Wave Motion & Superposition Notes | Interference, Doppler Effect, Resonance, Complete Notes with Question

December 28, 2025

Wave Motion & Superposition Notes | Interference, Doppler Effect, Resonance, Complete Notes with Question

Wave Motion

Wave motion is a type of motion in which a disturbance or vibration travels through a medium or space, transferring energy from one point to another without the actual transfer of matter. In wave motion, particles of the medium only oscillate about their mean positions, while the wave itself moves forward.

Wave motion plays a crucial role in physics, as it helps explain phenomena such as sound, light, water waves, earthquakes, and even quantum behavior. Understanding wave motion builds the foundation for studying acoustics, optics, and modern physics.

Nature of Wave Motion

In wave motion, the motion of individual particles and the motion of the wave are different. Each particle performs oscillatory motion, while the wave transfers energy through the medium. The particles do not travel with the wave; instead, they vibrate around their equilibrium positions.

For example, when a stone is dropped into water, circular ripples spread outward. The water molecules do not move outward with the ripple; they simply move up and down around their original positions.

Essential Conditions for Wave Motion

There must be a source of disturbance or vibration.
The medium must possess elasticity so that restoring forces can act.
The medium must have inertia so that particles can oscillate.

Without elasticity, particles cannot return to their mean positions, and without inertia, oscillations cannot continue. Hence, both are essential for wave propagation.

Types of Wave Motion

1. Mechanical Waves

Mechanical waves require a material medium for propagation. These waves cannot travel in vacuum. Examples include sound waves, water waves, and waves on a stretched string.

2. Non-Mechanical Waves

Non-mechanical waves do not require a medium and can propagate through vacuum. Electromagnetic waves such as light, radio waves, and X-rays fall into this category.

Transverse and Longitudinal Waves

Transverse Waves

In transverse waves, the particles of the medium vibrate perpendicular to the direction of wave propagation. Examples include waves on a stretched string and electromagnetic waves.

Longitudinal Waves

In longitudinal waves, the particles vibrate parallel to the direction of wave propagation. These waves consist of compressions and rarefactions. Sound waves in air are the best example.

Important Wave Parameters

Wavelength (λ): The distance between two consecutive particles in the same phase of vibration.

Frequency (f): The number of vibrations per second. Unit: hertz (Hz).

Time Period (T): Time taken to complete one vibration. T = 1/f.

Wave Velocity (v): Speed at which the wave propagates through the medium.

The basic wave equation is:
v = fλ

Energy Transfer in Wave Motion

Waves transfer energy without transferring matter. The amount of energy carried by a wave depends on its amplitude. Greater amplitude means greater energy.

In sound waves, louder sounds correspond to higher amplitudes, while in light waves, brightness depends on amplitude.

Examples of Wave Motion

Sound waves traveling through air
Waves on the surface of water
Seismic waves during earthquakes
Light waves traveling in space

Numerical Examples

Example 1: A wave has a frequency of 10 Hz and wavelength of 2 m. Find its velocity.

Solution: v = fλ = 10 × 2 = 20 m/s

Example 2: If the wavelength of a sound wave is 0.5 m and velocity is 340 m/s, find frequency.

Solution: f = v/λ = 340/0.5 = 680 Hz

Wave Motion – Practice Questions

Define wave motion.
Why is elasticity necessary for wave motion?
What is the difference between particle velocity and wave velocity?
Give two examples of transverse waves.
Can sound travel in vacuum? Explain.
Define wavelength and frequency.
What is the SI unit of frequency?
Explain why waves transfer energy but not matter.
Write the wave equation.
What happens to wave velocity if wavelength increases?
State two properties of longitudinal waves.
What determines the energy of a wave?
Define amplitude.
What type of wave is light?
Explain compressions and rarefactions.

Short Answer (with answers)

Q: What is wave velocity?
A: It is the speed with which a wave propagates through a medium.

Q: Name one mechanical wave.
A: Sound wave.

Q: What is frequency?
A: Number of oscillations per second.

Q: Does wave motion require a medium always?
A: No, electromagnetic waves do not require a medium.

