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Waves and Oscillations
Superposition of Waves
The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This fundamental principle explains phenomena such as interference, stationary waves, and beats.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic principle of superposition, graphical addition of waves, constructive and destructive interference, conditions for sustained interference, formation of standing waves in strings and air columns, and beats.
PGT extension
Analytical treatment of superposition of two waves with phase difference, intensity distribution in interference, superposition of perpendicular simple harmonic motions (Lissajous figures), and reflection and transmission of waves at boundaries.
Prerequisites
Simple Harmonic Motion (SHM) and phase relationships
Wave equation and wave parameters (wavelength, frequency, wave velocity)
Trigonometric identities for sum and difference of angles
Principle of Superposition
If two or more waves travel simultaneously through a medium, each wave acts independently of the others as if no other waves are present.
The net displacement y(x, t) at any point and at any instant is the algebraic sum of the individual displacements y_1(x, t) and y_2(x, t).
Applicable to linear wave equations where the differential operator governing wave propagation is linear.
Constructive and Destructive Interference
Constructive interference occurs when waves arrive in phase, resulting in a maximum displacement and maximum intensity.
Destructive interference occurs when waves arrive out of phase (phase difference of odd multiples of π), resulting in minimum displacement and minimum intensity.
The condition for constructive interference is path difference Δ$x = n\lambda$ and phase difference Δ$\phi = 2n\pi$.
Standing Waves (Stationary Waves)
Formed by the superposition of two identical waves traveling in opposite directions with the same frequency and amplitude.
Nodes are points of zero displacement and maximum pressure variation; antinodes are points of maximum displacement and zero pressure variation.
Energy is not propagated along the medium; it remains confined between nodes.
Beats Phenomenon
Produced by the superposition of two sound waves of slightly different frequencies traveling in the same direction.
The resultant intensity rises and falls periodically, creating an audible pulsing sound.
The beat frequency is equal to the absolute difference between the two component frequencies: $f_b = |f_1 - f_2|$.
Derivations
Superposition of Two Harmonic Waves with Phase Difference
Conditions: Two sinusoidal waves of equal amplitude (A) and frequency (ω) traveling in the same direction with a constant phase difference (φ).
Let the first wave be $y_1 = A \sin(\omegat)$ and the second wave be $y_2 = A \sin(\omegat + \phi)$.
Using the principle of superposition, the resultant displacement is $y = y_1 + y_2 = A [\sin(\omegat) + \sin(\omegat + \phi)]$.
Apply trigonometric identity for sum of sines: sin A + sin $B = 2 \sin((A+B)/2) \cos((A-B)/2)$.
Substitute the terms: $y = 2A \cos(\phi/2) \sin(\omegat + \phi/2)$.
Let $R = 2A \cos(\phi/2) be the amplitude of the resultant wave$.
Result: $y = R \sin(\omegat + \phi/2)$ where $R = 2A \cos(\phi/2)$
Intensity Distribution in Interference
Conditions: Two interfering waves with amplitudes A_1 and A_2 and phase difference φ.
Resultant amplitude squared is $R_2 = A_1^2 + A_2^2 + 2A_1 A_2 \cos(\phi) from vector addition$.
Since intensity I is proportional to the square of the amplitude (I ∝ A^2), substitute I_1 ∝ A_1^2 and I_2 ∝ A_2^2.
A_1, $A_2 = component amplitudes$, $\phi = phase difference$ m
Same frequency waves
Condition for Constructive Interference (Path Difference)
$$\Delta x = n\lambda$$
n = 0, 1, 2, 3..., $\lambda = wavelength$ m
In-phase sources
Condition for Destructive Interference (Path Difference)
$$\Delta x = (2n - 1)\frac{\lambda}{2}$$
n = 1, 2, 3..., $\lambda = wavelength$ m
In-phase sources
Beat Frequency
$$f_b = |f_1 - f_2|$$
f_1, $f_2 = frequencies of component waves$ Hz
|f_1 - f_2| \le 16 \text{ Hz} for human ear detection
Stationary Wave Equation
$$y(x,t) = 2A \sin(kx) \cos(\omega t)$$
A = component amplitude, k = wave number, $\omega = angular frequency$ m
Two identical waves traveling in opposite directions
Positions of Nodes in a Stationary Wave
$$x = \frac{n\lambda}{2}$$
n = 0, 1, 2, 3... m
Fixed boundary condition
Positions of Antinodes in a Stationary Wave
$$x = (2n - 1)\frac{\lambda}{4}$$
n = 1, 2, 3... m
Fixed boundary condition
Graphs and experiments
Graphical Superposition of Pulses
Draw: Two triangular wave pulses traveling towards each other on a string, showing overlap and temporary reinforcement/cancellation.
