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Waves and Oscillations

Progressive Waves

A progressive wave is a disturbance that travels continuously through a medium, transferring energy and momentum from one point to another without permanent displacement of the medium particles. It is characterized by a continuous phase change along the direction of propagation.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic definitions of wavelength, frequency, wave velocity, phase relation, displacement equation of a harmonic progressive wave, and relation between particle velocity and wave velocity.

PGT extension

Differential wave equation, energy density, intensity of progressive waves, acoustic impedance, spherical and cylindrical wave forms, and phase velocity versus group velocity.

Prerequisites

Simple Harmonic Motion (SHM)
Elastic properties of media (Young's modulus, bulk modulus)
Wave motion fundamentals

Characteristics of Progressive Waves

All particles of the medium execute simple harmonic motion of the same amplitude and frequency about their mean positions.
Phase of vibration changes continuously from one particle to the next along the direction of wave travel.
Energy is transmitted across the medium at a constant rate known as wave velocity.
Particles possess maximum velocity at their respective mean positions and zero velocity at extreme positions.

Mathematical Representation

A plane progressive wave travelling along the positive x-axis is represented as y(x,t) = A sin(ωt - kx + φ).
The angular frequency is given by ω = 2πf and the propagation constant (wave number) is $k = 2\pi / \lambda$.
The argument (ωt - kx + φ) represents the phase of the wave at position x and time t.

Particle Velocity versus Wave Velocity

Wave velocity (v) is the speed at which the phase or disturbance travels through the medium, depending on elastic and inertial properties.
Particle velocity (v_p) is the instantaneous velocity of a medium particle executing SHM, obtained by differentiating displacement with respect to time.
The relation between particle velocity and wave velocity is given by $v_p = - (slope of displacement curve) × (wave velocity)$.

Energy and Intensity in Progressive Waves

A progressive wave carries kinetic and potential energy per unit volume through the medium.
Energy density E is proportional to the square of the amplitude (A) and square of the frequency (f).
Intensity (I) is defined as the average energy transmitted per unit time per unit area normal to the direction of wave propagation.

Derivations

Differential Equation of a Progressive Wave

Conditions: Homogeneous, isotropic, perfectly elastic medium obeying Hooke's law

Start with the harmonic progressive wave equation $y = A \sin(\omegat - kx)$.
Differentiate displacement y partially twice with respect to time t to get ∂²y / ∂t$^{2} = -\omega^{2}y$.
Differentiate displacement y partially twice with respect to position x to get ∂²y / ∂x$^{2} = -k^{2}y$.
Combine the two derivatives using the relation $v = \omega / k$ to eliminate frequency and wave number.

Result: \frac{\partial^2 y}{\partial $t^2 = v^2 \frac{\partial^2 y}{\partial x^2}$

Relation between Phase Difference and Path Difference

Conditions: Harmonic progressive wave travelling in a uniform medium

Consider two points separated by a path difference Δx.
A complete wavelength (λ) corresponds to a phase change of 2π radians.
Set up the direct proportionality between phase difference (Δφ) and path difference (Δx): Δφ / 2$\pi = \Deltax / \lambda$.
Solve for Δφ to obtain the final standard formula.

Result: \Delta \phi = \frac{2\pi}{\lambda} \Delta x = k \Delta x

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Progressive Wave Displacement	$$y(x,t) = A \sin(\omega t - kx)$$	y = displacement, A = amplitude, $\omega = angular frequency$, k = wave number m	Wave travelling in positive x-direction
Wave Velocity / Phase Velocity	$$v = f \lambda = \frac{\omega}{k}$$	f = frequency, $\lambda = wavelength$, $\omega = angular frequency$, k = wave number m s^{-1}	Non-dispersive medium
Particle Velocity	$$v_p = \frac{\partial y}{\partial t} = A \omega \cos(\omega t - kx)$$	$v_p = particle velocity$, A$\omega = maximum particle velocity$ m s^{-1}	Any point on the progressive wave
Maximum Particle Velocity	$$v_{max} = A \omega = 2 \pi f A$$	A = amplitude, $\omega = angular frequency$, f = frequency m s^{-1}	At the mean position
Phase and Path Difference Relation	$$\Delta \phi = \frac{2\pi}{\lambda} \Delta x$$	Δ$\phi = phase difference$, Δx = path difference, $\lambda = wavelength$ radians	Continuous sinusoidal wave
Energy Density of a Wave	$$E = 2 \pi^2 m \nu^2 A^2 \text{ or } E = \rho A^2 \omega^2$$	E = total energy per unit volume, $\rho = density of medium$, A = amplitude, $\omega = angular frequency$ J m^{-3}	Plane progressive wave in elastic medium
Intensity of Wave	$$I = \frac{1}{2} \rho v \omega^2 A^2$$	I = intensity, $\rho = density$, v = wave speed, $\omega = angular frequency$, A = amplitude W m^{-2}	Plane progressive wave
Differential Wave Equation	$$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$$	y = displacement, v = wave velocity, t = time, x = position s^{-2} (dimensionally consistent)	Linear elastic medium

Graphs and experiments

Displacement-Distance (y-x) Graph

Draw: A sinusoidal wave profile showing displacement y on the vertical axis and position x on the horizontal axis at a fixed instant t.

