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Waves and Oscillations
Wave Motion
Wave motion refers to the propagation of disturbances transferring energy and momentum through a medium or vacuum without bulk transport of matter. It is mathematically characterized by space-time periodic functions obeying the linear wave equation.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic definitions of transverse and longitudinal waves, relation between wave velocity, frequency, and wavelength (v = fλ), particle velocity versus wave velocity, reflection of waves, and stationary waves.
PGT extension
Differential wave equation in 1D and 3D, phase velocity, group velocity, energy density and intensity of wave motion, acoustic impedance, and Doppler effect in sound.
Prerequisites
Simple Harmonic Motion (SHM)
Hooke's Law and Elasticity
Partial Differential Equations
Kinematics of Progressive Waves
A progressive wave travels continuously in a given direction without attenuation in an ideal medium.
The displacement y(x,t) of a particle at position x and time t for a sinusoidal wave is given by y(x,t) = A sin(kx - ωt + φ).
The phase difference Δφ between two points separated by distance Δx is given by Δ$\phi = k\Deltax = (2\pi/\lambda)\Deltax$.
Particle Velocity vs Wave Velocity
Wave velocity (v) is the speed at which the phase of the wave propagates through the medium, given by $v = f\lambda = \omega/k$.
Particle velocity (u) is the instantaneous velocity of a medium particle executing SHM, given by $u = \partial y/\partial t = -\omegaA \cos(kx - \omegat + \phi)$.
The maximum particle velocity is $u_max = \omegaA$, which is distinct from the wave velocity v.
Energy and Intensity in Wave Motion
A wave element of mass d$m = \rho dx possesses kinetic$ and potential energy as it oscillates.
The average energy density E (energy per unit volume) of a sinusoidal wave is $E = 2\pi^2 f^2 A^2 \rho = (1/2)\rho\omega^2 A^2$.
Intensity (I) is the average power transmitted per unit area perpendicular to the wave propagation direction, given by $I = Ev = (1/2)\rho\omega^2 A^2 v$.
Reflection and Boundary Conditions
Reflection from a rigid (fixed) boundary introduces a phase change of π (or 180°) and a sign inversion in the reflected displacement wave.
Reflection from a free (open) boundary occurs with zero phase change, meaning the reflected displacement wave is in phase with the incident wave.
Superposition of incident and reflected waves of equal amplitude forms standing (stationary) waves with distinct nodes and antinodes.
Derivations
One-Dimensional Classical Wave Equation
Conditions: Ideal, linear, homogeneous elastic medium obeying Newton's second law.
Consider a small string element of length dx and linear mass density μ. The tension at x is T and at x+dx is T + (∂T/∂x)dx.
Applying Newton's second law to the vertical transverse motion: (μ dx)(∂^2 y / ∂$t^2) = (T + (\partial T/\partial x)dx) \sin(\theta + d\theta) - T \sin(\theta)$.
For small angles, sin(θ) ≈ tan(θ) = ∂y/∂x, and assuming uniform tension T, the vertical force simplifies to T (∂^2 y / ∂x^2) dx.
Equating inertial force to restoring force: μ dx (∂^2 y / ∂$t^2) = T (\partial ^2 y / \partial x^2) dx$.
Z = characteristic acoustic impedance, $\rho = density$, v = wave velocity kg/(m^2 s)
Plane wave propagation in medium
Graphs and experiments
Displacement-Distance (y-x) Snapshot Graph
Draw: A sinusoidal wave curve plotted with displacement y on the vertical axis and position x on the horizontal axis at a fixed time t.
Interpret: Shows the spatial profile of the wave at one instant. The horizontal distance between successive crests represents the wavelength (λ).
Exam use: Tested for identifying wavelength, amplitude, and spatial phase relationships.
Displacement-Time (y-t) History Graph
Draw: A sinusoidal curve plotted with displacement y on the vertical axis and time t on the horizontal axis for a fixed position x.
Interpret: Shows the temporal oscillation of a specific particle in the medium. The time period (T) corresponds to the interval between successive identical phase states.
Exam use: Used to determine particle frequency, time period, and instantaneous particle velocity (∂y/∂t).
Melde's Experiment Apparatus
Draw: A taut string connected to an electrically driven tuning fork, passing over a pulley to a pan holding variable masses (tension adjustment).
Interpret: Demonstrates the formation of transverse stationary waves and verifies the relation between wave frequency, tension, and linear mass density ($v = \sqrt{T/\mu})$.
Exam use: Direct questions on transverse stationary wave modes (p-loops) under parallel and perpendicular tuning fork configurations.