Interpret: At the instant of overlap, the vertical heights add algebraically. After passing each other, both pulses continue unchanged in shape and velocity.
Exam use: Tested in TGT to check conceptual understanding of wave independence after interaction.
Standing Wave Profile
Draw: Envelopes of a string fixed at both ends showing multiple loops with clearly marked nodes (N) and antinodes (A).
Interpret: Demonstrates spatial confinement of energy with alternating points of zero and maximum vibration amplitude.
Exam use: Numerical and conceptual problems on harmonics in strings.
Quincke's Tube Experiment
Draw: U-shaped sliding tube apparatus with a sound source and detector demonstrating acoustic interference.
Interpret: Changing the path length of one tube shifts the sound from loud to soft, proving the wave nature of sound and interference.
Exam use: Questions based on path difference calculation to observe intensity minima/maxima.
Solved numericals
TGT core
Two sound waves given by equations $y_1 = 0.3 \sin(500\pit)$ and $y_2 = 0.4 \cos(500\pit) superimpose$. Find the amplitude of the resulting wave.
Rewrite y_2 in sine form: $y_2 = 0.4 \sin(500\pit + \pi/2)$.
A tuning fork of unknown frequency produces 4 beats per second with a standard fork of frequency 384 Hz. When a little wax is added to the unknown fork, the beat frequency becomes 6 beats per second. Find the original frequency of the unknown fork.
Let the unknown frequency be f. Given |f - 384| = 4, so f = 388 Hz or 380 Hz.
Adding wax increases the mass, which decreases the frequency of the unknown fork.
If original frequency was 388 Hz, decreasing it would make the new frequency closer to or further from 384 Hz. Test new frequency: if f' < 388, say 386 Hz, beat frequency with 384 Hz is 2 Hz (decreased). But question states beat frequency becomes 6 Hz.
If original frequency was 380 Hz, decreasing it (e.g., to 378 Hz) increases the beat frequency with 384 Hz to |378 - 384| = 6 Hz.
Therefore, the original frequency must be 380 Hz.
Answer: 380 Hz
PGT extension
The intensity ratio of two coherent waves is 9:1. Find the ratio of maximum to minimum intensity in the interference pattern.
Let intensities be I_1 and I_2, such that $I_1/I_2 = 9/1$.
Since intensity is proportional to amplitude squared, the amplitude ratio $A_1/A_2 = \sqrt{9/1} = 3/1$.
The principle of superposition strictly holds for linear systems where the restoring force is strictly proportional to displacement (Hooke's Law).
Non-linear effects (such as shock waves or finite-amplitude acoustic waves) violate simple linear superposition and lead to harmonic generation and wave steepening.
Common misconceptions
Common error: Waves bounce off each other when they collide during superposition.
Correct: Waves pass right through each other without altering their individual identities or velocities.
Interference affects only the displacement at the point of overlap during the encounter, not the permanent state of the individual propagating waves.
Common error: Destructive interference destroys energy.
Correct: Energy is conserved; it is redistributed from points of destructive interference (minima) to points of constructive interference (maxima).
Total integrated energy across the interference pattern remains constant.
The net displacement during wave superposition is given by the algebraic sum of individual displacements $y(x,t) = y_1(x,t) + y_2(x,t)$ in a linear medium.
Constructive interference occurs when path difference is $\Delta x = n\lambda$ and phase difference is $\Delta\phi = 2n\pi$.
Destructive interference occurs when path difference is $\Delta x = (2n - 1)\frac{\lambda}{2}$ and phase difference is odd multiples of $\pi$.
Standing waves are formed by the superposition of two identical waves traveling in opposite directions, with nodes at zero displacement and antinodes at maximum displacement.
The beat frequency is given by $f_b = |f_1 - f_2|$, representing the periodic rise and fall of sound intensity.
Energy is conserved during interference and is merely redistributed from minima to maxima.
