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Derivation

Dispersion, Phase Velocity and Group Velocity

Statement

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.

Why it matters

A monochromatic wave is an idealisation: it is infinitely long and carries no information, because a pure sinusoid that has existed for all time transmits nothing new. Every real signal is a packet built from a band of wavenumbers, and once \(\omega/k\) varies with \(k\) the different components run at different phase speeds. The rate at which the packet as a whole — its crest of amplitude — moves is the group velocity, and it is what a stopwatch measures for a pulse, a wave train's leading edge, or a quantum particle.

The distinction is not academic. It is the difference between the pattern speed of ocean swell and the speed at which wave energy actually reaches a shore; between the superluminal phase speed of a wave in a waveguide or plasma and the strictly sub-luminal speed at which it can deliver a bit; and between the phase of a de Broglie wave and the velocity of the electron it describes. Getting \(v_p\) and \(v_g\) straight is the entry point to group-velocity dispersion, pulse broadening in optical fibre, and the whole machinery of wave-packet dynamics.

Assumptions
Linearity of the medium.If the governing equation is nonlinear, superposition fails: components do not evolve independently, harmonics are generated, and coherent structures (solitons, shocks) can travel at speeds set by amplitude rather than by \(d\omega/dk\). Homogeneity and time-invariance.If the medium's properties depend on position or time, \(k\) and \(\omega\) are no longer good labels (no plane-wave eigenmodes), and the dispersion relation \(\omega(k)\) does not exist in the simple form used here. A single-valued, differentiable branch \(\omega=\omega(k)\).If \(\omega(k)\) is multivalued (several modes at one \(k\)) or non-differentiable (band edge, cutoff), \(d\omega/dk\) must be taken branch-by-branch and may vanish or diverge; a single group velocity is then ill-defined. Narrow bandwidth about a carrier \(k_0\).If the spectrum is broad, the first-order expansion \(\omega(k)\approx\omega_0+\omega'(k_0)(k-k_0)\) is insufficient: second-order group-velocity dispersion \(\omega''\) spreads and reshapes the packet, so no single speed transports it undistorted. Negligible dissipation over the region of interest.If \(\omega(k)\) is appreciably complex (strong absorption, anomalous-dispersion resonance), \(v_g=d\omega/dk\) can exceed \(c\) or become negative and ceases to equal the energy- or signal-velocity; the stationary-phase argument breaks down.
Derivation
1
\[ \psi(x,t)=A\cos(kx-\omega t),\qquad kx-\omega t=\text{const} \]
Track a surface of constant phase for one Fourier component. A crest is a locus of fixed phase, so we impose \(kx-\omega t=\phi_0\). A
2
\[ k\,dx-\omega\,dt=0 \;\Longrightarrow\; v_p\equiv\frac{dx}{dt}=\frac{\omega}{k} \]
Differentiate the constant-phase condition and solve for the crest's speed. This defines the phase velocity purely from the kinematics of one sinusoid. A