Solved numericals
TGT core
A progressive wave is represented by $y = 0.05 \sin(100\pit - 2\pix)$, where x and y are in meters and t in seconds. Find the frequency and wavelength of the wave.
Compare the given equation $y = 0.05 \sin(100\pit - 2\pix) with the standard wave equation y = A \sin(\omegat - kx)$.
Identify angular frequency $\omega = 100\pi rad/s$ and wave number $k = 2\pi rad/m$.
The tension in a stretched string is increased by 21%. If the mass per unit length remains constant, calculate the percentage increase in the wave velocity.
The velocity of a transverse wave in a string is given by $v = \sqrt{T / \mu}$.
Differentiating for small fractional changes: Δv / $v = (1/2)(\DeltaT / T)$.
Given ΔT / T = 0.21, substitute this into the relation: Δv / $v = (1/2) \times 21% = 10.5%$.
Answer: 10.5% increase
PGT extension
A sound wave of frequency 500 Hz travels in air with a velocity of 340 m/s. What is the phase difference between two points on the wave separated by a distance of 34 cm?
Calculate the wavelength using $\lambda = v / f = 340 / 500 = 0.68 m = 68 cm$.
Use the phase difference formula Δ$\phi = (2\pi / \lambda) \Deltax$.
Substitute the given path difference Δx = 34 cm and $\lambda = 68 cm: \Delta\phi = (2\pi / 68) \times 34$.
Simplify the expression: Δ$\phi = \pi radians$.
Answer: π radians (or 180 degrees)
Assumptions and limitations
Assumption: The amplitude of oscillation is infinitesimally small compared to the wavelength, ensuring the linear approximation (sin θ ≈ tan θ) holds true.
Limitation: Nonlinear wave effects (such as shock wave formation and harmonic distortion) are neglected in the standard linear wave equation.
Common misconceptions
Common error: Particle velocity and wave velocity are the same physical quantity.
Correct: Particle velocity is the periodic velocity of a medium particle in SHM, whereas wave velocity is the speed at which the wave profile propagates.
Particle velocity depends on time and position within the wave cycle (maximum at equilibrium), while wave velocity is a constant property determined by the elastic and inertial properties of the medium.
Common error: Energy travels along with the individual particles in a progressive wave.
Correct: Only energy and momentum are transported forward; the medium particles only oscillate about their equilibrium mean positions.
Wave motion is a collective disturbance propagation without net displacement of matter.
Quick revision
Wave velocity depends strictly on medium properties ($v = \sqrt{T/\mu} or v = \sqrt{\gammaP/\rho})$, whereas frequency depends solely on the source.
Progressive Wave Equation: y(x,t) = A \sin(kx - \omega t)
Wave Velocity: v = f\lambda = \frac{\omega}{k}
Must remember
The wave velocity is given by $v = f\lambda = \frac{\omega}{k}$, depending entirely on the medium properties, whereas particle velocity $u = -\omega A \cos(kx - \omega t)$ is the instantaneous velocity of a medium particle executing SHM.
Maximum particle velocity ($u_{\max} = \omega A$) is independent of the wave propagation velocity, and for transverse waves in a string, $v = \sqrt{\frac{T}{\mu}}$.
Reflection from a rigid boundary introduces a phase change of $\pi$ ($180^\circ$) and a sign inversion, whereas reflection from a free boundary occurs with zero phase change.
The intensity of a harmonic progressive wave is proportional to the square of the amplitude and square of the angular frequency: $I = \frac{1}{2}\rho \omega^2 A^2 v$.
The relation between phase difference $\Delta \phi$ and path difference $\Delta x$ is expressed as $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$.
Glossary
Wave Velocity ($v$)The speed at which the phase of a wave propagates through space, determined by $v = f\lambda = \frac{\omega}{k}$.
Particle Velocity ($u$)The instantaneous transverse or longitudinal velocity of a medium particle as it oscillates about its mean position, given by $u = \frac{\partial y}{\partial t}$.
Acoustic Impedance ($Z$)The product of medium density $\rho$ and wave propagation velocity $v$, defined as $Z = \rho v$.
Wave Number ($k$)The spatial angular frequency defined as $k = \frac{2\pi}{\lambda}$, representing the phase change per unit distance.
Intensity ($I$)The average power transmitted per unit area perpendicular to the direction of wave propagation, $I = \frac{1}{2}\rho \omega^2 A^2 v$.
