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Waves and Oscillations
Plane Waves
A plane wave is a wave of constant frequency whose wavefronts are infinite parallel planes normal to the direction of travel. Its physical description relies on solving the linear wave equation in three dimensions.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic wave kinematics, phase velocity, wavelength, frequency, relation between particle velocity and wave velocity in transverse waves, intensity and energy density.
PGT extension
Derivation of the 3D wave equation from Maxwell's equations or elastic continuum, complex wave representation, acoustic impedance, energy flux vector, and boundary conditions for plane waves.
Prerequisites
Simple Harmonic Motion (SHM)
Partial differential equations and chain rule
Hooke's law and elasticity moduli
Definition and Wavefront Geometry
Wavefronts are flat, parallel planes perpendicular to the propagation vector k.
All points on a given plane share the identical phase at any instant of time t.
Amplitude and phase remain constant across any plane perpendicular to the direction of motion.
Mathematical Representation
Displacement is expressed as y(x,t) = A sin(kx - omega t + phi) in one dimension.
In three dimensions, the spatial phase term kx is replaced by the dot product k·r.
The wave vector k has a magnitude equal to 2π/λ and points in the direction of propagation.
Particle Velocity versus Wave Velocity
Wave velocity ($v = \omega/k)$ is the speed at which the phase propagates through the medium.
Particle velocity ($v_p = \partial y/\partial t)$ is the instantaneous transverse speed of a medium element.
The maximum particle velocity is given by v_{p,max} = Aomega, which can exceed wave velocity depending on amplitude.
Energy Transport and Intensity
Energy density (E) in a plane wave is the sum of kinetic and potential energy densities, equal to 2pi^2 m' A^2 \nu^2.
Intensity (I) is defined as the average power transmitted per unit area normal to the wave vector.
Intensity is directly proportional to the square of the wave amplitude (I propto A^2) and square of frequency (\nu^2).
Derivations
Derivation of the One-Dimensional Wave Equation
Conditions: Ideal uniform string or acoustic medium undergoing small transverse displacements
Consider a small segment of a string of mass per unit length mu under constant tension T.
Apply Newton's second law to the vertical forces: sum $F_y = T \sin(\theta + d\theta) - T \sin\theta = (\mu dx) \frac{\partial^2 y}{\partial t^2}$.
For small angles, substitute sin\theta approx \tan\theta = \frac{partial y}{partial x}, yielding T left[ left(\frac{partial y}{partial x}\right)_{x+dx} - left(\frac{partial y}{partial x}\right)_x \right] = mu dx \frac{partial^2 y}{partial t^2}.
Divide by T dx and take the limit as dx \to 0 to obtain \frac{partial^2 y}{partial $x^2 = \frac{\mu}{T} \frac{\partial^2 y}{\partial t^2}$.
Result: The 1D wave equation \frac{partial^2 y}{partial $x^2 = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$ where $v = \sqrt{T/\mu}$.
Relation between Particle Velocity and Wave Velocity
Differentiate the displacement equation with respect to time t to find particle velocity: $v_p = \frac{\partial y}{\partial t} = -Aomega \cos(kx - \omega t)$.
Differentiate the displacement equation with respect to space x to find spatial gradient: \frac{partial y}{partial x} = kA cos(kx - omega t).
Compare the two expressions by noting that Aomega cos(kx - omega t) = -v (kA cos(kx - omega t)) since $v = \omega/k$.
y = displacement, A = amplitude, k = wave number, omega = angular frequency, phi = initial phase m
Linear homogeneous medium
Wave Number
$$k = \frac{2\pi}{\lambda}$$
k = angular wave number, lambda = wavelength rad/m
Uniform medium
Phase Velocity
$$v = \frac{\omega}{k} = \nu \lambda$$
v = phase velocity, omega = angular frequency, \nu = frequency, lambda = wavelength m/s
Non-dispersive medium
Particle Velocity
$$v_p = -A\omega \cos(kx - \omega t)$$
$v_p = particle velocity$, A = amplitude, omega = angular frequency m/s
Transverse harmonic wave
Wave Speed on a Stretched String
$$v = \sqrt{\frac{T}{\mu}}$$
T = tension, mu = mass per unit length m/s
Flexible string
Speed of Sound in Solids (Longitudinal Plane Wave)
$$v = \sqrt{\frac{Y}{\rho}}$$
Y = Young's modulus, \rho = density m/s
Extended solid medium
Intensity of a Plane Wave
$$I = 2\pi^2 \rho v \nu^2 A^2$$
I = intensity, \rho = density, v = wave speed, \nu = frequency, A = amplitude W/m^2
Harmonic progressive wave
Acoustic Impedance for Plane Waves
$$Z = \rho v$$
Z = characteristic impedance, \rho = density, v = wave speed kg/(m^2\cdot s)
Plane acoustic wave
Graphs and experiments
Displacement vs Position Snapshot (y vs x at t = 0)
Draw: A sinusoidal curve plotted along the x-axis with peak amplitude +A and trough -A, labeling wavelength lambda between successive crests.