3
\[ \omega=\omega(k) \]
A linear, homogeneous, time-invariant medium admits plane-wave eigenmodes \(e^{i(kx-\omega t)}\); substituting into the field equation fixes an allowed relation between \(\omega\) and \(k\), the dispersion relation. For the loaded string in its long-wave limit this is \(\omega=ck\); in general it is not linear in \(k\). B
4
\[ \psi=A\cos\!\big((k+\Delta k)x-(\omega+\Delta\omega)t\big)+A\cos\!\big((k-\Delta k)x-(\omega-\Delta\omega)t\big) \]
Build the simplest packet: two equal-amplitude components symmetric about the carrier \((k,\omega)\), with \(\Delta\omega=\omega'(k)\,\Delta k\) to leading order. Superposition is legal because the medium is linear. A
5
\[ \cos\alpha+\cos\beta=2\cos\!\frac{\alpha+\beta}{2}\,\cos\!\frac{\alpha-\beta}{2} \]
\[ \psi=2A\,\underbrace{\cos(kx-\omega t)}_{\text{carrier}}\;\underbrace{\cos(\Delta k\,x-\Delta\omega\,t)}_{\text{envelope}} \]
Apply the sum-to-product identity with \(\alpha=(k+\Delta k)x-(\omega+\Delta\omega)t\) and \(\beta=(k-\Delta k)x-(\omega-\Delta\omega)t\). The field factors into a fast carrier times a slow envelope. A
6
\[ v_{\text{carrier}}=\frac{\omega}{k}=v_p,\qquad v_{\text{envelope}}=\frac{\Delta\omega}{\Delta k} \]
Each factor is itself a travelling wave; read its speed from the ratio of its temporal to spatial frequency, exactly as in Steps 1–2. The envelope — the beat pattern of amplitude — moves at \(\Delta\omega/\Delta k\), distinct from the carrier speed. B
7
\[ v_g\equiv\lim_{\Delta k\to 0}\frac{\Delta\omega}{\Delta k}=\frac{d\omega}{dk} \]
Take the narrow-band limit so the two components merge into a continuum peaked at the carrier. The envelope speed becomes the derivative of the dispersion relation. B
8
\[ \psi(x,t)=\int_{-\infty}^{\infty} A(k)\,e^{i\left(kx-\omega(k)t\right)}\,dk,\qquad \omega(k)\approx\omega_0+\omega'(k_0)(k-k_0)+\tfrac12\omega''(k_0)(k-k_0)^2 \]
\[ \psi\approx e^{i(k_0x-\omega_0 t)}\!\int A(k)\,e^{i(k-k_0)\left(x-\omega'(k_0)t\right)}\,dk \;=\; e^{i(k_0x-\omega_0 t)}\,\mathcal{E}\!\left(x-\omega'(k_0)\,t\right) \]
For a general packet sharply peaked at \(k_0\), expand \(\omega(k)\) and keep the linear term. The integral then depends on position and time only through the combination \(x-\omega'(k_0)t\): the envelope \(\mathcal{E}\) rigidly translates at \(v_g=\omega'(k_0)\). This is the stationary-phase statement of the group velocity and shows Step 7 is not special to two waves. The retained \(\omega''\) term is the leading correction (group-velocity dispersion), responsible for spreading. C
9
\[ v_g=\frac{d\omega}{dk}=\frac{d(v_p\,k)}{dk}=v_p+k\,\frac{dv_p}{dk}=v_p-\lambda\,\frac{dv_p}{d\lambda} \]
Write \(\omega=v_p k\) and differentiate by the product rule; convert to wavelength with \(k=2\pi/\lambda\), \(dk=-2\pi\lambda^{-2}d\lambda\). This exposes group velocity as phase velocity corrected by how \(v_p\) itself varies across the band. B
Result
\[ v_p=\frac{\omega}{k},\qquad v_g=\frac{d\omega}{dk}=v_p+k\,\frac{dv_p}{dk} \]

