Phase velocity is the rate at which the phase of a wave propagates in space. It represents the velocity at which a point of constant phase (such as a crest or trough) travels along the wave.
Definition of phase velocity, relation between wave velocity, frequency, and wavelength, and simple numerical problems involving sinusoidal wave equations.
Derivation of phase velocity from the wave equation, relation between phase velocity and group velocity, dispersion relations, and dispersive media.
Conditions: Homogeneous, one-dimensional linear wave equation
Result: $v_p = \frac{\omega}{k}$
Conditions: Dispersive medium where ω is a non-linear function of k
Result: $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$
| Formula / law | Equation | Variables and units | Conditions |
|---|---|---|---|
| Phase Velocity Definition | $$v_p = \frac{\omega}{k}$$ | $\omega = angular frequency$, k = angular wave number m s^{-1} | General traveling wave |
| Frequency-Wavelength Form | $$v_p = \nu \lambda$$ | $\nu = frequency$, $\lambda = wavelength$ m s^{-1} | Sinusoidal waves |
| Angular Wave Number | $$k = \frac{2\pi}{\lambda}$$ | $\lambda = wavelength$ rad m^{-1} | Periodic spatial waves |
| Angular Frequency | $$\omega = 2\pi \nu$$ | $\nu = frequency$ rad s^{-1} | Periodic temporal waves |
| Group Velocity from Phase Velocity | $$v_g = v_p - \lambda \frac{dv_p}{d\lambda}$$ | $v_g = group velocity$, $v_p = phase velocity$, $\lambda = wavelength$ m s^{-1} | Dispersive media |
| Phase Velocity in Stretched String | $$v_p = \sqrt{\frac{T}{\mu}}$$ | T = tension in string, $\mu = mass per unit length$ m s^{-1} | Transverse waves on a string |
| Phase Velocity in Gas | $$v_p = \sqrt{\frac{\gamma P}{\rho}}$$ | $\gamma = adiabatic index$, P = pressure, $\rho = density$ m s^{-1} | Longitudinal acoustic waves |
| Phase Velocity in Electromagnetic Wave | $$v_p = \frac{1}{\sqrt{\mu \epsilon}}$$ | $\mu = permeability$, $\epsilon = permittivity of medium$ m s^{-1} | Free space or dielectric medium |
Draw: Displacement y versus position x at two successive times t and t + Δt, showing the forward shift of a wave crest.
Interpret: The spatial displacement of the crest divided by Δt directly illustrates the phase velocity.
Exam use: Used in conceptual multiple-choice questions identifying crest travel speed.
Draw: A plot of angular frequency ω on the y-axis against wave number k on the x-axis. A straight line passing through the origin represents non-dispersive waves (vp = vg), while a curved plot represents dispersive waves.
Interpret: The slope of the secant line from the origin gives the phase velocity (vp = ω/k), and the local tangent slope gives the group velocity (vg} = dω/dk).
Exam use: Frequently tested in PGT to evaluate understanding of dispersive characteristics.
Draw: Experimental setup showing a horizontal glass tube with lycopodium powder, a vibrating piston at one end, and a fixed reflecting boundary.
Interpret: Forms standing waves where the distance between consecutive heaps of powder equals half the wavelength (λ/2), allowing experimental determination of phase velocity $v = \nu\lambda$.
Exam use: Directly tested as a classical acoustic experiment to measure wave speed in gases and solids.
Answer: 50 m s^-1
Answer: 220 Hz
Answer: a
Common error: Phase velocity can always exceed the speed of light in vacuum without physical restriction.
Correct: While phase velocity (vp = ω/k) can exceed c in certain anomalous dispersion regions or wave guides, it transmits no energy or information, thus not violating relativity.
Information and energy travel at the group velocity or signal velocity, which never exceeds c.
Common error: Phase velocity and group velocity are always identical.
Correct: They are identical only in non-dispersive media where phase velocity is independent of frequency.
In dispersive media, different frequency components travel at different speeds, making vp different from vg.
| Parameter | Phase Velocity ($v_p$) | Group Velocity ($v_g$) |
|---|---|---|
| Definition | Speed of constant phase point | Speed of wave packet envelope and energy transport |
| Formula | $v_p = \frac{\omega}{k} = \nu \lambda$ | $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$ |
| Dispersive Medium | Depends on wavelength/frequency | Differs from phase velocity |
| Relativity Limit | Can exceed speed of light $c$ in anomalous dispersion | Never exceeds speed of light $c$ for information/energy transfer |
Choose an option to reveal the explanation.
Explanation: Phase velocity is defined as the ratio of angular frequency to angular wave number, $v_p = \frac{\omega}{k}$.
Explanation: Using $\omega = 2\pi \nu$ and $k = \frac{2\pi}{\lambda}$, the phase velocity becomes $v_p = \frac{2\pi \nu}{2\pi / \lambda} = \nu \lambda$.
Explanation: In a non-dispersive medium, all frequency components travel at the exact same speed, meaning phase velocity is independent of frequency or wavelength.