Interpret: Represents the spatial profile of the wave at a single instant in time.
Exam use: Tested for identifying wavelength, amplitude, and spatial nodes/antinodes.
Displacement vs Time History (y vs t at x = 0)
Draw: A sinusoidal curve plotted along the time axis t, marking the time period T between successive identical phase points.
Interpret: Shows the simple harmonic motion executed by a single fixed particle in the medium as the wave passes.
Exam use: Used to evaluate frequency \nu and angular frequency omega.
Kundt's Tube Experiment
Draw: A horizontal glass tube containing lycopodium powder, with a piston at one end and a sounding rod clamped at the other, showing standing wave nodes and antinodes.
Interpret: Demonstrates the formation of stationary plane waves by superposition of incident and reflected sound waves, used to measure the wavelength of sound in air or rods.
Exam use: Core experimental setup for calculating speed of sound in gases.
Solved numericals
TGT core
A plane progressive wave is represented by $y = 0.05 \sin(100pi t - 2pi x)$, where x and y are in meters and t in seconds. Calculate the frequency and wavelength of the wave.
Compare the given equation $y = 0.05 \sin(100pi t - 2pi x) with the standard form y = A \sin(\omega t - kx)$.
Equate angular frequency: omega = 100pi. Since omega = 2pi\nu, we get 2pi\nu = 100pi implies \nu = 50 \text{ Hz}.
Equate wave number: k = 2pi. Since $k = 2pi/\lambda$, we get 2pi/lambda = 2pi implies lambda = 1 \text{ m}.
Answer: Frequency = 50 Hz, Wavelength = 1 m
TGT core
Find the speed of a transverse wave on a stretched string of length 2 m and total mass 0.04 kg under a tension of 16 N.
Calculate the mass per unit length (mu): m$u = M / L = 0.04 \text{ kg} / 2 \text{ m} = 0.02 \text{ kg/m}$.
Substitute the values of tension T = 16 N and mu into the formula $v = \sqrt{T/\mu}$.
A plane sound wave in air has an intensity of 1.0 \times 10^{-6} \text{ W/m}^2. If the density of air is 1.2 kg/m^3 and the speed of sound is 340 m/s, calculate the amplitude of particle displacement at a frequency of 1000 Hz.
Use the intensity formula for plane waves: $I = 2pi^2 \rho v \nu^2 A^2$.
Rearrange to solve for amplitude A: $A = \sqrt{\frac{I}{2pi^2 \rho v \nu^2}}$.
The medium is assumed to be infinite, homogeneous, and isotropic unless boundary conditions are explicitly introduced.
Displacements are assumed to be infinitesimally small so that linear wave equations hold true without nonlinear acoustic effects.
Attenuation and absorption losses due to viscosity and thermal conduction are neglected in ideal plane wave formulations.
Common misconceptions
Common error: Particles of the medium travel along with the plane wave from one point to another.
Correct: Only energy and momentum propagate forward; individual medium particles oscillate about their equilibrium positions.
Wave motion is a propagation of state and disturbance, not bulk transport of matter.
Common error: Particle velocity and wave velocity are the same physical quantity.
Correct: Wave velocity is the speed of phase propagation, whereas particle velocity is the instantaneous vibrational speed of a mass element.
Wave velocity depends on elastic and inertial properties of the medium, while particle velocity depends on wave amplitude and time.
Common error: The amplitude of an ideal plane wave decreases with distance from the source.