Reading. The phase velocity is the speed of an individual crest and is set pointwise by the ratio \(\omega/k\). The group velocity is the speed of the packet's envelope and is set by the local slope of the dispersion curve. They agree only when that curve is a straight line through the origin (\(\omega=ck\)); every departure from proportionality — every real medium — makes them differ, and \(v_g\) is the one that transports amplitude, energy, and information to leading order.

Units check. \(\omega\) has units of \(\mathrm{rad\,s^{-1}}\) and \(k\) of \(\mathrm{rad\,m^{-1}}\); rad is dimensionless, so \(\omega/k\) is \(\mathrm{m\,s^{-1}}\). The derivative \(d\omega/dk\) is a ratio of the same two quantities and is likewise \(\mathrm{m\,s^{-1}}\). The correction term \(k\,dv_p/dk\) has units \(\mathrm{m^{-1}}\cdot(\mathrm{m\,s^{-1}}/\mathrm{m^{-1}})=\mathrm{m\,s^{-1}}\). Consistent.

Limiting cases
  • Non-dispersive line, \(\omega=ck\): \(v_p=v_g=c\). The dispersion curve is a ray through the origin; packets travel undistorted. This is the long-wave loaded string and the ideal string of the wave-equation result.
  • Flat phase speed, \(dv_p/dk=0\) at \(k_0\): \(v_g=v_p\) there even though the medium is dispersive elsewhere — the packet momentarily rides with its carrier.
  • Deep-water gravity waves, \(\omega=\sqrt{gk}\): \(v_g=\tfrac12 v_p\). The envelope lags the crests; individual waves appear at the back of a group, sweep forward, and vanish at the front.
  • Monatomic lattice at the zone boundary, \(k=\pi/a\): \(v_g=a\sqrt{K/m}\cos(ka/2)=0\). The mode is a standing wave (alternate masses in antiphase), consistent with the normal-mode picture; no energy propagates.
  • Relativistic / plasma branch, \(\omega^2=\omega_p^2+c^2k^2\): \(v_p\,v_g=c^2\) with \(v_p>c\) and \(v_g<c\); the superluminal phase speed carries nothing.
Breaks when
  • Anomalous dispersion / strong absorption. Near a resonance \(\omega(k)\) becomes complex and \(v_g=d\omega/dk\) can exceed \(c\), diverge, or turn negative. It no longer equals the speed at which energy or a detectable signal arrives; one must instead use the energy velocity or the Sommerfeld–Brillouin front velocity, which stays \(\le c\).
  • Broad-band packets. When the spectrum is not narrow, the neglected \(\tfrac12\omega''(k-k_0)^2\) term dominates: the packet spreads, chirps, and develops sub-structure, so a single translation speed no longer describes it. Group-velocity dispersion, not \(v_g\) alone, governs the evolution.
  • Nonlinear media. With amplitude-dependent response, superposition and the linear dispersion relation both fail. Wave speed can depend on amplitude (solitons, shock fronts), and \(d\omega/dk\) evaluated from the linear branch is not the propagation speed.
  • Inhomogeneous or non-stationary media. If material properties vary in \(x\) or \(t\), \(k\) and \(\omega\) are not conserved labels; \(\omega(k)\) is only local, and rays must be traced with the eikonal/WKB equations rather than a global \(v_g\).
Failure modes
  • Reporting \(\omega/k\) as the signal speed. Students quote the phase velocity for how fast a pulse arrives; energy and information travel at \(v_g\) (in transparent media), which can be very different.
  • Computing \(v_g\) as \(\omega/k\) then differentiating the number. \(v_g\) is \(d\omega/dk\), the slope of the \(\omega\)-vs-\(k\) curve, not the derivative of the already-formed ratio \(\omega/k\). Use \(v_g=v_p+k\,dv_p/dk\) if you must go through \(v_p\).
  • Dropping the \(-\lambda\) factor when switching to wavelength. \(v_g=v_p-\lambda\,dv_p/d\lambda\), not \(v_p+\lambda\,dv_p/d\lambda\); the sign flip comes from \(k=2\pi/\lambda\).
  • Claiming \(v_p>c\) or \(v_g>c\) violates relativity. Neither phase nor (anomalous) group velocity is the signal velocity; only the front velocity is bounded by \(c\).
  • Pairing the wrong \(\Delta\omega\) with \(\Delta k\). In the two-wave beat, the envelope carries the difference frequencies \((\Delta k,\Delta\omega)\) and the carrier the mean \((k,\omega)\); swapping them inverts which factor is fast.
  • Assuming \(v_g=\tfrac12 v_p\) universally. That factor is specific to \(\omega\propto\sqrt{k}\) (deep water). Each dispersion relation gives its own ratio.
Discussion

The physical content of the derivation is that a wave carries two independent speeds because it has two independent structures: the microscopic ripple (the carrier) and the macroscopic modulation (the envelope). Only the modulation can encode a message — switch the amplitude on and off and you have sent a bit — so the modulation's speed, \(v_g\), is the physically operative one. In a transparent medium \(v_g\) also equals the velocity of energy transport, because energy density and energy flux are both quadratic in the field and are carried by the envelope. This is why an oceanographer computing when swell energy reaches a coast uses \(v_g\), and why the "wave you surf" (a crest, moving at \(v_p\)) outruns the group it belongs to.