Q: What is the relation between T and f?
A: T = 1/f

Principle of Superposition

The principle of superposition states that when two or more waves travel through the same medium simultaneously, the resultant displacement at any point is equal to the algebraic sum of the individual displacements produced by each wave.

This principle applies only to linear systems, where wave amplitudes are small and the medium behaves elastically. Superposition explains many wave phenomena such as interference, beats, and standing waves.

Mathematical Statement

If two waves produce displacements y₁ and y₂ at a point, then the resultant displacement y is:

y = y₁ + y₂

Superposition in Transverse Waves

When two transverse waves overlap, their displacements add algebraically. If both displacements are in the same direction, the resultant amplitude increases. If they are in opposite directions, cancellation occurs.

Constructive and Destructive Superposition

Constructive Superposition

When two waves meet in phase (crest meets crest), their amplitudes add, producing a larger resultant wave.

Destructive Superposition

When a crest meets a trough, partial or complete cancellation occurs, reducing or eliminating displacement.

Interference of Waves

Interference is the phenomenon resulting from superposition of two coherent waves. It leads to regions of maximum intensity (constructive interference) and minimum intensity (destructive interference).

Conditions for Sustained Interference

Sources must be coherent
Waves must have same frequency
Phase difference must be constant

Standing Waves

Standing waves are formed due to superposition of two waves of same frequency and amplitude traveling in opposite directions. Nodes and antinodes are formed in standing waves.

Nodes are points of zero displacement, while antinodes have maximum displacement.

Applications of Superposition

Noise cancellation headphones
Interference patterns in optics
Musical instruments
Standing waves in strings

Numerical Examples

Example: Two waves of amplitude 3 cm overlap in phase. Find resultant amplitude.

Solution: Resultant amplitude = 3 + 3 = 6 cm

Superposition – Practice Questions

State the principle of superposition.
What is meant by constructive interference?
Define destructive interference.
What is coherence?
Can superposition occur in all media?
What happens when two equal waves meet out of phase?
Define standing waves.
What are nodes?
What are antinodes?
Give one application of superposition.
What condition is necessary for interference?
Is energy conserved in superposition?
What is the role of phase difference?
Can superposition occur in sound waves?
What is linear superposition?

Short Answer (with answers)

Q: What is superposition?
A: Addition of displacements of overlapping waves.

Q: What causes interference?
A: Superposition of coherent waves.

Q: What are nodes?
A: Points of zero displacement.

Q: Name one device using superposition.
A: Noise-canceling headphones.

Q: Does superposition violate energy conservation?
A: No, total energy remains conserved.

Superposition of Two Harmonic Waves

Consider two simple harmonic waves of the same frequency and wavelength traveling in the same direction through a medium. When these waves overlap, the resultant displacement at any point depends on the phase difference between the two waves.

Let the two waves be represented as:

y₁ = a sin(ωt − kx)
y₂ = a sin(ωt − kx + φ)

According to the principle of superposition, the resultant displacement is:

y = y₁ + y₂

Using trigonometric identities, the resultant wave can be written as:

y = 2a cos(φ/2) sin(ωt − kx + φ/2)

This equation shows that the resultant wave is also a harmonic wave with the same frequency and wavelength, but its amplitude depends on the phase difference φ.

Resultant Amplitude due to Superposition

The amplitude of the resultant wave is given by:

A = 2a cos(φ/2)

From this relation, we observe:

When φ = 0, A = 2a (maximum amplitude)
When φ = π, A = 0 (complete cancellation)
When 0 < φ < π, partial interference occurs

Thus, the phase difference plays a crucial role in determining the nature of superposition.

Interference of Waves

Interference is a phenomenon that occurs due to the superposition of two or more waves of the same frequency that maintain a constant phase difference. As a result, regions of reinforcement and cancellation are formed.

Interference does not violate the law of conservation of energy. Energy is redistributed in space rather than destroyed.

Types of Interference

Constructive Interference: Occurs when waves meet in phase. The resultant amplitude is maximum.

Destructive Interference: Occurs when waves meet out of phase by π. The resultant amplitude becomes minimum or zero.

Path Difference and Phase Difference

Path difference is the difference in distances traveled by two waves before reaching a point. Phase difference depends on the path difference.

