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Waves and Oscillations
Group Velocity in Waves and Oscillations
Group velocity is the velocity with which the overall shape of the amplitudes of a wave packet—known as the envelope—propagates through space. It represents the speed at which wave energy and information are transmitted in a dispersive medium.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of phase and group velocity, formula derivation for non-dispersive media where v_g = v_p, and simple numerical problems involving wave number and angular frequency.
PGT extension
Advanced relationship between group velocity and phase velocity in dispersive media, expression involving wavelength (v_g = v_p - lambda \frac{dv_p}{dlambda}), relation with refractive index, and wave packet spreading.
Prerequisites
Simple Harmonic Motion and Wave Equation
Superposition Principle of Waves and Beats
Partial Differentiation
Definition and Physical Significance
Group velocity is defined as the velocity of the envelope of a wave packet formed by the superposition of two or more waves of slightly different frequencies.
It represents the speed of energy propagation, whereas phase velocity represents the speed of individual phase fronts.
In a non-dispersive medium, all frequency components travel at the same speed, making the group velocity equal to the phase velocity.
Dispersive vs Non-Dispersive Media
A medium is termed dispersive if the phase velocity depends on the frequency or wavelength of the wave.
In normal dispersion, phase velocity increases with wavelength (dv_p/dlambda < 0), causing group velocity to be less than phase velocity (v_g < v_p).
In anomalous dispersion, phase velocity decreases with wavelength (dv_p/dlambda > 0), leading to a group velocity greater than phase velocity (v_g > v_p).
Mathematical Formulation
Expressed mathematically as $v_g = domega/dk$, where omega is the angular frequency and k is the angular wave number.
Relates to phase velocity $v_p = \omega/k via the differential operation$.
For electromagnetic waves in an ionosphere or plasma, the product of group velocity and phase velocity is constant: v_g $v_p = c^2$.
Wave Packet and Dispersion
A localized wave packet is constructed using Fourier components spanning a small range of wave numbers Delta k.
Due to different frequency components traveling at different velocities in a dispersive medium, the wave packet broadens or spreads out over time.
The spatial width of the wave packet is inversely proportional to the spread in wave number, satisfying the uncertainty principle Delta x Delta k ge 1/2.
Derivations
Relation between Group Velocity and Phase Velocity
Conditions: Superposition of two waves of equal amplitude, slightly differing frequencies (omega, omega + domega) and wave numbers (k, k + dk).
Write the displacement equations for the two interfering waves: $y_1 = a \cos(\omega t - kx)$ and $y_2 = a \cos((\omega + domega)t - (k + dk)x)$.
Apply the superposition principle $y = y_1 + y_2$ and use the trigonometric identity for the sum of cosines.
The resulting displacement is given by $y = 2a \cos\left(\frac{domega}{2} t - \frac{dk}{2} x\right) \cos(\omega t - kx)$.
Identify the envelope term traveling at group velocity $v_g = \frac{domega}{dk}$ and the carrier wave term traveling at phase velocity $v_p = \frac{\omega}{k}$.
Result: $v_g = \frac{domega}{dk}$
Alternative Expression for Group Velocity in Terms of Wavelength
Conditions: Wave propagation in a dispersive medium where phase velocity depends on wavelength lambda.
Start from the fundamental definition of group velocity in terms of angular frequency and wave number: $v_g = \frac{domega}{dk}$.
Express angular frequency as omeg$a = k v_p$, where v_p is a function of k.
Differentiate with respect to k: $v_g = \frac{d}{dk}(kv_p) = v_p + k \frac{dv_p}{dk}$.
Substitute the relation between wave number and wavelength, k = \frac{2pi}{lambda}, and apply the chain rule: \frac{dv_p}{dk} = \frac{dv_p}{dlambda} \frac{dlambda}{dk} = -\frac{lambda^2}{2pi} \frac{dv_p}{dlambda}.
Substitute back to obtain the final expression: $v_g = v_p - \lambda \frac{dv_p}{dlambda}$.