Correct: An ideal plane wave maintains constant amplitude in all parallel planes as it propagates through a lossless medium.
Spherical and cylindrical waves experience geometric attenuation (1/r or 1/sqrt{r}), whereas plane waves do not spread out geometrically.
Quick revision
Plane wave wavefronts are infinite parallel planes perpendicular to the wave vector k.
The fundamental wave equation is partial^2 y / partial $x^2 = (1/v^2) \partial^2 y / \partial t^2$.
Phase velocity is given by $v = \omega/k = ulambda$.
Maximum particle velocity equals Aomega, which is generally different from wave speed v.
Intensity of a plane progressive wave is proportional to A^2 and \nu^2.
Must remember
Wavefronts of a plane wave are flat, parallel planes perpendicular to the propagation vector $\mathbf{k}$.
The phase velocity of a plane wave is given by $v = \frac{\omega}{k} = \nu \lambda$.
Maximum particle velocity in a harmonic plane wave is $v_{p,\text{max}} = A\omega$, which is distinct from wave velocity.
Intensity of a plane wave is directly proportional to the square of the amplitude and square of the frequency ($I \propto A^2 \nu^2$).
Ideal plane waves do not suffer geometric attenuation; their amplitude remains constant across planes perpendicular to propagation.
Glossary
Wave Vector ($\mathbf{k}$)A vector whose magnitude is $k = \frac{2\pi}{\lambda}$ and whose direction indicates the direction of wave propagation.
Phase Velocity ($v$)The speed at which a point of constant phase of the wave propagates through the medium, given by $v = \frac{\omega}{k}$.
Particle Velocity ($v_p$)The instantaneous transverse or longitudinal speed of a medium element about its equilibrium position, given by $v_p = \frac{\partial y}{\partial t}$.
Acoustic Impedance ($Z$)The product of medium density $\rho$ and wave speed $v$, expressed as $Z = \rho v$ for plane acoustic waves.
Intensity ($I$)The average power transmitted per unit area normal to the direction of wave propagation.
Wave Velocity vs. Particle Velocity
Parameter
Wave Velocity ($v$)
Particle Velocity ($v_p$)
Nature
Speed of propagation of phase/disturbance
Instantaneous vibrational speed of a medium particle
Dependence
Depends on elastic and inertial properties ($v = \sqrt{T/\mu}$ or $\sqrt{Y/\rho}$)
Depends on wave amplitude, frequency, and time ($v_p = -A\omega \cos(kx - \omega t)$)
Transport
Represents transport of energy and momentum
Represents simple harmonic oscillation around a fixed equilibrium point
Value
Constant for a uniform, non-dispersive medium
Varies continuously from zero to a maximum value of $A\omega$
Practice Quiz
Choose an option to reveal the explanation.
1. What is the shape of wavefronts for an ideal plane wave in a homogeneous medium?
Explanation: Wavefronts of a plane wave are defined as flat, parallel planes perpendicular to the propagation vector $\mathbf{k}$, where all points share an identical phase.
2. In a three-dimensional plane wave, how is the spatial phase term $kx$ modified?
Explanation: In three dimensions, the one-dimensional spatial phase term $kx$ is replaced by the dot product $\mathbf{k} \cdot \mathbf{r}$ of the wave vector and position vector.
3. What is the relationship between the magnitude of the wave vector $k$ and the wavelength $\lambda$?
Explanation: The magnitude of the wave vector is defined as $k = \frac{2\pi}{\lambda}$, representing the phase change per unit distance.
4. Which of the following expressions correctly gives the maximum particle velocity $v_{p,\text{max}}$ for a transverse harmonic plane wave?
Explanation: Particle velocity is given by $v_p = \frac{\partial y}{\partial t} = -A\omega \cos(kx - \omega t)$, making the maximum particle velocity $v_{p,\text{max}} = A\omega$.
5. How does the intensity $I$ of an ideal plane wave vary with its amplitude $A$ and frequency $\nu$?
Explanation: The intensity of a plane wave is given by $I = 2\pi^2 \rho v \nu^2 A^2$, which shows that $I \propto A^2 \nu^2$.
6. What happens to the amplitude of an ideal plane wave as it propagates through a lossless medium?
Explanation: Unlike spherical or cylindrical waves which experience geometric attenuation, an ideal plane wave maintains a constant amplitude across any plane perpendicular to the direction of motion.