Conditions:

Constructive interference: Path difference = nλ
Destructive interference: Path difference = (2n + 1)λ/2

where n = 0, 1, 2, 3, ...

Beats

Beats are produced when two sound waves of slightly different frequencies but nearly equal amplitudes superpose. Due to interference, the sound intensity alternately increases and decreases with time.

The phenomenon of beats is commonly observed in tuning instruments.

Beat Frequency

Beat frequency is defined as the number of beats produced per second and is equal to the absolute difference of the two frequencies.

f_beats = |f₁ − f₂|

For example, if two tuning forks of frequencies 256 Hz and 260 Hz are sounded together, the beat frequency is:

|260 − 256| = 4 beats per second

Energy Distribution in Beats

In beats, energy is not lost. Instead, it alternates between maximum and minimum intensity regions. When constructive interference occurs, sound is loud; when destructive interference occurs, sound is weak.

Standing Waves (Stationary Waves)

Standing waves are formed due to the superposition of two identical waves traveling in opposite directions with the same frequency and amplitude.

Unlike progressive waves, standing waves do not transport energy from one place to another.

Nodes and Antinodes

Node: A point where the displacement is always zero.

Antinode: A point where the displacement is maximum.

The distance between two consecutive nodes or two consecutive antinodes is λ/2.

Standing Waves on a Stretched String

When a string is fixed at both ends, standing waves are formed under certain conditions. The simplest mode of vibration is called the fundamental mode.

In the fundamental mode:

Length of string = λ/2
Frequency is minimum

Higher modes are called harmonics.

Applications of Standing Waves

Musical instruments like guitar and violin
Microwave ovens
Resonance tubes
Engineering structures

Important Numerical Examples

Example 1: Two waves of equal amplitude interfere destructively. What is the resultant amplitude?

Answer: Zero

Example 2: Find the beat frequency if two waves have frequencies 500 Hz and 504 Hz.

Answer: 4 Hz

Wave Motion & Superposition – Long Answer Questions

Explain wave motion and distinguish it from particle motion.
Derive the expression for resultant amplitude due to superposition.
Explain constructive and destructive interference.
Describe the phenomenon of beats with examples.
What are standing waves? Explain their formation.
Discuss nodes and antinodes.
Explain why standing waves do not transfer energy.
Derive the condition for maxima and minima in interference.
Explain the role of phase difference in superposition.
Discuss applications of superposition in daily life.

Wave Motion & Superposition – Short Questions (with Answers)

Q: What causes beats?
A: Superposition of two waves of slightly different frequencies.

Q: What is a node?
A: A point of zero displacement in a standing wave.

Q: Define beat frequency.
A: Difference between the frequencies of two waves.

Q: Do standing waves carry energy?
A: No.

Q: What is the distance between two consecutive nodes?
A: λ/2

Doppler Effect

The Doppler Effect is a phenomenon observed when there is a relative motion between the source of a wave and the observer. It results in an apparent change in the frequency of the wave as perceived by the observer, even though the actual frequency emitted by the source remains constant.

The Doppler Effect is commonly observed in sound waves, such as the change in pitch of a siren when an ambulance passes by. It is also applicable to light waves and plays a crucial role in astronomy and modern physics.

Basic Explanation of Doppler Effect

When a sound source moves towards an observer, the wavefronts get compressed, resulting in a shorter wavelength and higher frequency. When the source moves away, the wavefronts spread out, increasing the wavelength and decreasing the frequency.

Similarly, if the observer moves towards the source, the observer encounters wavefronts more frequently, increasing the apparent frequency.

Important Assumptions in Doppler Effect

The medium is stationary and uniform.
The velocity of sound in the medium remains constant.
The velocities of the source and observer are much smaller than the wave velocity.

Doppler Effect for Sound Waves

Case 1: Source Moving, Observer Stationary

If the source moves with velocity v_s towards a stationary observer, the apparent frequency heard by the observer increases.