\omega = angular frequency, k = angular wave number m/s
General wave motion
Group Velocity (Derivative Form)
$$v_g = \frac{d\omega}{dk}$$
\omega = angular frequency, k = angular wave number m/s
Superposition of waves forming a packet
Group Velocity in terms of Phase Velocity and Wavelength
$$v_g = v_p - \lambda \frac{dv_p}{d\lambda}$$
$v_p = phase velocity$, \lambda = wavelength m/s
Dispersive medium
Group Velocity in terms of Refractive Index
$$v_g = \frac{c}{n + \nu \frac{dn}{d\nu}}$$
c = speed of light in vacuum, n = refractive index, \nu = frequency m/s
Electromagnetic waves in material media
Angular Wave Number
$$k = \frac{2\pi}{\lambda}$$
\lambda = wavelength rad/m
Plane harmonic waves
Relation for Deep Water Gravity Waves
$$v_g = \frac{1}{2} v_p$$
$v_p = phase velocity of water waves$ m/s
Surface waves on deep water (dispersive)
Product of Group and Phase Velocity in Waveguides
$$v_g v_p = c^2$$
c = speed of light in vacuum m^2/s^2
Electromagnetic waves in plasma or waveguides
Group Velocity for Non-Dispersive Medium
$$v_g = v_p$$
$v_p = constant phase velocity$ m/s
Vacuum or non-dispersive media (\frac{dv_p}{dlambda} = 0)
Graphs and experiments
Formation of a Wave Packet (Envelope and Carrier)
Draw: A high-frequency sinusoidal carrier wave modulated by a slower, broader envelope curve.
Interpret: Shows how localized wave packets are formed by constructive and destructive interference of neighboring frequencies.
Exam use: Visual identification of phase velocity (speed of inner crests) vs group velocity (speed of envelope peak).
Dispersion Curve (omega vs k)
Draw: A non-linear curve of angular frequency omega plotted against wave number k, with a tangent line drawn at a specific point.
Interpret: The slope of the tangent at any point gives the group velocity ($v_g = domega/dk)$, while the slope of the secant line from the origin gives the phase velocity ($v_p = \omega/k)$.
Exam use: Graphical determination of v_g and v_p to classify media as normal or anomalous.
Wave Packet Spreading in a Dispersive Medium
Draw: A series of wave packet amplitude profiles at successive time intervals t_1, t_2, t_3 showing both translation and horizontal broadening.
Interpret: Demonstrates that different frequency components travel at different speeds, causing the pulse to lose its sharpness over distance.
Exam use: Explaining limitations in signal transmission and communication channel bandwidth.
Solved numericals
TGT core
If the angular frequency of a wave is given by omeg$a = 5k^2 + 2k$, find the group velocity at wave number $k = 2 rad/m$.
Recall the formula for group velocity: $v_g = \frac{domega}{dk}$.
Differentiate the given dispersion relation omeg$a = 5k^2 + 2k with respect$ to k: \frac{domega}{dk} = 10k + 2.
Substitute the given value $k = 2 into the derivative: v_g = 10(2) + 2 = 20 + 2 = 22 m/s$.
Answer: 22 m/s
PGT extension
For deep water gravity waves, the phase velocity is given by $v_p = \sqrt{\frac{glambda}{2pi}}$. Calculate the group velocity of these waves.
Use the relation between group velocity and phase velocity: $v_g = v_p - \lambda \frac{dv_p}{dlambda}$.
Find the derivative of phase velocity with respect to wavelength: \frac{dv_p}{dlambda} = \frac{d}{dlambda}left(left(\frac{g}{2pi}\right)^{1/2} lambda^{1/2}\right) = \frac{1}{2} left(\frac{g}{2pi}\right)^{1/2} lambd$a^{-1/2} = \frac{1}{2lambda} \sqrt{\frac{glambda}{2pi}} = \frac{v_p}{2lambda}$.
Substitute the derivative back into the formula: $v_g = v_p - \lambda \left(\frac{v_p}{2lambda}\right) = v_p - \frac{v_p}{2} = \frac{1}{2} v_p$.