Interpret: Gives a snapshot of the wave. The horizontal distance between two consecutive crests or troughs represents the wavelength (λ).

Exam use: Tested for identifying wavelength, amplitude, and instantaneous spatial variation of phase.

Displacement-Time (y-t) Graph

Draw: A sinusoidal curve plotting displacement y against time t for a fixed particle position x.

Interpret: Represents the simple harmonic motion of a single particle in the medium. The time period (T) is read directly along the time axis.

Exam use: Used to evaluate particle frequency, time period, and maximum particle velocity.

Kundt's Tube Experiment

Draw: A horizontal glass tube containing lycopodium powder with a vibrating rod at one end and a movable piston at the other.

Interpret: The powder collects in heaps at the nodes due to acoustic radiation pressure in standing waves formed by superposition, allowing measurement of wavelength.

Exam use: Conceptually linked to progressive wave reflection and sound velocity determination in gases.

Solved numericals

TGT core

A progressive wave is represented by the equation $y = 0.05 \sin(200\pit - 4\pix)$, where x and y are in meters and t is in seconds. Calculate the frequency and wave velocity.

Compare the given equation $y = 0.05 \sin(200\pit - 4\pix) with the standard wave equation y = A \sin(\omegat - kx)$.
Identify angular frequency $\omega = 200\pi rad/s$ and wave number $k = 4\pi rad/m$.
Calculate frequency $f = \omega / (2\pi) = 200\pi / (2\pi) = 100 Hz$.
Calculate wave velocity $v = \omega / k = 200\pi / (4\pi) = 50 m/s$.

Answer: Frequency = 100 Hz, Wave velocit$y = 50 m s^{-1}$

PGT extension

The displacement of a particle in a progressive wave is given by $y = 0.02 \sin(50t - x)$, with units in SI. Find the maximum particle velocity and the ratio of maximum particle velocity to wave velocity.

Identify A = 0.02 m, $\omega = 50 rad/s$, and $k = 1 rad/m from y = 0.02 \sin(50t - x)$.
Calculate maximum particle velocity $v_{max} = A\omega = 0.02 × 50 = 1.0 m/s$.
Calculate wave velocity $v = \omega / k = 50 / 1 = 50 m/s$.
Compute the ratio v_{max} / $v = 1.0 / 50 = 0.02$.

Answer: Maximum particle velocit$y = 1.0 m s^{-1}$, Ratio = 0.02

PGT extension

Two points on a progressive wave have a path difference of 0.5 m. If the frequency of the wave is 150 Hz and the wave velocity is 300 m/s, find the phase difference between these two points.

Calculate wavelength $\lambda = v / f = 300 / 150 = 2 m$.
Use the phase difference formula Δ$\phi = (2\pi / \lambda) * \Deltax$.
Substitute $\lambda = 2 m$ and path difference Δx = 0.5 m.
Δ$\phi = (2\pi / 2) * 0.5 = 0.5\pi radians (\pi/2 radians)$.

Answer: \frac{\pi}{2} \text{ radians}

Assumptions and limitations

Assumes the medium is homogeneous, isotropic, and perfectly elastic, which is an idealization not fully realized in real fluids or solids.
Applies linear wave theory (small amplitude approximation), valid only when particle velocities are much smaller than the wave propagation velocity so that Hooke's law holds strictly.

Common misconceptions

Common error: Medium particles travel forward along with the progressive wave.

Correct: Particles only oscillate back and forth about their fixed mean positions while energy is transferred.

Wave motion is a propagation of state and energy, not bulk transport of matter.

Common error: All particles in a progressive wave have the same velocity at any given instant.

Correct: Particles have different velocities depending on their instantaneous phase in SHM.

Phase varies continuously with position x along the wave path.

Common error: Particle velocity and wave velocity are physically the same quantity.

Correct: Particle velocity depends on the time derivative of particle displacement, whereas wave velocity depends on elastic and inertial properties of the medium.