The apparent frequency is given by:

f' = f × (v / (v − v_s))

If the source moves away from the observer:

f' = f × (v / (v + v_s))

Case 2: Observer Moving, Source Stationary

If the observer moves towards the source with velocity v_o, the apparent frequency increases:

f' = f × ((v + v_o) / v)

If the observer moves away from the source:

f' = f × ((v − v_o) / v)

Case 3: Both Source and Observer in Motion

When both source and observer are moving, the apparent frequency is given by:

f' = f × ((v ± v_o) / (v ∓ v_s))

The sign convention depends on the direction of motion.

Practical Examples of Doppler Effect

Change in pitch of ambulance or police siren
Radar speed detection
Astronomical red shift and blue shift
Medical ultrasound imaging

Numerical Examples on Doppler Effect

Example 1: A siren of frequency 600 Hz approaches a stationary observer with a speed of 20 m/s. If the speed of sound is 340 m/s, find the apparent frequency.

Solution:
f' = 600 × (340 / (340 − 20)) = 600 × (340 / 320) ≈ 637.5 Hz

Example 2: An observer moves towards a stationary sound source of frequency 500 Hz at 10 m/s. Find the apparent frequency.

Solution:
f' = 500 × ((340 + 10) / 340) ≈ 514.7 Hz

Limitations of Doppler Effect

Not accurate when velocities are comparable to wave speed
Assumes stationary medium
Classical Doppler formulas fail for light at high speeds

Resonance

Resonance is a phenomenon in which a system vibrates with maximum amplitude when the frequency of an externally applied periodic force matches the natural frequency of the system.

Resonance occurs due to efficient transfer of energy from the external source to the vibrating system. This phenomenon is closely related to wave motion and oscillations.

Natural Frequency

Every vibrating system has a natural frequency at which it prefers to oscillate. When forced to vibrate at this frequency, resonance occurs.

Forced Oscillations

When an external periodic force acts on a system, the system undergoes forced oscillations. The frequency of oscillation is determined by the external force, not by the system itself.

The amplitude of forced oscillations depends on the frequency of the applied force.

Condition for Resonance

Resonance occurs when:

Frequency of applied force = Natural frequency of system

At resonance:

Amplitude becomes maximum
Energy absorption is maximum
Phase difference between force and displacement is π/2

Examples of Resonance

Resonance in tuning forks
Breaking of glass by sound
Radio tuning circuits
Resonance in bridges and buildings

Resonance in Air Columns

In air columns, resonance occurs when the length of the air column matches certain conditions depending on whether the pipe is open or closed.

Open Pipe

Both ends are antinodes
Fundamental length = λ/2
All harmonics are present

Closed Pipe

One end node, one end antinode
Fundamental length = λ/4
Only odd harmonics present

Importance and Applications of Resonance

Musical instrument tuning
Communication systems
Medical imaging
Mechanical engineering design

Numerical Example on Resonance

Example: A closed organ pipe resonates at 170 Hz. If the speed of sound is 340 m/s, find the length of the pipe.

Solution:
λ = v / f = 340 / 170 = 2 m
Length = λ / 4 = 0.5 m

Wave Motion – Doppler Effect & Resonance: Practice Questions

What is Doppler Effect?
Why does pitch change in a moving siren?
Write Doppler formula when source moves towards observer.
What assumptions are made in Doppler Effect?
Define natural frequency.
What is resonance?
Why is resonance dangerous in bridges?
What happens to amplitude at resonance?
Define forced oscillations.
Why are only odd harmonics present in closed pipes?
What is the phase difference at resonance?
State one limitation of Doppler Effect.
Give two applications of Doppler Effect.
What is resonance frequency?
How is resonance useful in radio tuning?
What is red shift?
What is blue shift?
Does Doppler Effect apply to light?
Why does glass break at resonance?
What determines the loudness at resonance?
What happens if damping is high?
Explain resonance in air columns.
What is forced vibration?
Define amplitude.
Why is resonance avoided in structures?

Short Answer Questions (with Answers)

Q: What causes Doppler Effect?
A: Relative motion between source and observer.

Q: When is apparent frequency maximum?
A: When source and observer move towards each other.

Q: What is resonance?
A: Maximum vibration at natural frequency.

Q: Is Doppler Effect observed in stationary source and observer?
A: No.

Q: What is the condition for resonance?
A: Driving frequency equals natural frequency.