The relation \(v_g=v_p+k\,dv_p/dk\) organises all dispersive behaviour into two regimes. In normal dispersion the phase speed falls as \(k\) rises (\(dv_p/dk<0\)), so \(v_g<v_p\); this is the optical case, blue light slower than red in glass, and it makes short pulses spread predictably. In anomalous dispersion \(dv_p/dk>0\) and \(v_g>v_p\), occurring near absorption resonances, where the tidy identification of \(v_g\) with a signal speed collapses. The same formula underlies group-velocity dispersion in optical fibre, where the residual \(\omega''\) determines how a data pulse broadens over kilometres and sets the bit-rate ceiling.

The quantum connection is the most striking. A de Broglie wave has \(E=\hbar\omega\) and \(p=\hbar k\); for a free non-relativistic particle \(E=p^2/2m\) gives \(\omega=\hbar k^2/2m\), so \(v_g=d\omega/dk=\hbar k/m=p/m\), exactly the classical particle velocity, while \(v_p=\omega/k=\hbar k/2m\) is half of it and physically inert. The particle is the packet, and it moves at the group velocity — this is how wave mechanics reproduces Newtonian trajectories in the appropriate limit (Ehrenfest's theorem). For a relativistic particle the same construction yields \(v_p\,v_g=c^2\), tying the superluminal phase speed to the sub-luminal particle speed.

The deepest subtlety is what happens to "speed" when \(\omega(k)\) is complex. Sommerfeld and Brillouin showed, by contour integration of the packet integral, that although \(v_p\) and even \(v_g\) may exceed \(c\) in an absorbing anomalous-dispersion band, the very first disturbance — the front, associated with the highest-frequency components where any real medium behaves like vacuum (\(n\to 1\)) — always propagates at exactly \(c\). Causality is preserved not by any of the "obvious" velocities but by the analyticity of the response function (Kramers–Kronig relations), which forbids a signal front from preceding its cause. The group velocity is thus a leading-order kinematic approximation valid only where dispersion is weak and absorption negligible; outside that window it is a bookkeeping quantity, not the speed of anything real.

Common misconceptions. Phase velocity is not "the wave's speed" and group velocity is not merely "the average phase velocity." \(v_p\) is a per-component crest speed with no upper bound and no obligation to transport anything; \(v_g\) is the envelope speed and equals the energy/signal speed only in transparent, weakly dispersive media. When the two differ, watching a single crest tells you nothing about when the disturbance arrives.

Worked examples

Example 1 — Deep-water gravity waves.

1
\[ \omega(k)=\sqrt{gk},\qquad \lambda=100\ \mathrm{m}\ \Rightarrow\ k=\frac{2\pi}{\lambda} \]
State the dispersion relation for gravity waves on deep water and convert the given wavelength to wavenumber. A
2
\[ k=\frac{2\pi}{100\ \mathrm{m}}=6.28\times10^{-2}\ \mathrm{rad\,m^{-1}} \]
Insert numbers. A
3
\[ v_p=\frac{\omega}{k}=\frac{\sqrt{gk}}{k}=\sqrt{\frac{g}{k}}=\sqrt{\frac{9.81}{6.28\times10^{-2}}}\ \mathrm{m\,s^{-1}} \]
Form the phase velocity symbolically first, then substitute \(g=9.81\ \mathrm{m\,s^{-2}}\). A
4
\[ v_g=\frac{d\omega}{dk}=\frac{d}{dk}\sqrt{gk}=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}v_p \]
Differentiate the dispersion relation; the half-power law gives the factor \(\tfrac12\) exactly. B
\[ v_p=12.5\ \mathrm{m\,s^{-1}},\qquad v_g=6.25\ \mathrm{m\,s^{-1}} \]

Reading. Individual crests travel at 12.5 m/s but the group — and its energy — advances at only 6.25 m/s, so crests continually overtake the group, appearing at its trailing edge and disappearing at its front. Units check. \(\sqrt{(\mathrm{m\,s^{-2}})/(\mathrm{m^{-1}})}=\sqrt{\mathrm{m^2\,s^{-2}}}=\mathrm{m\,s^{-1}}\).

