarization
In order to completely specify the EM wave, it is sufficient to specify te electric field since the magnetic field and Poynting vector can be determined once E is known.
The direction of the electric field is known as the polarization of the wave.
e.g.
According to Maxwell’s equations:
and the Poynting vector would be:
The polarization determines the direction of the force that the EM wave exerts on charged particles in the path of the wave:
unless v is a significant fraction of c, electric force will be much larger than that of the magnetic force.
the elctric force on the charged particle is along the direction of polarization
many optical applications depend critically on the nature and manipulation of the polarization of EM waves.

23. Linear Polarization
Evolution of the electric field vector over one period at a fixed plane z=0

24. Circular Polarization (special case of Elliptical Polarization)
Evolution of the electric field vector over one period at a fixed plane z=0
the component of the electric field along the y-direction is always π/2 out of phase with the x-component.

25. Unpolarized Light
individual atoms from a source, at any given instant, emit light with differing random polarizations
light coming from such a source is a superposition of the EM fields with differing and randomly distributed polarizations.
 randomly polarized or unpolarized light
if a certain EM field consists of the superposition of fields with many different polarizations, of which one or more polarization predominates  partially polarized

26. SUPERPOSITION OF WAVES
Definition of Superposition principle
If 1 and 2 are independent solutions of the wave equation,
where a and b are constants,
is also a solution.
then the linear combination
Thus, the resultant displacement is the sum of the separate displacements of the constituent waves, that is, for two separate waves 1 and 2, the resultant is
For EM wave, E = E1 + E2 and B = B1 + B2
(does not apply to nonlinear effects – when light of very large amplitude interacts with matter)
(We shall treat electric fields as scalar quantities at the moment)

27. (a) Superposition of waves of the same frequency:
Consider 2 harmonic waves of same frequency (but different amplitude and phase) combining to form resultant wave disturbance.
Starting with wave of form:
and set k r = constant as we wish to examine waves at fixed point in space.
Thus,
where constant phase angle
Two such waves intersecting at fixed point, may differ in phase by:
path difference initial phase difference

28. The two waves at given point are:
and
The resultant field at the point is (superposition principle):
From trigonometric identity: sin (A+B) = sin A cos B + cos A sin B

29. from the phasor diagrams:
(a) Adding two harmonic waves
(b) Phasor components of the 2 waves
Components of resultant (vectorally – fig (b)) are:
Therefore, the resultant becomes
which is
The resultant is another harmonic wave of same frequency , with amplitude E0 and phase 

30. From fig (a), cosine law gives
;  = 2  1
From fig (b), phase angle is
Similarly, applying to N harmonic waves:
(e.g. of 4 harmonic waves)

31. (from Pythagorus’ theorem):
expanding each term:
Adding them:
= 1
= cos (i  j)
Thus,
Sum of N harmonic waves of identical frequency is again a harmonic wave of the same frequency, with amplitude given by Eo and phase 

32. E.g. Harmonic waves:
E1 = 7 sin(t + /3) V/m
E2 = 12 cos(t + /4) V/m
E3 = 20 sin(t + /5) V/m
To make all angles consistent, change cos wave to sin wave:
E2 = 12 cos(t + /4) = 12 sin (t + /4 +/2) = 12 sin (t + 3/4) =
=  E0 = 28.6 V/m

33. Or =
Phase angle of resultant from: =
Thus,  = 1.17 radians or 0.372 or

34. For sound waves, (propagate through material medium)
Doppler effect:
For sound waves, (propagate through material medium)
frequency shift due to moving source is based physically on change in transmitted wavelength
frequency shift due to moving observer is based physically on the change in speed of sound waves relative to observer
Both the effects are distinct and described by different equations.
In light waves, (propagates in vacuum)
because the medium of propagation is removed, there is no physical basis for distinction between moving observer and moving source
There is only one relative motion between them that will determine the frequency shift
Derivation of Doppler effect for light requires theory of special relativity (not discussed here) and the Doppler-shifted
wavelength is given as:
where v = relative velocity between source and observer
+v when both are approaching one another

35.  When v << c, it is approximated to:
Doppler effect used to find the speed of astronomical sources emitting em radiation
Red shift = shift in wavelength of radiation toward longer wavelengths, due to a relative speed of source away from us
 Doppler broadening of spectral lines  fast-moving atoms of gas radiate light with both increases and decreases in frequency due to their random motion toward and away from the observer (using spectroscopic detector).

36. thus, v =  0.0234c =  7020 km/s  moving away from earth
Example:
Light from a distant galaxy shows the characteristic lines of Oxygen spectrum, except that the wavelengths are shifted from their values as measured using laboratory sources. The line expected at 513 nm shows up at 525 nm. What is the speed of the galaxy relative to the earth?
 = 513 nm, shifted ’ = 525 nm.
From , ,
thus, v =  c =  7020 km/s  moving away from earth
As the shifted wavelength is larger (i.e. the frequency is smaller), the galaxy is moving away from the earth with a speed of about 7020 km/s