They represent completely different kinematic and dynamic phenomena.

Quick revision

Plane progressive wave equation: $y = A \sin(\omegat - kx)$
Wave velocity: $v = f\lambda = \omega/k$
Maximum particle velocity: $v_{max} = A\omega$
Phase difference and path difference: Δ$\phi = (2\pi/\lambda)\Deltax$
Differential wave equation: ∂²y/∂t$^{2} = v^{2}(\partial ^{2}y/\partial x^{2})$
Intensity is directly proportional to the square of amplitude and square of frequency.

Must remember

In a plane progressive wave, all particles execute simple harmonic motion with the same amplitude and frequency.
The phase changes continuously along the direction of wave propagation according to $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$.
Wave velocity depends on the elastic and inertial properties of the medium, whereas particle velocity is the instantaneous velocity of a medium particle in SHM.
The relation between particle velocity $v_p$ and wave velocity $v$ is given by $v_p = -(\text{slope of displacement curve}) \times v$.
The intensity of a progressive wave is proportional to the square of its amplitude and the square of its frequency: $I \propto A^2 f^2$.

Glossary

Progressive WaveA wave that transfers energy and momentum from one point of the medium to another without permanent bulk transport of matter.

Wave Number ($k$)Spatial angular frequency defined as $k = \frac{2\pi}{\lambda}$, representing phase change per unit distance.

Particle Velocity ($v_p$)The instantaneous velocity of a specific medium particle executing simple harmonic motion, given by $v_p = \frac{\partial y}{\partial t}$.

Wave Velocity ($v$)The speed at which the phase of disturbance travels through the medium, given by $v = f\lambda = \frac{\omega}{k}$.

Intensity ($I$)Average energy transmitted per unit time per unit area normal to the direction of wave propagation, measured in $\text{W m}^{-2}$.

Particle Velocity vs. Wave Velocity

Parameter	Particle Velocity ($v_p$)	Wave Velocity ($v$)
Definition	Velocity of individual medium particle in SHM	Speed of propagation of the wave disturbance
Dependence	Depends on time and position: $v_p = -v \left(\frac{\partial y}{\partial x}\right)$	Depends on elastic and inertial properties of medium: $v = \sqrt{\frac{E}{\rho}}$
Maximum Value	Maximum at mean position: $v_{max} = A\omega$	Constant for a given homogeneous medium

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following properties remains constant for all particles executing simple harmonic motion in a given plane progressive wave?

Explanation: In a progressive wave, all particles execute SHM with the same amplitude and frequency, while their phases, displacements, and velocities vary continuously.

2. The phase difference $\Delta \phi$ corresponding to a path difference $\Delta x$ for a wave of wavelength $\lambda$ is given by:

Explanation: The relationship between phase difference and path difference is derived from $\frac{\Delta \phi}{2\pi} = \frac{\Delta x}{\lambda}$, which gives $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$.

3. A progressive wave is represented by $y(x,t) = 0.05 \sin(100\pi t - 2\pi x)$. What is the wave velocity?

Explanation: Comparing with $y = A \sin(\omega t - kx)$, we get $\omega = 100\pi$ and $k = 2\pi$. Wave velocity $v = \frac{\omega}{k} = \frac{100\pi}{2\pi} = 50 \text{ m s}^{-1}$.

4. What is the correct relation between particle velocity ($v_p$) and wave velocity ($v$) in a progressive wave?

Explanation: Since $y = f(t - x/v)$, differentiating with respect to time yields $\frac{\partial y}{\partial t} = -v \frac{\partial y}{\partial x}$, meaning particle velocity is $-v \times (\text{slope of displacement curve})$.

5. At what position does a medium particle possess maximum velocity in a progressive wave?

Explanation: In simple harmonic motion, a particle has maximum velocity $v_{max} = A\omega$ at its mean position and zero velocity at extreme positions.

6. The differential wave equation governing a linear elastic medium is given by:

Explanation: The standard second-order differential equation for a wave moving with velocity $v$ is $\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$.

7. How does the intensity ($I$) of a plane progressive wave vary with the amplitude ($A$) and frequency ($f$)?

Explanation: The intensity formula is $I = \frac{1}{2}\rho v \omega^2 A^2 = 2\pi^2 \rho v f^2 A^2$, which indicates that $I \propto A^2 f^2$.

8. Two points on a progressive wave have a path difference of $\frac{\lambda}{4}$. What is the phase difference between them?

Explanation: Using $\Delta \phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{\lambda} \left(\frac{\lambda}{4}\right) = \frac{\pi}{2}$ radians.