Wave Velocity vs. Particle Velocity
Feature
Wave Velocity ($v$)
Particle Velocity ($u$)
Nature
Speed of propagation of disturbance/phase
Instantaneous velocity of medium particles in SHM
Dependence
Depends on elastic and inertial properties of the medium
Depends on time, position, amplitude, and angular frequency
Maximum Value
Constant for a given medium ($v = \sqrt{T/\mu}$ or $\sqrt{\gamma P/\rho}$)
$u_{\max} = \omega A$
Reflection Boundaries
Boundary Type
Phase Change
Displacement Behavior
Rigid / Fixed Boundary
$\pi$ radians ($180^\circ$)
Sign inversion; forms a node at the boundary
Free / Open Boundary
Zero ($0$)
In phase with incident wave; forms an antinode at the boundary
Practice Quiz
Choose an option to reveal the explanation.
1. What is the phase change experienced by a transverse wave when it is reflected from a rigid (fixed) boundary?
Explanation: Reflection from a rigid or fixed boundary introduces a phase reversal of $\pi$ ($180^\circ$) along with a sign inversion in the displacement wave.
2. The maximum particle velocity $u_{\max}$ for a sinusoidal wave given by $y(x,t) = A \sin(kx - \omega t)$ is:
Explanation: Particle velocity is $u = \frac{\partial y}{\partial t} = -\omega A \cos(kx - \omega t)$. Its maximum magnitude is $u_{\max} = \omega A$.
3. If the path difference between two points on a progressive wave is $\frac{\lambda}{4}$, the corresponding phase difference is:
4. The velocity of a transverse wave on a stretched string depends on tension $T$ and mass per unit length $\mu$ as:
Explanation: Dimensional analysis and derivation from Newton's laws for a string segment yield $v = \sqrt{\frac{T}{\mu}}$.
5. Which of the following physical quantities is transported by a progressive wave through a medium?
Explanation: A progressive wave transports energy and momentum across the medium without any net transfer of mass or medium particles.
6. The intensity $I$ of a harmonic progressive wave is proportional to which power of the amplitude $A$?
Explanation: Wave intensity is given by $I = \frac{1}{2} \rho \omega^2 A^2 v$, showing that intensity is directly proportional to the square of the amplitude ($A^2$).
7. What is the characteristic acoustic impedance $Z$ of a medium of density $\rho$ and wave velocity $v$?
Explanation: Acoustic impedance is defined as the product of medium density and wave velocity, $Z = \rho v$.
8. Reflection of a wave from a free (open) boundary occurs with what phase change?
Explanation: At a free boundary, the medium is free to move transversely, and the reflected wave is in phase with the incident wave, meaning zero phase change.
9. In a progressive sinusoidal wave, the ratio of maximum particle velocity to wave velocity is equal to:
Explanation: Maximum particle velocity is $u_{\max} = \omega A$, and wave velocity is $v = \frac{\omega}{k}$. Thus, $\frac{u_{\max}}{v} = \frac{\omega A}{\omega/k} = k A$.
10. The Newton-Laplace formula for the speed of sound in a gas is $v = \sqrt{\frac{\gamma P}{\rho}}$. Here, $\gamma$ represents:
Explanation: In the Newton-Laplace correction, sound propagation in gases is treated as an adiabatic process, where $\gamma = \frac{C_p}{C_v}$ is the adiabatic index.
11. Which statement correctly distinguishes particle velocity from wave velocity?
Explanation: Particle velocity relates to the local oscillatory motion of particles, while wave velocity is the speed at which the wave form travels through the medium.
12. The average energy density $E$ of a sinusoidal wave in a medium of density $\rho$ is given by:
Explanation: The average energy density (energy per unit volume) for a sinusoidal wave is $E = \frac{1}{2}\rho \omega^2 A^2$.
13. What happens to the velocity of sound in an ideal gas if the absolute temperature $T$ is increased by a factor of 4?
Explanation: Since $v = \sqrt{\frac{\gamma R T}{M}}$, speed is proportional to $\sqrt{T}$. If temperature increases by 4 times, velocity increases by $\sqrt{4} = 2$.
14. The differential wave equation in one dimension is represented as:
Explanation: The standard 1D linear wave equation is $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$.
15. For a standing wave formed by the superposition of two identical waves traveling in opposite directions, nodes are points where:
Explanation: Nodes are points of destructive interference where the displacement amplitude is permanently zero.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the tension in a stretched string is increased by $21\%$, keeping its mass per unit length constant, the percentage change in the transverse wave velocity is:
2. Two waves are represented by $y_1 = A \sin(\omega t - kx)$ and $y_2 = A \cos(\omega t - kx)$. The phase difference between them is:
Explanation: Rewriting $y_2 = A \cos(\omega t - kx) = A \sin(\omega t - kx + \frac{\pi}{2})$, the phase difference is $\frac{\pi}{2}$ radians.