Glossary
Principle of SuperpositionThe principle stating that when two or more waves overlap, the net displacement at any point is the sum of the individual displacements.
NodeA point along a standing wave where the displacement is zero and pressure variation is maximum.
AntinodeA point along a standing wave where the displacement amplitude is maximum and pressure variation is zero.
Beat FrequencyThe frequency at which the loudness of sound rises and falls when two sound waves of slightly different frequencies interfere.
Interference vs Standing Waves
Parameter
Interference
Standing Waves
Wave Direction
Superposition of waves traveling in the same or crossing directions
Superposition of identical waves traveling in opposite directions
Energy Transport
Energy is transported along the medium
Energy is confined; net energy transport across nodes is zero
Displacement Amplitude
Uniform or varying based on phase difference across space
Fixed spatial pattern of zero (nodes) and maximum (antinodes) amplitudes
Practice Quiz
Choose an option to reveal the explanation.
1. What is the net displacement of a medium when two waves overlap according to the principle of superposition?
Explanation: The principle of superposition states that the net displacement $y(x,t)$ at any point is the algebraic sum of the individual displacements $y_1(x,t)$ and $y_2(x,t)$.
2. Constructive interference between two coherent waves occurs when their path difference $\Delta x$ is equal to:
Explanation: For constructive interference, the path difference must be an integral multiple of wavelength, i.e., $\Delta x = n\lambda$ where $n = 0, 1, 2, \dots$
3. What is the phase difference corresponding to destructive interference?
Explanation: Destructive interference occurs when waves arrive out of phase with a phase difference of odd multiples of $\pi$, expressed as $(2n - 1)\pi$.
4. Two sound waves of frequencies $450\text{ Hz}$ and $456\text{ Hz}$ are sounded together. What is the beat frequency?
Explanation: The beat frequency is given by $f_b = |f_1 - f_2| = |456 - 450| = 6\text{ Hz}$.
5. In a stationary wave, nodes are points where:
Explanation: Nodes are points of zero displacement and maximum pressure variation in standing waves.
6. What is the distance between consecutive nodes in a stationary wave of wavelength $\lambda$?
Explanation: Nodes occur at $x = \frac{n\lambda}{2}$, so the distance between two consecutive nodes is $\frac{\lambda}{2}$.
7. Which of the following equations represents a standing wave?
Explanation: The stationary wave equation is $y(x,t) = 2A \sin(kx) \cos(\omega t)$, representing the combination of two identical oppositely traveling waves.
8. What happens to the energy during destructive interference?
Explanation: Interference does not violate the conservation of energy; energy is redistributed from points of destructive interference (minima) to constructive interference (maxima).
9. What is the resultant amplitude of two interfering waves with component amplitudes $A_1$ and $A_2$ and phase difference $\phi$?
Explanation: The resultant amplitude is derived via vector addition of amplitudes as $R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)}$.
10. Under what condition is the principle of superposition strictly applicable?
Explanation: The superposition principle is applicable to linear wave equations where the differential operator governing wave propagation is linear.
11. What is the position of the first antinode from a fixed boundary in a standing wave?
Explanation: Antinodes are located at $x = (2n - 1)\frac{\lambda}{4}$. For $n = 1$, the first antinode is at $x = \frac{\lambda}{4}$.
12. When two identical waves interfere destructively, what is the resultant intensity?
Explanation: Destructive interference of identical waves results in minimum displacement and zero intensity.
13. What is the maximum frequency difference (in Hz) that the human ear can typically detect as distinct beats?
Explanation: The human ear can typically detect beats when $|f_1 - f_2| \le 16\text{ Hz}$.
14. What happens to two waves when they collide during superposition?
Explanation: Waves pass right through each other without altering their individual identities or velocities; superposition only affects the net displacement during overlap.
15. Which parameter determines whether two interfering waves produce constructive or destructive interference?
Explanation: The relative phase difference or path difference between the interfering waves dictates whether interference is constructive or destructive.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Two waves represented by $y_1 = A \sin(\omega t)$ and $y_2 = A \sin(\omega t + \frac{\pi}{3})$ superimpose in a medium. What is the resultant amplitude of the combined wave?