7. What does wave velocity ($v = \frac{\omega}{k}$) physically represent?
Explanation: Wave velocity is the speed at which a state of constant phase propagates through the medium, distinct from particle velocity.
8. What is the formula for the speed of transverse waves on a stretched string?
Explanation: The wave speed on a flexible string under tension $T$ with mass per unit length $\mu$ is $v = \sqrt{\frac{T}{\mu}}$.
9. What is the formula for the speed of longitudinal plane waves in an extended solid medium?
Explanation: The longitudinal wave speed in an extended solid depends on Young's modulus $Y$ and density $\rho$, given by $v = \sqrt{\frac{Y}{\rho}}$.
10. What is the acoustic impedance $Z$ for a plane acoustic wave in terms of density $\rho$ and wave speed $v$?
Explanation: Characteristic acoustic impedance for plane waves is defined as $Z = \rho v$, representing the resistance of the medium to acoustic wave propagation.
11. Which of the following statements about medium particles in a progressive plane wave is correct?
Explanation: Wave motion propagates energy and momentum without bulk transport of matter; individual particles only oscillate about their equilibrium positions.
12. If the frequency of a plane wave is doubled while keeping the medium properties constant, what happens to its phase velocity?
Explanation: In a non-dispersive medium, phase velocity is determined by the elastic and inertial properties of the medium and remains constant independent of frequency (since wavelength adjusts inversely).
13. What is the SI unit of acoustic impedance $Z$?
Explanation: Acoustic impedance is $Z = \rho v$, having units of density multiplied by velocity, which equals $\text{kg}/(\text{m}^2 \cdot \text{s})$ or equivalently $\text{Pa} \cdot \text{s}/\text{m}$ (or $\text{N} \cdot \text{s}/\text{m}^3$). Wait, $\text{kg}/(\text{m}^3) \times \text{m}/\text{s} = \text{kg}/(\text{m}^2 \cdot \text{s})$. Let's check unit compatibility: options b and c are standard.
14. In the one-dimensional displacement equation $y(x,t) = A \sin(kx - \omega t + \phi)$, what role does $\phi$ play?
Explanation: The parameter $\phi$ is the initial phase constant, establishing the initial state of oscillation at the origin.
15. Can the maximum particle velocity in a wave exceed the wave velocity?
Explanation: Since $v_{p,\text{max}} = A\omega$ and wave velocity $v = \frac{\omega}{k}$, their ratio is $A\omega k$, which can be greater than 1 for large amplitudes.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The displacement of a plane progressive wave is given by $y = 0.05 \sin(200\pi t - 4\pi x)$, where $x$ is in meters and $t$ in seconds. What is the phase velocity of the wave?
Explanation: Comparing with $y = A \sin(\omega t - kx)$, we have $\omega = 200\pi \text{ rad/s}$ and $k = 4\pi \text{ rad/m}$. Phase velocity $v = \frac{\omega}{k} = \frac{200\pi}{4\pi} = 50 \text{ m/s}$.
2. If the tension in a stretched string is increased by a factor of 4, what happens to the transverse wave speed on the string?
Explanation: Wave speed on a string is $v = \sqrt{\frac{T}{\mu}}$. If tension $T$ becomes $4T$, the new speed is $v' = \sqrt{\frac{4T}{\mu}} = 2\sqrt{\frac{T}{\mu}} = 2v$.
3. A plane wave is represented in 3D by the displacement function $\psi(\mathbf{r}, t) = A \cos(\mathbf{k} \cdot \mathbf{r} - \omega t)$. What is the physical significance of the vector $\mathbf{k}$?
Explanation: The wave vector $\mathbf{k}$ is normal to the flat parallel planes of constant phase (wavefronts), and its magnitude is $k = \frac{2\pi}{\lambda}$.
4. The maximum particle velocity of a harmonic plane wave is three times the wave velocity. If the wavelength is $\lambda$, what is the amplitude $A$ of the wave?
Explanation: We are given $v_{p,\text{max}} = 3v$, which means $A\omega = 3 \left(\frac{\omega}{k}\right)$. Therefore, $A = \frac{3}{k} = \frac{3}{2\pi/\lambda} = \frac{3}{2\pi}\lambda$.
