Dispersion: Phase and Group Velocity
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You should be comfortable with the superposition principle for linear waves and the basic calculus of partial derivatives. Familiarity with the wave equation and its plane-wave solutions is assumed.
Why this matters
In dispersive media, different frequency components travel at different speeds, fundamentally altering how wave packets evolve.
Up to now, we have often treated waves as if all frequencies travel at the same speed. This is true for non-dispersive media like ideal strings or sound in air at low frequencies. However, in many physical systems—light in glass, water waves, or quantum matter waves—the speed depends on frequency. Understanding this distinction between how fast a single wave crest moves (phase velocity) and how fast the overall shape or energy of a pulse moves (group velocity) is crucial for interpreting signal propagation and energy transport.
The Intuition Behind Dispersion
Imagine a long train of identical sine waves stretching to infinity. If you mark a specific crest, you can track it as it moves along the medium. This speed is the phase velocity. Now, imagine you modulate this train by turning the amplitude up and down slowly, creating a "packet" of waves. The envelope of this packet—the region where the waves are strongest—moves at a different speed if the medium is dispersive. In a non-dispersive medium, the crest and the envelope move together. In a dispersive medium, the crests may appear to race through the envelope, entering at the back and exiting at the front, while the envelope itself drifts along at the group velocity. This separation is the hallmark of dispersion.
Defining Phase and Group Velocity
Consider a linear, homogeneous medium where plane-wave solutions obey a dispersion relation \(\omega=\omega(k)\). A single sinusoidal component is given by \(\psi=A\cos(kx-\omega t)\). The surfaces of constant phase satisfy \(kx-\omega t = \text{constant}\). Differentiating with respect to time, we find the speed at which these surfaces advance:
This is the phase velocity. It tells us how fast a single frequency component propagates. However, physical signals are rarely single frequencies; they are superpositions of many nearby frequencies. The envelope of such a narrow-band superposition is determined by the slow variation of the amplitude. The speed at which this envelope, and thus the energy and information, propagates is the group velocity:
These two velocities coincide only when \(\omega\) is strictly proportional to \(k\), meaning the medium is non-dispersive. In general, they differ, and it is \(v_g\) that carries the modulation and energy to leading order.
Relating the Two Velocities
We can express the group velocity in terms of the phase velocity and its variation with wavenumber. Starting from \(v_p = \omega/k\), we have \(\omega = k v_p\). Differentiating with respect to \(k\):
Thus, the group velocity is the phase velocity plus a correction term that accounts for how the phase velocity changes with wavenumber. If \(v_p\) is constant (non-dispersive), the second term vanishes, and \(v_g = v_p\). If \(v_p\) increases with \(k\), then \(v_g > v_p\). If \(v_p\) decreases with \(k\), then \(v_g < v_p\). This relationship highlights that dispersion is fundamentally about the frequency dependence of the wave speed.
Physical Interpretation and Energy Transport
In a dispersive medium, the phase velocity can sometimes exceed the speed of light in vacuum, which might seem to violate relativity. However, this is not a contradiction because phase velocity does not carry information or energy. In ordinary (normal) dispersion the group velocity, which then also gives the speed of energy transport, stays below the speed of light. In regions of anomalous dispersion it can exceed \(c\), or even be negative, without any violation of causality, because the pulse is strongly reshaped; the front of a signal, which is what carries new information, never travels faster than light. Therefore, when analyzing signal propagation, we must always refer to the group velocity, not the phase velocity.
The derivation
Dispersion, Phase Velocity and Group Velocity
For a linear, homogeneous medium whose plane-wave solutions obey a dispersion relation \(\omega=\omega(k)\), a single sinusoidal component \(\psi=A\cos(kx-\omega t)\) advances its surfaces of constant phase at the phase velocity \(v_p=\omega/k\), whereas the envelope of a narrow-band superposition of such components advances at the group velocity \(v_g=\dfrac{d\omega}{dk}\). The two coincide only when \(\omega\) is strictly proportional to \(k\); otherwise the medium is dispersive and it is \(v_g\), not \(v_p\), that carries the modulation, energy, and signal to leading order.
A common misconception is that the group velocity is always the speed of energy transport. While this is true for narrow-band pulses in weakly dispersive media, in strongly dispersive or anomalous dispersion regimes, the peak of the pulse can move faster than \(v_g\) due to reshaping. The true signal velocity is defined by the front velocity, which is always causal. C
Worked example
Problem. Waves in a certain medium obey the dispersion relation \(\omega(k) = c_0 k + \alpha k^2\), with \(c_0 = 300\ \mathrm{m/s}\) and \(\alpha = 2.0 \times 10^{-3}\ \mathrm{m^2/s}\). Calculate the phase velocity and the group velocity for a wave with wavenumber \(k = 1.0 \times 10^4\ \mathrm{rad/m}\).
- Angular frequency: \(\omega = c_0 k + \alpha k^2 = (300)(1.0 \times 10^4) + (2.0 \times 10^{-3})(1.0 \times 10^4)^2 = 3.0 \times 10^6 + 2.0 \times 10^5 = 3.2 \times 10^6\ \mathrm{rad/s}\).
- Phase velocity: \(v_p = \omega/k = (3.2 \times 10^6)/(1.0 \times 10^4) = 320\ \mathrm{m/s}\).
- Group velocity: \(v_g = d\omega/dk = c_0 + 2\alpha k = 300 + 2(2.0 \times 10^{-3})(1.0 \times 10^4) = 340\ \mathrm{m/s}\). As expected, \(v_p\) increases with \(k\) here, so \(v_g > v_p\).
Units check. \(\omega\) is in \(\mathrm{rad/s}\), \(k\) is in \(\mathrm{rad/m}\), so \(v_p\) is in \(\mathrm{m/s}\). \(d\omega/dk\) is \(\mathrm{rad/s} / \mathrm{rad/m} = \mathrm{m/s}\). Both velocities have correct units.
Three quick tests
- For a non-dispersive medium, what is the relationship between \(v_p\) and \(v_g\)?
Answer
They are equal: \(v_p = v_g\). - If \(v_p\) increases with \(k\), is \(v_g\) greater than or less than \(v_p\)?
Answer
Greater than \(v_p\), because \(v_g = v_p + k \frac{dv_p}{dk}\) and the second term is positive. - Which velocity carries the energy of a wave packet in a dispersive medium?
Answer
The group velocity \(v_g\).
Problem set for PU-103 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.