Explanation: Using $R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)}$ with $A_1 = A_2 = A$ and $\phi = \frac{\pi}{3}$: $R = \sqrt{A^2 + A^2 + 2A^2(0.5)} = \sqrt{3}A$.
2. A string of length $L$ fixed at both ends vibrates in its fundamental mode. What is the wavelength of the standing wave formed?
Explanation: In the fundamental mode of a string fixed at both ends, the length of the string is equal to half the wavelength ($L = \frac{\lambda}{2}$), so $\lambda = 2L$.
3. If the path difference between two interfering light waves of wavelength $\lambda$ is $5\lambda$, what type of interference is observed at that point?
Explanation: A path difference of $\Delta x = 5\lambda$ satisfies the condition $\Delta x = n\lambda$ (with $n = 5$), which corresponds to constructive interference.
4. A tuning fork of unknown frequency gives $4\text{ beats/s}$ with a standard tuning fork of frequency $340\text{ Hz}$. When a small piece of wax is added to the unknown fork, the beat frequency reduces to $2\text{ beats/s}$. What was the original frequency of the unknown fork?
Explanation: Initial frequency is either $340 + 4 = 344\text{ Hz}$ or $340 - 4 = 336\text{ Hz}$. Adding wax decreases the frequency of the fork. Since the beat frequency decreased from $4$ to $2$, the original frequency must be $344\text{ Hz}$ (as loading $344$ reduces it toward $342$, lowering the difference to $2$).
5. What is the distance between a node and the adjacent antinode in a stationary wave of wavelength $\lambda$?
Explanation: Nodes are at $x = \frac{n\lambda}{2}$ and antinodes at $x = (2n - 1)\frac{\lambda}{4}$. The distance between a node and an adjacent antinode is $\frac{\lambda}{4}$.
6. The equation of a standing wave in a string is given by $y = 0.04 \sin(5\pi x) \cos(40\pi t)$. What is the velocity of the component traveling waves?
Explanation: Here $k = 5\pi$ and $\omega = 40\pi$. The wave velocity $v = \frac{\omega}{k} = \frac{40\pi}{5\pi} = 8\text{ m/s}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Two coherent sources of intensity $I_0$ interfere. What is the maximum intensity in the interference pattern?
Explanation: Intensity is proportional to the square of amplitude. Maximum amplitude is $A_1 + A_2 = 2A_0$, so maximum intensity is $(2A_0)^2 = 4A_0^2 = 4I_0$.
2. A standing wave is set up in a 1.5 m long string fixed at both ends. If it vibrates in the 3rd harmonic, how many nodes are present along the string (including boundaries)?
Explanation: For the $n$-th harmonic in a string fixed at both ends, there are $n+1$ nodes. For the 3rd harmonic ($n=3$), there are $3+1 = 4$ nodes.
3. What is the phase difference between two points on a wave separated by a path difference of $\frac{\lambda}{3}$?
5. Which of the following characteristics distinguishes stationary waves from progressive waves?
Explanation: Stationary waves feature permanent nodes (zero amplitude) and antinodes (maximum amplitude) fixed in space, unlike progressive waves where all points share the same amplitude sequentially.
6. If the frequency of a tuning fork is $256\text{ Hz}$, what is the period of the beats produced when it is sounded with a fork of $260\text{ Hz}$?
Explanation: Beat frequency $f_b = |260 - 256| = 4\text{ Hz}$. The period of beats is $T_b = \frac{1}{f_b} = \frac{1}{4} = 0.25\text{ s}$.
7. What is the condition for destructive interference in terms of phase difference $\Delta\phi$?
Explanation: Destructive interference occurs when waves arrive out of phase with a phase difference of odd multiples of $\pi$, i.e., $(2n - 1)\pi$.
8. In a closed organ pipe, what type of waves are formed?
Explanation: Sound waves in organ pipes are longitudinal waves, and due to reflection at boundaries, they form longitudinal stationary waves.
9. A wave pulse is represented by $y = \frac{1}{1 + (x - vt)^2}$. What does the linear nature of the medium imply for two such pulses meeting?
Explanation: In a linear medium, superimposing pulses add algebraically during overlap and emerge unaltered in shape and velocity afterward.
10. The resultant intensity at a point in an interference pattern is $I$. If the phase difference between the two equal component waves (each of intensity $I_0$) is $\frac{\pi}{2}$, what is $I$?