Example 2 — Monatomic lattice (loaded string, one mass per cell).

1
\[ \omega(k)=2\sqrt{\frac{K}{m}}\left|\sin\!\frac{ka}{2}\right|,\qquad K=15\ \mathrm{N\,m^{-1}},\ m=4.0\times10^{-26}\ \mathrm{kg},\ a=3.0\times10^{-10}\ \mathrm{m} \]
Use the dispersion relation obtained from the loaded-string equation of motion \(m\ddot u_n=K(u_{n+1}+u_{n-1}-2u_n)\) with \(u_n=A e^{i(kna-\omega t)}\). Evaluate at \(k=\dfrac{\pi}{2a}\), i.e. \(ka/2=\pi/4\). A
2
\[ \sqrt{\frac{K}{m}}=\sqrt{\frac{15}{4.0\times10^{-26}}}=1.94\times10^{13}\ \mathrm{s^{-1}},\qquad \sin\frac{\pi}{4}=0.707 \]
Compute the natural frequency scale and the trigonometric factor. A
3
\[ \omega=2(1.94\times10^{13})(0.707)=2.74\times10^{13}\ \mathrm{rad\,s^{-1}},\qquad k=\frac{\pi}{2a}=5.24\times10^{9}\ \mathrm{rad\,m^{-1}} \]
Assemble \(\omega\) and the wavenumber at the chosen point. A
4
\[ v_p=\frac{\omega}{k}=\frac{2.74\times10^{13}}{5.24\times10^{9}}\ \mathrm{m\,s^{-1}} \]
Phase velocity from the ratio. A
5
\[ v_g=\frac{d\omega}{dk}=a\sqrt{\frac{K}{m}}\cos\!\frac{ka}{2}=(3.0\times10^{-10})(1.94\times10^{13})(0.707)\ \mathrm{m\,s^{-1}} \]
Differentiate \(\omega(k)\): \(\dfrac{d}{dk}\,2\sqrt{K/m}\,\sin(ka/2)=a\sqrt{K/m}\cos(ka/2)\). B
\[ v_p=5.2\times10^{3}\ \mathrm{m\,s^{-1}},\qquad v_g=4.1\times10^{3}\ \mathrm{m\,s^{-1}} \]

Reading. At quarter-zone the lattice is already dispersive: \(v_g<v_p\), and both fall below the long-wave sound speed \(c=a\sqrt{K/m}=5.8\times10^{3}\ \mathrm{m\,s^{-1}}\), which \(v_g\) recovers as \(k\to0\) and abandons (\(v_g\to0\)) at the zone boundary. Units check. \(a\sqrt{K/m}\): \(\mathrm{m}\cdot\sqrt{(\mathrm{N\,m^{-1}})/\mathrm{kg}}=\mathrm{m}\cdot\sqrt{\mathrm{s^{-2}}}=\mathrm{m\,s^{-1}}\).