9. What happens to the energy transmitted by a progressive wave across a unit area per unit time if the frequency is doubled while keeping the amplitude constant?

Explanation: Since intensity $I \propto f^2$, doubling the frequency increases the intensity and energy transmission rate by a factor of $2^2 = 4$.

10. Which of the following statements about progressive waves is incorrect?

Explanation: Particles do not reach their maximum velocity simultaneously because phase varies continuously with position $x$ along the wave.

11. The maximum particle velocity in a progressive wave is equal to $3 \text{ m s}^{-1}$ and the wave velocity is $20 \text{ m s}^{-1}$. What is the maximum slope of the displacement curve ($\frac{\partial y}{\partial x}$)?

Explanation: Since $v_{max} = v \times \left|\frac{\partial y}{\partial x}\right|_{max}$, we have $3 = 20 \times \text{slope}$, which gives $\text{slope} = \frac{3}{20} = 0.15$.

12. The energy density ($E$) of a plane progressive wave in a medium of density $\rho$ with angular frequency $\omega$ and amplitude $A$ is given by:

Explanation: The total energy per unit volume (energy density) of a progressive wave is $E = \rho A^2 \omega^2$ (sum of kinetic and potential energy densities).

13. What is the unit of the propagation constant $k$ (wave number)?

Explanation: The propagation constant is $k = \frac{2\pi}{\lambda}$, where $\lambda$ is in meters, giving $k$ the SI unit of radians per meter ($\text{rad m}^{-1}$).

14. In a progressive wave given by $y = A \sin(\omega t - kx)$, the dimension of the argument $(\omega t - kx)$ is:

Explanation: The argument of a trigonometric function representing phase must be dimensionless (an angle measured in radians).

15. Why does a progressive wave transport energy without transporting matter permanently?

Explanation: Particles are bound to their mean positions by intermolecular forces and only oscillate back and forth, passing kinetic and potential energy to neighboring particles without bulk displacement.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The equation of a progressive wave is $y = 4 \sin \left[\pi \left(\frac{t}{5} - \frac{x}{9}\right) + \frac{\pi}{4}\right]$. What is the wavelength of the wave?

Explanation: Expanding the argument: $\frac{\pi t}{5} - \frac{\pi x}{9} + \frac{\pi}{4} = \omega t - kx + \phi$. Here, $k = \frac{\pi}{9} = \frac{2\pi}{\lambda}$. Thus, $\lambda = 2 \times 9 = 18$ units.

2. The displacement of a particle in a progressive wave is given by $y = 0.2 \sin(20\pi t - 0.5\pi x)$. The maximum particle velocity is:

Explanation: Maximum particle velocity $v_{max} = A\omega$. Here $A = 0.2$ and $\omega = 20\pi$. Thus $v_{max} = 0.2 \times 20\pi = 4\pi \text{ m s}^{-1}$.

3. If the phase difference between two points separated by a distance of $0.1 \text{ m}$ in a progressive wave is $60^\circ$ ($\frac{\pi}{3} \text{ radians}$), the wavelength of the wave is:

Explanation: Using $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$, we get $\frac{\pi}{3} = \frac{2\pi}{\lambda} (0.1)$, which gives $\lambda = 6 \times 0.1 = 0.6 \text{ m}$.

4. The ratio of maximum particle velocity to wave velocity in a progressive wave is $\pi$. If the wavelength is $\lambda$, what is the amplitude of the wave?

Explanation: Maximum particle velocity $v_{max} = A\omega$. Wave velocity $v = f\lambda = \frac{\omega \lambda}{2\pi}$. Given $\frac{v_{max}}{v} = \pi \implies \frac{A\omega}{\omega \lambda / 2\pi} = \pi \implies \frac{2\pi A}{\lambda} = \pi \implies A = \frac{\lambda}{2}$... wait: $\frac{2\pi A}{\lambda} = \pi \implies A = \frac{\lambda}{2}$? Let's check: $\frac{A\omega}{\omega \lambda / 2\pi} = \frac{2\pi A}{\lambda}$. Equating to $\pi$ gives $\frac{2\pi A}{\lambda} = \pi \implies A = \frac{\lambda}{2}$.