Explanation: The fundamental relation linking group velocity to phase velocity is $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$.
Explanation: The phase velocity of transverse waves on a stretched string is derived from wave mechanics as $v_p = \sqrt{\frac{T}{\mu}}$.
Explanation: In a dispersive medium, the phase velocity depends on wavelength, meaning its derivative with respect to wavelength is non-zero ($\frac{dv_p}{d\lambda} \neq 0$).
Explanation: In normal dispersion, $\frac{dv_p}{d\lambda} < 0$, which makes $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$ result in $v_p > v_g$.
Explanation: Using Newton-Laplace correction, the phase velocity of sound waves in a gas is $v_p = \sqrt{\frac{\gamma P}{\rho}}$.
Explanation: The angular wave number is defined as $k = \frac{2\pi}{\lambda}$, representing radians per unit length.
Explanation: From Maxwell's equations, the phase velocity of an electromagnetic wave in a dielectric medium is given by $v_p = \frac{1}{\sqrt{\mu \epsilon}}$.
Explanation: Comparing with $y = A \sin(\omega t - kx)$, we get $\omega = 50$ rad/s and $k = 2$ rad/m. Thus, $v_p = \frac{\omega}{k} = \frac{50}{2} = 25$ m/s.
Explanation: Phase velocity corresponds to the movement of a mathematical phase point and does not carry physical energy or information; hence, no relativistic limit is violated.
Explanation: Since $v_p = \frac{\omega}{k}$, if $\omega$ is doubled and $k$ is constant, $v_p$ is directly doubled.
Explanation: Deep water gravity waves are dispersive because their phase velocity depends on wavelength ($v_p = \sqrt{\frac{g\lambda}{2\pi}}$).
Explanation: In a non-dispersive medium, phase velocity is independent of wavelength and frequency; therefore, changing wavelength while frequency adjusts accordingly leaves $v_p$ unchanged.
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
Explanation: If $\omega = c k$, then $v_p = \frac{\omega}{k} = c$. The group velocity is $v_g = \frac{d\omega}{dk} = c$. Thus, $v_p = v_g = c$ in this non-dispersive linear relation.
Explanation: Using $v_p = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{100}{0.25}} = \sqrt{400} = 20 \text{ m/s}$.
Explanation: Differentiating $\omega^2 = c^2 k^2 + \omega_0^2$ with respect to $k$ gives $2\omega \frac{d\omega}{dk} = 2c^2 k$, so $v_g = \frac{d\omega}{dk} = \frac{c^2 k}{\omega}$.
Explanation: Here $\frac{dv_p}{d\lambda} = \frac{A}{2\sqrt{\lambda}}$. Then $v_g = v_p - \lambda \frac{dv_p}{d\lambda} = v_p - \lambda \left(\frac{A}{2\sqrt{\lambda}}\right) = v_p - \frac{1}{2}v_p = 0.5 v_p$. Thus $\frac{v_g}{v_p} = 0.5$.
Explanation: Here $\omega = 100\pi$ and $k = 2\pi$. Phase velocity $v_p = \frac{\omega}{k} = \frac{100\pi}{2\pi} = 50 \text{ m/s}$.
Explanation: Since $v_p \propto \lambda^{1/2}$, $\frac{dv_p}{d\lambda} = \frac{1}{2} \frac{v_p}{\lambda}$. Then $v_g = v_p - \lambda \left(\frac{1}{2} \frac{v_p}{\lambda}\right) = v_p - 0.5 v_p = 0.5 v_p$.
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Explanation: Both definitions are valid since $k = \frac{2\pi}{\lambda}$ and from $v_p = \frac{\omega}{k}$, we get $k = \frac{\omega}{v_p}$.
Explanation: In anomalous dispersion, phase velocity increases with increasing wavelength, making $\frac{dv_p}{d\lambda} > 0$.
Explanation: Using $v_p = \nu \lambda = 50 \times 4 = 200 \text{ m/s}$.
Explanation: When $v_g > v_p$, $\frac{dv_p}{d\lambda} > 0$, which defines anomalous dispersion.
Explanation: Frequency is a characteristic of the source and remains invariant when a wave crosses boundaries between media, while wavelength and phase velocity change.
Explanation: Using $\omega = v_p k = 300 \times 1.5 = 450 \text{ rad/s}$.
Explanation: Since $v_p = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}$, phase velocity depends on the adiabatic index $\gamma$, absolute temperature $T$, and molecular weight $M$.
Explanation: Since $v_p = \sqrt{\frac{T}{\mu}}$, scaling tension $T$ by 4 increases $v_p$ by a factor of $\sqrt{4} = 2$. Thus, $50 \times 2 = 100 \text{ m/s}$.
Explanation: The term $\lambda \frac{dv_p}{d\lambda}$ accounts for the correction due to medium dispersion when relating group velocity to phase velocity.
Explanation: When phase velocity is independent of wavelength (non-dispersive medium), $\frac{dv_p}{d\lambda} = 0$, leading to $v_g = v_p$.