PHY104 - General Physics II: WEEK 3

(2-Credit Units)
Department of Physics and Materials Science.
Kwara State University, Malete, Nigeria.
Lecturer: DR. G. B. EGBEYALE
June 05, 2024

Course Outline: Superposition of waves; Interference,

diffraction, dispersion, polarization, Solved problems

Superposition of Waves
Introduction
The superposition principle is key to understanding wave interactions. When two or
more waves meet, their combined effect is the algebraic sum of their individual
displacements. This principle leads to phenomena such as interference, diffraction,
dispersion, and polarization.

Superposition Principle
The superposition principle states that when multiple waves overlap, the resultant
displacement at any point is the sum of the displacements of the individual waves at
that point.

I.Interference
Interference occurs when two or more waves superpose to form a resultant wave.
There are two types of interference:

1. Constructive Interference:

Consider two waves that arrive in phase as shown in Figure 1. Their crests arrive at
exactly the same time. Hence, they interfere constructively. A resultant wave is
produced, which has a crest much larger than the two individual waves, and the
troughs are deeper. If the two incoming waves that are in phase have amplitude of A,
then the resultant wave has an amplitude of 2A. The frequency of the resultant is the
same as that of the incoming waves.
 Figure 1: Constructive Interference

2. Destructive Interference:

Consider two waves that arrive in antiphase (with a phase difference of π or 180°) as
illustrated in Figure 2. The crest of one wave and the trough of another wave arrive at
exactly the same time. Hence, they interfere destructively. A resultant wave is
produced, which has a smaller amplitude. If the two incoming waves that are in
antiphase have amplitude of A, then the resultant wave has an amplitude of zero. The
frequency of the resultant is the same as that of the incoming waves.
 Figure 2: Destructive Interference

It can be noted that as the displacement is a vector quantity, the direction of the wave
must be noted when the individual waves are added. Stationary waves are formed by
two waves with the same frequency travelling in opposite directions.

Figure 3 shows the interference of waves from two point sources, S1 and S2. The
point C is equidistant from S1 and S2 and hence, the path difference is 0. If the waves
start in phase at S1 and S2, they will arrive at C combining constructively, producing
maximum disturbance at C.

Points D and E are at different distances from S1 and S2 and hence there is a path
difference between the waves. If the path difference is a whole number of
wavelengths (1λ, 2λ, 3λ, and so on), the waves are in phase and combine
constructively. A resultant with maximum displacement is formed. If the path
difference is an odd number of half-wavelengths (1λ/2, 3λ/2, 5λ/2, and so on), the
waves are in antiphase and combine destructively. A resultant with minimum
displacement is formed. The maximum and minimum displacements are called
fringes. This collection of fringes produced by superposition of overlapping waves is
called an interference pattern.

Path difference, ∆ λ = d sinθ

For constructive interference, nλ

For destructive interference, (n+1)λ

Figure 3: Path Difference

Summary
  The principle of superposition states that when two or more waves meet at a
point, the resultant displacement at that point is equal to the sum of the
displacements of the individual waves at that point.
 If the two incoming waves that are in phase have amplitude of A, then the
resultant wave has an amplitude of 2A.
 If the two incoming waves that are in antiphase have amplitude of A, then the
resultant wave has an amplitude of zero.
 Stationary waves are formed by two waves with the same frequency travelling
in opposite directions.

Solved Example:
1. If and y2
Then Resultant displacement , y = 6sin(x-2t+cos(x-2t)
2.

3. Two waves, y1 = 2cos(x-t) and y2 = 2cos(x-t + π/2), interfere. Find the resultant
wave.

Solution:
The waves are π/2 out of phase:
y = 2cos(x−t) + 2cos(x−t+π/2)
Using trigonometric identities:
y =2cos(x−t) + 2sin(x−t) = 2√ 2 cos(x−t−π/4)

3. Diffraction: Diffraction refers to the bending or spreading of waves as they

encounter an obstacle or aperture. It occurs when waves encounter an obstruction that
is comparable in size to their wavelength. Diffraction is most noticeable when the
obstacle or aperture is of similar size to the wavelength of the waves.
Single-Slit Diffraction: When a wave passes through a single narrow slit or aperture,
it spreads out, forming a pattern of alternating bright and dark fringes on a screen
placed behind the slit.
Double-Slit Diffraction: When a wave passes through two closely spaced slits, it
undergoes interference between the waves emerging from each slit, resulting in an
interference pattern of alternating bright and dark fringes.