3. The equation of a wave is given by $y = 10 \sin(0.02x - 2t)$, where $x$ is in centimeters and $t$ in seconds. The wave velocity is:
Explanation: Comparing with $y = A \sin(kx - \omega t)$, we have $k = 0.02$ and $\omega = 2$. Wave velocity $v = \frac{\omega}{k} = \frac{2}{0.02} = 100 \text{ cm/s}$.
4. The ratio of velocities of sound in a monatomic gas ($\gamma = 5/3$) and a diatomic gas ($\gamma = 7/5$) at the same temperature is approximately:
Explanation: Speed $v = \sqrt{\frac{\gamma R T}{M}}$. For identical temperature, $\frac{v_1}{v_2} = \sqrt{\frac{\gamma_1 M_2}{\gamma_2 M_1}}$. Assuming standard molecular weights ($M_1=4, M_2=28$ for He and $N_2$ or similar comparison), calculating gives $\sqrt{\frac{5/3}{7/5} \times \frac{28}{4}} = \sqrt{\frac{25}{21} \times 7} = \sqrt{\frac{25}{3}} \approx 2.88$ ... wait, let's use equal molar mass or standard gas formula where only $\gamma$ and $M$ matter. For helium ($M=4$) and hydrogen ($M=2$), ratio is $\sqrt{\frac{5/3 / 4}{7/5 / 2}} = \sqrt{\frac{5}{12} \times \frac{10}{7}} = \sqrt{\frac{25}{42}} \approx 0.77$. If comparing same molar mass, ratio of $\sqrt{\gamma}$ is $\sqrt{\frac{5/3}{7/5}} = \sqrt{\frac{25}{21}} \approx 1.09$. Let's select $1.08$ based on $\sqrt{25/21} \approx 1.09$ or standard $\sqrt{\gamma_1/\gamma_2}$.
5. The displacement of a particle in a wave is given by $y = 0.25 \sin(3\pi t - \pi x)$. The maximum particle velocity is:
Explanation: Here $A = 0.25$ and $\omega = 3\pi$. Maximum particle velocity $u_{\max} = \omega A = (3\pi)(0.25) = 0.75\pi \text{ m/s}$.
6. At what temperature will the velocity of sound in air be double its value at $0^\circ\text{C}$?
Explanation: Since $v \propto \sqrt{T}$, to double the velocity ($v_2 = 2v_1$), temperature must increase by a factor of $2^2 = 4$. $T_2 = 4 \times 273\text{ K} = 1092\text{ K}$, which means $1092 - 273 = 819^\circ\text{C}$ ($819 \text{ K}$ absolute temperature is $4 \times 273$). Wait: $4 \times 273 = 1092 \text{ K}$, so $T_2 = 1092 \text{ K}$. Let's verify option values. Option C is $819 \text{ K}$? No, $4 \times 273 = 1092 \text{ K}$. Let's use $1092 \text{ K}$ as correct if listed, or correct the index. Let's fix correct index to match $1092 \text{ K}$ if it's option D.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The equation of a progressive wave is $y = 5 \sin \pi (t - 0.01x)$. The wavelength of the wave is:
Explanation: Expanding the argument: $y = 5 \sin(\pi t - 0.01\pi x)$. Here $k = 0.01\pi = \frac{2\pi}{\lambda}$. Thus $\lambda = \frac{2\pi}{0.01\pi} = 200 \text{ cm}$.
2. The acoustic impedance of air having density $1.2 \text{ kg/m}^3$ and sound speed $340 \text{ m/s}$ is:
7. A wave pulse is reflected from a rigid wall. The reflected pulse undergoes:
Explanation: Reflection from a rigid wall results in a phase inversion of $\pi$ radians, meaning the displacement is inverted.
8. The transverse displacement of a string is given by $y(x,t) = 0.05 \sin(10\pi t - 2\pi x)$. The wave velocity is:
Explanation: Here $\omega = 10\pi$ and $k = 2\pi$. Wave velocity $v = \frac{\omega}{k} = \frac{10\pi}{2\pi} = 5 \text{ m/s}$.
9. The energy density of a progressive wave is directly proportional to:
Explanation: Average energy density $E = \frac{1}{2}\rho \omega^2 A^2$, which is directly proportional to $A^2$.
10. Which of the following wave phenomena demonstrates that sound waves are longitudinal?
Explanation: Polarization is exclusively exhibited by transverse waves. Longitudinal waves like sound cannot be polarized because their oscillations are parallel to the direction of propagation.