Problems
  1. (A) Non-dispersive check. A transverse wave on an ideal string obeys \(\omega=ck\) with \(c=340\ \mathrm{m\,s^{-1}}\). Find \(v_p\) and \(v_g\), and state whether a pulse spreads.
    Solution \(v_p=\omega/k=c=340\ \mathrm{m\,s^{-1}}\). \(v_g=d\omega/dk=c=340\ \mathrm{m\,s^{-1}}\). Since \(\omega\) is exactly linear in \(k\), every component travels at the same speed, so the pulse propagates without spreading (no group-velocity dispersion, \(\omega''=0\)).
  2. (B) Half-speed law. A medium has phase velocity \(v_p=B\,k^{-1/2}\) (constant \(B\)). Using \(v_g=v_p+k\,dv_p/dk\), show \(v_g=\tfrac12 v_p\).
    Solution \(\dfrac{dv_p}{dk}=-\tfrac12 B\,k^{-3/2}\). Then \(k\dfrac{dv_p}{dk}=-\tfrac12 B\,k^{-1/2}=-\tfrac12 v_p\). Hence \(v_g=v_p-\tfrac12 v_p=\tfrac12 v_p\). (This is the deep-water case, since \(\omega=\sqrt{gk}\Rightarrow v_p=\sqrt{g}\,k^{-1/2}\).)
  3. (B) Plasma / waveguide branch. For \(\omega^2=\omega_p^2+c^2k^2\), show \(v_p v_g=c^2\). Then with a signal at \(\omega=2\omega_p\), evaluate \(v_p\) and \(v_g\) in units of \(c\).
    Solution Differentiate implicitly: \(2\omega\,d\omega=2c^2k\,dk\Rightarrow v_g=d\omega/dk=c^2k/\omega=c^2/v_p\), so \(v_p v_g=c^2\). At \(\omega=2\omega_p\): \(k=\sqrt{\omega^2-\omega_p^2}/c=\sqrt{3}\,\omega_p/c\). Thus \(v_p=\omega/k=2\omega_p/(\sqrt3\,\omega_p/c)=\tfrac{2}{\sqrt3}c=1.15c\), and \(v_g=c^2/v_p=\tfrac{\sqrt3}{2}c=0.87c\). The phase speed is superluminal but carries nothing; the group speed is sub-luminal, and their product is exactly \(c^2\).
  4. (B) Two-wave beat. Two components have \(k_1=10.0,\ k_2=10.2\ \mathrm{rad\,m^{-1}}\) and \(\omega_1=2000,\ \omega_2=2060\ \mathrm{rad\,s^{-1}}\). Find the carrier (phase) speed, the envelope (group) speed, and the envelope wavelength.
    Solution Carrier: \(\bar k=10.1\ \mathrm{rad\,m^{-1}}\), \(\bar\omega=2030\ \mathrm{rad\,s^{-1}}\), so \(v_p=\bar\omega/\bar k=2030/10.1=201\ \mathrm{m\,s^{-1}}\). Envelope: \(v_g\approx\Delta\omega/\Delta k=(2060-2000)/(10.2-10.0)=60/0.20=300\ \mathrm{m\,s^{-1}}\). Writing the beat as \(2A\cos(\bar k x-\bar\omega t)\cos(\delta k\,x-\delta\omega\,t)\) with half-differences \(\delta k=0.10\ \mathrm{rad\,m^{-1}}\), the envelope wavelength is \(2\pi/\delta k=2\pi/0.10=63\ \mathrm{m}\).
  5. (C) de Broglie wave. For a free non-relativistic electron, \(\omega=\hbar k^2/2m\). Show that the group velocity equals the particle velocity and the phase velocity is half of it. Then for \(v=1.0\times10^{6}\ \mathrm{m\,s^{-1}}\) give \(v_g\), \(v_p\), and the de Broglie wavelength.
    Solution \(v_g=d\omega/dk=\hbar k/m=p/m=v\), the classical speed; \(v_p=\omega/k=\hbar k/2m=v/2\). Numerically \(v_g=1.0\times10^{6}\ \mathrm{m\,s^{-1}}\), \(v_p=5.0\times10^{5}\ \mathrm{m\,s^{-1}}\). Momentum \(p=mv=(9.11\times10^{-31})(1.0\times10^{6})=9.11\times10^{-25}\ \mathrm{kg\,m\,s^{-1}}\); \(\lambda=h/p=(6.63\times10^{-34})/(9.11\times10^{-25})=7.3\times10^{-10}\ \mathrm{m}\). The electron travels at the group velocity of its own wave — the packet is the particle.