When waves encounter an obstacle or a slit that is comparable in size to their

wavelength, they bend and spread out.
The amount of diffraction increases with increasing wavelength.
Solved Example:
1. A wave of wavelength 500 nm passes through a slit of width 1 μm. Describe the
diffraction pattern.
Solution:
Since the slit width (1 μm) is comparable to the wavelength (500 nm), significant
diffraction occurs. The wave spreads out, creating a pattern of alternating dark and
light bands on a screen behind the slit.

2. Light of wavelength 600 nm passes through a single slit of width 2 μm. Calculate
the angular width of the central maximum.
Solution:
Using the formula for single-slit diffraction,
sin θ=¿λ/a

= 0.3
The angular width 2θ = 34.8o

3. Dispersion
Dispersion occurs when the speed of a wave depends on its frequency, causing
different frequencies to travel at different speeds. (Hint: Dispersion of white light)

In optics, dispersion causes different colors of light to spread out because each color
travels at a different speed in a medium.
Example: A prism disperses white light into its constituent colors (ROYGBIV). Red
light has the highest wavelength while violet has the smallest (λR = 700 nm and λB =
450 nm

4. Polarization
Polarization is the orientation of the oscillations of a transverse wave, such as light, in
a particular direction.
Unpolarized light has oscillations in all directions perpendicular to the direction of
propagation.
Polarized light has oscillations in one direction.
Polarization of Light:
Polarization refers to the orientation of the electric field vector of light waves. Light
waves are transverse electromagnetic waves, meaning the electric and magnetic fields
oscillate perpendicular to the direction of propagation. Polarization specifies the
direction in which the electric field oscillates.
Unpolarized Light: In unpolarized light, the electric field vectors oscillate randomly in
all possible directions perpendicular to the direction of propagation. Natural light
sources, such as the Sun or incandescent bulbs, typically emit unpolarized light.
Polarization Mechanisms:
Transmission: Certain materials can selectively absorb light waves vibrating in
certain directions perpendicular to the direction of propagation, thus polarizing the
transmitted light. Polarizing filters (polarizers) utilize this mechanism.
Reflection: When light reflects off a surface, it often becomes partially polarized,
with the electric field oriented in a specific direction parallel to the surface. This is
known as glare, and polarizing sunglasses can effectively block horizontally polarized
light to reduce glare.
Scattering: When light scatters off particles or molecules in the atmosphere, it can
become partially polarized. This is commonly observed in the blue sky, where the
scattered light is predominantly polarized perpendicular to the direction of sunlight.

Types of Polarization:
1. Linear Polarization: The electric field vectors oscillate in a single plane
perpendicular to the direction of propagation.
2. Circular Polarization: The electric field vectors rotate in a circular manner as the
light propagates, with a constant amplitude.
3. Elliptical Polarization: A combination of linear and circular polarization, where
the electric field vectors trace out an ellipse as the light propagates.

Applications of Polarization
It is applied in various fields, such as optical communications, photography, 3D
movie projection, and materials science.

Solved Example:
If unpolarized light passes through a polarizing filter, what will be the intensity of the
transmitted light?

Solution:
The intensity I of the transmitted light is half the intensity Io of the incident
unpolarized light:
I =Io/2

Tutorial
1. If y1 = 2sin(3x – 2t) & y2 = 3sin(3x – 2t + π/4). Find the expression for resultant
displacement caused by superposition of the two waves.
a) 3.6sin(2x-3t+3π/7)
b) 4.63sin(3x-2t+3π/5)
c) 3.6cos(3x-2t+3π/7)
d) 4.63cos(3x-2t+3π/5)
Answer: b
Explanation: y = y1 + y2= 2sin(3x-2t) + 3sin(3x-2t+π/4)
= Asin(3x-2t+φ).

∴ A = sqrt(4+9+12/√2) = 4.63m.
A2 = A12 + A22 + 2A1A2cosθ, where θ = π/4, A1 = 2 & A2 = 3.

= (3/√2)/(2+3/√2) = 0.51. ∴ φ = 27rad.

Tanφ = A2sinθ / (A1 + A2cosθ)

∴ y = 4.63sin(3x-2t+8.6π) = 4.63sin(3x-2t+3π/5).

2. A wave having an amplitude of 3cm is to be superimposed with another wave of

the same amplitude. What should be their phase difference if the net amplitude has to
be 0?
a) π/2
b) π
c) 2π
d) 0
Answer: b
Explanation: The net amplitude is given by: A2 = A12 + A22 + 2A1A2cosθ,
where θ is the phase difference.

∴ θ = π.
For A to be zero 9 + 9 + 2(9)(9)cosθ = 0