5. Two plane waves of identical frequency travel in the same medium. Wave 1 has double the amplitude of Wave 2 ($A_1 = 2A_2$). What is the ratio of their intensities ($I_1 / I_2$)?
Explanation: Intensity is directly proportional to the square of the amplitude ($I \propto A^2$). Thus, $\frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 = (2)^2 = 4$.
6. Which of the following properties distinguishes plane waves from spherical waves as they propagate through a lossless medium?
Explanation: Plane waves do not experience geometric spreading attenuation, so their amplitude remains constant, unlike spherical waves where intensity and amplitude drop with distance due to wavefront area expansion.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In a continuous elastic medium, a longitudinal plane wave propagates with speed $v$. If the bulk modulus of the medium is $B$ and density is $\rho$, what is $v$?
Explanation: The speed of longitudinal waves in a fluid medium (or bulk medium) is given by $v = \sqrt{\frac{B}{\rho}}$, where $B$ is the bulk modulus and $\rho$ is density.
2. What is the phase difference between two points on a plane wave separated by a distance $\Delta x$ along the direction of propagation?
Explanation: The spatial phase change over a distance $\Delta x$ in a plane wave is given by $\Delta \theta = k \Delta x$, since the phase term is $(kx - \omega t)$.
3. If the particle velocity in a plane wave is given by $v_p = -A\omega \cos(kx - \omega t)$, at what position within a wavelength is the particle acceleration maximum?
Explanation: Particle acceleration is $a_p = \frac{\partial v_p}{\partial t} = -A\omega^2 \sin(kx - \omega t)$, which has its maximum magnitude when displacement $y = A \sin(kx - \omega t)$ is at its maximum (extremity of oscillation).
4. A plane wave travels in a medium with characteristic acoustic impedance $Z$. If the particle velocity amplitude is $u_0$, what is the acoustic intensity $I$?
Explanation: Acoustic intensity for a plane wave is related to characteristic impedance $Z = \rho v$ and particle velocity amplitude $u_0$ by $I = \frac{1}{2} \rho v u_0^2 = \frac{1}{2} Z u_0^2$.
5. Which of the following represents a valid mathematical wave function for a harmonic plane wave traveling in the negative $x$-direction?
Explanation: A wave traveling in the negative $x$-direction is represented by replacing $kx$ with $kx + \omega t$ (or $kx + \omega t + \phi$), giving $y(x,t) = A \sin(kx + \omega t)$.
6. What is the total energy density $E$ of a plane progressive wave of amplitude $A$, frequency $\nu$, and medium density $\rho$?
Explanation: As stated in the study notes, energy density $E$ in a plane wave is $2\pi^2 m' A^2 \nu^2$, which in terms of mass density $\rho$ corresponds to $2\pi^2 \rho A^2 \nu^2$ (sum of kinetic and potential energy densities).
7. If the wavelength of a plane wave is halved and its frequency is doubled, what happens to the phase velocity in a non-dispersive medium?
Explanation: Phase velocity $v = \nu \lambda$. If frequency is doubled ($2\nu$) and wavelength is halved ($1/2 \lambda$), their product $2\nu \times \frac{1}{2}\lambda = \nu \lambda$ remains constant.
8. Why does the concept of a strictly infinite plane wave remain an idealized model in physics?
Explanation: An ideal plane wave extending infinitely across every plane perpendicular to propagation would possess infinite total energy, which is physically impossible. Real sources produce wave packets or beams that approximate plane waves locally.
9. How are particle velocity ($v_p$) and wave velocity ($v$) related in a transverse harmonic wave $y(x,t) = A \sin(kx - \omega t)$?
Explanation: Since $\frac{\partial y}{\partial x} = Ak \cos(kx - \omega t)$ and $v_p = -A\omega \cos(kx - \omega t)$, and noting $\omega = vk$, we have $v_p = -v \left(\frac{\partial y}{\partial x}\right)$.
10. What does a constant phase surface $\mathbf{k} \cdot \mathbf{r} - \omega t = \text{constant}$ represent geometrically in three dimensions?
Explanation: The equation $\mathbf{k} \cdot \mathbf{r} = \text{constant}$ represents a set of flat, parallel planes perpendicular to the wave vector $\mathbf{k}$.