Answer: 0.5 v_p (or half the phase velocity)
PGT extension
A medium has a dispersion relation omeg$a^2 = c^2 k^2 + \omega_p^2$, where c and omega_p are constants. Find the product of group velocity and phase velocity.
Assumes the wave packet is formed by a narrow band of frequencies, so that dispersion across the band is negligible.
Does not account for severe attenuation or absorption in the medium, which would alter the amplitude envelope independently of dispersion.
Common misconceptions
Common error: Group velocity is always greater than or equal to phase velocity.
Correct: Group velocity can be less than, greater than, or equal to phase velocity depending on whether the dispersion is normal or anomalous.
In normal dispersion, dv_p/dlambda < 0 makes v_g < v_p.
Common error: Phase velocity can never exceed the speed of light in a vacuum.
Correct: Phase velocity can exceed c (e.g., in wave guides or anomalous dispersion regions), but group velocity (representing energy flow) never exceeds c.
Phase velocity does not transmit physical signals or energy, whereas group velocity does.
Common error: Group velocity and phase velocity are identical for all mechanical waves.
Correct: They are identical only in non-dispersive media (like sound waves in air at constant temperature); most physical media are dispersive.
Frequency dependence of wave speed inherently separates group and phase velocities.
Quick revision
Group velocity represents the speed of energy transmission and is given by $v_g = domega/dk$.
In a dispersive medium with normal dispersion, v_g < v_p.
For deep water gravity waves, group velocity is half of phase velocity.
In waveguides or plasma, the product of group and phase velocities equals the square of the speed of light in a vacuum (v_g $v_p = c^2)$.
Must remember
Group velocity is defined as $v_g = \frac{d\omega}{dk}$, representing the speed of energy propagation of a wave packet.
Phase velocity is given by $v_p = \frac{\omega}{k}$, representing the speed of individual phase fronts.
In a non-dispersive medium, $v_g = v_p$ because all frequency components travel at the same speed.
In normal dispersion, phase velocity increases with wavelength ($\frac{dv_p}{d_{\lambda}} < 0$), leading to $v_g < v_p$.
In anomalous dispersion, phase velocity decreases with wavelength ($\frac{dv_p}{d_{\lambda}} > 0$), leading to $v_g > v_p$.
The relation between group velocity and phase velocity in terms of wavelength is $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$.
For electromagnetic waves in plasma or ionosphere, the product of group and phase velocities is constant: $v_g v_p = c^2$.
The spatial width and wave number spread of a wave packet satisfy the uncertainty principle $\Delta x \Delta k \ge \frac{1}{2}$.
Glossary
Group Velocity ($v_g$)The velocity with which the envelope or the energy of a wave packet propagates through a medium.
Phase Velocity ($v_p$)The speed at which a single phase of a wave (such as a crest or trough) travels through space.
Angular Frequency ($\omega$)The rate of change of phase of a sinusoidal wave, measured in radians per second.
Angular Wave Number ($k$)The spatial frequency of a wave, defined as $k = \frac{2\pi}{\lambda}$, measured in rad/m.
Dispersive MediumA medium in which the phase velocity of a wave depends on its frequency or wavelength.
Normal DispersionA dispersion regime where phase velocity increases with wavelength, causing $v_g < v_p$.
Anomalous DispersionA dispersion regime where phase velocity decreases with wavelength, causing $v_g > v_p$.
Phase Velocity vs Group Velocity
Feature
Phase Velocity ($v_p$)
Group Velocity ($v_g$)
Definition
Speed of a single phase front
Speed of the wave packet envelope
Formula
$v_p = \frac{\omega}{k}$
$v_g = \frac{d\omega}{dk}$
Signal/Energy Transmission
Does not transmit information or energy
Transmits physical signals and energy
In Non-dispersive Medium
Constant speed
Equal to phase velocity ($v_g = v_p$)
Speed Limit
Can exceed the speed of light $c$
Never exceeds the speed of light $c$
Practice Quiz
Choose an option to reveal the explanation.