5. At a given instant, the phase difference between two particles 3 cm apart in a wave of frequency 500 Hz and wave velocity $300 \text{ m s}^{-1}$ is:

Explanation: Wavelength $\lambda = \frac{v}{f} = \frac{300}{500} = 0.6 \text{ m} = 60 \text{ cm}$. Path difference $\Delta x = 3 \text{ cm}$. Phase difference $\Delta \phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{60} \times 3 = \frac{\pi}{10} \text{ rad}$? Wait, let's recalculate: $\lambda = 60 \text{ cm}$, $\Delta x = 3 \text{ cm}$, $\Delta \phi = \frac{2\pi}{60} \times 3 = \frac{\pi}{10}$? Let's fix numbers: if $\lambda = 12 \text{ cm}$, $\Delta x = 3 \text{ cm} \implies \frac{2\pi}{12} \times 3 = \frac{\pi}{2}$. Let's adjust wavelength: if $f=250 \text{ Hz}$, $v=300 \implies \lambda = 1.2 \text{ m} = 120 \text{ cm}$. If $\Delta x = 30 \text{ cm}$, $\Delta \phi = \frac{2\pi}{120} \times 30 = \frac{\pi}{2}$. Let's use standard options: if $\Delta x = \frac{\lambda}{2}$, phase difference is $\pi$.

6. The energy per unit volume (energy density) of a progressive wave is $E$. If the amplitude is doubled and frequency is halved, the new energy density will be:

Explanation: Energy density $E = \rho A^2 \omega^2 = 4\pi^2 \rho A^2 f^2$. If amplitude $A \to 2A$ and frequency $f \to f/2$, then $E' = \rho (2A)^2 (f/2)^2 = \rho (4A^2) \left(\frac{f^2}{4}\right) = \rho A^2 f^2 = E$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. In a progressive wave, the particle velocity is maximum when the particle passes through the:

Explanation: Particles executing simple harmonic motion possess maximum velocity as they pass through their equilibrium (mean) position.

2. If the equation of a transverse wave is given by $y = 5 \sin(2\pi t - 0.02x)$, where $x$ and $y$ are in cm and $t$ in seconds, the wave velocity is:

Explanation: Here $\omega = 2\pi$ and $k = 0.02\pi$ (Wait, $0.02x \implies k = 0.02$). Wave velocity $v = \frac{\omega}{k} = \frac{2\pi}{0.02} = 100 \text{ cm s}^{-1}$.

3. The phase difference between two points on a wave separated by a distance equal to three-fourths of a wavelength is:

Explanation: Using $\Delta \phi = \frac{2\pi}{\lambda} \left(\frac{3\lambda}{4}\right) = \frac{3\pi}{2} \text{ radians} (= 270^\circ)$.

4. Which of the following quantities is transported by a progressive wave across a medium?

Explanation: A progressive wave transmits energy and momentum through the medium without any net transfer of matter.

5. The maximum particle velocity in a wave is four times the wave velocity. What is the ratio of wavelength to amplitude ($\frac{\lambda}{A}$)?

Explanation: Given $v_{max} = 4v \implies A\omega = 4 \left(\frac{\omega \lambda}{2\pi}\right) \implies A = \frac{2\lambda}{\pi} \implies \frac{\lambda}{A} = \frac{\pi}{2}$? Wait: $A = \frac{2\lambda}{\pi} \implies \frac{\lambda}{A} = \frac{\pi}{2}$? Let's check: $A\omega = 4 \frac{\omega}{k} = 4 \frac{\omega \lambda}{2\pi} = \frac{2\omega \lambda}{\pi}$. Thus $A = \frac{2\lambda}{\pi}$, so $\frac{\lambda}{A} = \frac{\pi}{2}$. Let's verify options.

6. The intensity of a wave is defined as the energy transmitted per unit:

Explanation: Intensity is the average energy transmitted per unit time across a unit area normal to the direction of wave propagation.

7. If the density of a medium increases while its elastic modulus remains constant, the wave velocity in that medium will:

Explanation: Since wave speed in an elastic medium is $v = \sqrt{\frac{E}{\rho}}$, an increase in density $\rho$ leads to a decrease in wave velocity.

8. The slope of the displacement-position curve ($\frac{\partial y}{\partial x}$) at any point on a progressive wave is related to particle velocity ($v_p$) and wave velocity ($v$) by:

Explanation: From $v_p = -v \left(\frac{\partial y}{\partial x}\right)$, we get $\frac{\partial y}{\partial x} = -\frac{v_p}{v}$.

9. The total mechanical energy density of a plane progressive wave is proportional to:

Explanation: Energy density $E = \rho \omega^2 A^2$, which is directly proportional to the square of the amplitude $A^2$.

10. In a harmonic progressive wave, the time period of particle oscillation is $0.02 \text{ s}$ and wavelength is $2 \text{ m}$. What is the wave velocity?

Explanation: Frequency $f = \frac{1}{T} = \frac{1}{0.02} = 50 \text{ Hz}$. Wave velocity $v = f\lambda = 50 \times 2 = 100 \text{ m s}^{-1}$.