1. What does group velocity represent in a wave packet?
Explanation: Group velocity $v_g$ represents the speed at which the envelope of a wave packet travels, which corresponds to the propagation of wave energy and signals.
2. What is the mathematical definition of group velocity $v_g$?
Explanation: Group velocity is defined mathematically as the derivative of angular frequency with respect to angular wave number: $v_g = \frac{d\omega}{dk}$.
3. In a non-dispersive medium, what is the relationship between group velocity ($v_g$) and phase velocity ($v_p$)?
Explanation: In a non-dispersive medium, all frequency components travel at the same speed, making group velocity equal to phase velocity ($v_g = v_p$).
4. What condition defines a normal dispersion medium in terms of phase velocity and wavelength?
Explanation: In normal dispersion, phase velocity increases with wavelength, which means the derivative of phase velocity with respect to wavelength is negative ($\frac{dv_p}{d\lambda} < 0$).
5. How does group velocity compare to phase velocity in a medium with normal dispersion?
Explanation: In normal dispersion, since $\frac{dv_p}{d\lambda} < 0$, the relation $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$ implies that $v_g < v_p$.
6. Which of the following expressions correctly relates group velocity to phase velocity and wavelength?
Explanation: The standard formula connecting group velocity and phase velocity in a dispersive medium is $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$.
7. For electromagnetic waves in an ionosphere or plasma, what is the product of group velocity and phase velocity?
Explanation: In a plasma or ionosphere waveguide, the dispersion relation yields the constant product $v_g v_p = c^2$.
8. What happens to a localized wave packet as it travels through a dispersive medium over time?
Explanation: Because different frequency components travel at different velocities in a dispersive medium, the wave packet broadens or spreads out spatially over time.
9. Which uncertainty relation governs the spatial width ($\Delta x$) and wave number spread ($\Delta k$) of a wave packet?
Explanation: The spatial width of a wave packet and its wave number spread satisfy the Fourier transform property leading to the uncertainty relation $\Delta x \Delta k \ge \frac{1}{2}$.
10. What is the relation between group velocity and phase velocity for deep water gravity waves?
Explanation: For deep water gravity waves, the phase velocity depends on wavelength as $v_p \propto \sqrt{\lambda}$, which results in group velocity being half of the phase velocity ($v_g = \frac{1}{2}v_p$).
11. Can phase velocity exceed the speed of light in a vacuum ($c$)?
Explanation: Phase velocity does not transmit physical signals or energy, so it can exceed $c$ in certain conditions like waveguides or anomalous dispersion. Group velocity, however, cannot exceed $c$.
12. What is the angular wave number $k$ for a plane harmonic wave of wavelength $\lambda$?
Explanation: The angular wave number is defined as $k = \frac{2\pi}{\lambda}$, representing radians per unit distance.
13. In an anomalous dispersion medium, how does group velocity relate to phase velocity?
Explanation: In anomalous dispersion, phase velocity decreases with wavelength ($\frac{dv_p}{d\lambda} > 0$), making the group velocity greater than phase velocity ($v_g > v_p$).
14. If the angular frequency of a wave is given by $\omega = 3k^2$, what is its group velocity?
Explanation: Group velocity is $v_g = \frac{d\omega}{dk} = \frac{d}{dk}(3k^2) = 6k$.
15. Which of the following statements is a common misconception about group velocity?
Explanation: It is a misconception that group velocity is always greater than or equal to phase velocity; it can be smaller when normal dispersion is present ($v_g < v_p$).
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The phase velocity $v_p$ of a wave is related to its wavelength $\lambda$ in a dispersive medium. The group velocity $v_g$ is given by:
Explanation: Using the relation between angular frequency, wave number, and wavelength, group velocity in terms of phase velocity and wavelength is expressed as $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$.
2. For a wave motion where the dispersion relation is given by $\omega^2 = c^2 k^2 + \omega_p^2$, what is the product of group velocity ($v_g$) and phase velocity ($v_p$)?
Explanation: Differentiating $\omega^2 = c^2 k^2 + \omega_p^2$ with respect to $k$ gives $2\omega \frac{d\omega}{dk} = 2c^2 k$, so $v_g = \frac{d\omega}{dk} = \frac{c^2 k}{\omega}$. Since phase velocity is $v_p = \frac{\omega}{k}$, their product is $v_g v_p = \left(\frac{c^2 k}{\omega}\right) \left(\frac{\omega}{k}\right) = c^2$.
3. In a dispersive medium, if phase velocity increases with an increase in wavelength, the medium exhibits:
Explanation: If phase velocity increases with wavelength, $\frac{dv_p}{d\lambda} < 0$, which defines normal dispersion. Consequently, $v_g = v_p - \lambda \frac{dv_p}{d\lambda} < v_p$.
4. Deep water gravity waves have a phase velocity $v_p \propto \sqrt{\lambda}$. What is the ratio of group velocity to phase velocity $\frac{v_g}{v_p}$?
5. What physical quantity does the group velocity of a wave packet fundamentally characterize?
Explanation: Group velocity is the velocity at which the overall shape of the wave amplitudes—the envelope—propagates, which represents the speed of energy and information transmission.
6. If a wave packet has a spread in wave number given by $\Delta k$, what is the minimum spatial width $\Delta x$ of the wave packet according to the uncertainty principle?
Explanation: From the Fourier uncertainty relation $\Delta x \Delta k \ge \frac{1}{2}$, the minimum spatial width is $\Delta x = \frac{1}{2\Delta k}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the angular frequency and wave number satisfy the relation $\omega = c k$, what is the group velocity of the wave?
Explanation: Given $\omega = ck$, the group velocity is $v_g = \frac{d\omega}{dk} = c$. This is a non-dispersive wave system.
2. A medium is termed non-dispersive for acoustic waves if:
Explanation: By definition, a non-dispersive medium is one in which the phase velocity does not depend on frequency or wavelength (i.e., $\frac{dv_p}{d\lambda} = 0$).
3. In an anomalous dispersion region, how does the phase velocity change with an increase in wavelength?
Explanation: In anomalous dispersion, phase velocity decreases with increasing wavelength, meaning $\frac{dv_p}{d\lambda} > 0$.
4. For a wave packet, what causes it to broaden or spread out over time as it propagates through a dispersive medium?
Explanation: Dispersion causes different frequency components of the wave packet to travel at different velocities, leading to temporal and spatial spreading of the packet envelope.
5. Which of the following wave velocities can potentially exceed the velocity of light in a vacuum ($c$)?
Explanation: Phase velocity can exceed $c$ because it does not carry physical energy or information. Group velocity, representing energy flow, can never exceed $c$ in relativity.
6. Calculate the group velocity of a wave packet if $\omega = 4k^3$ at $k = 2\text{ rad/m}$.
Explanation: Group velocity $v_g = \frac{d\omega}{dk} = 12k^2$. Substituting $k = 2$, we get $v_g = 12(2)^2 = 12(4) = 48\text{ m/s}$.
7. What is the physical significance of the product $v_g v_p = c^2$ found in plasma and waveguide propagation?
Explanation: The relation $v_g v_p = c^2$ indicates an inverse relationship between group and phase velocities in these media, ensuring that energy propagation ($v_g$) remains less than or equal to $c$ while $v_p$ exceeds $c$.
8. If $\lambda$ represents wavelength and $v_p$ is phase velocity, under what condition is group velocity equal to phase velocity?
Explanation: From $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$, group velocity equals phase velocity when $\frac{dv_p}{d\lambda} = 0$ (non-dispersive medium).
9. Which of the following statements about wave packets and Fourier components is correct?
Explanation: A localized wave packet is constructed by superimposing Fourier components spanning a narrow range of wave numbers $\Delta k$.
10. In terms of refractive index $n$ and frequency $\nu$, which expression correctly represents group velocity in a material medium?
Explanation: The group velocity in a refractive material medium is given by $v_g = \frac{c}{n + \nu \frac{dn}{d\nu}}$.