Wave Packets, Group Velocity and Spreading

Derivation

Wave Packets, Group Velocity and Spreading

D-072 Home PU-103 Threads waves Depends on Dispersion, Phase Velocity and Group Velocity, Fourier Decomposition into Normal Modes

Statement

A localized disturbance built from a narrow band of plane waves centred on wavenumber \(k_0\), \(\ \psi(x,t)=\int A(k)\,e^{i(kx-\omega(k)t)}\,dk\), has an envelope that translates rigidly at the group velocity \(v_g=\left.\frac{d\omega}{dk}\right|_{k_0}\), and — once the second-order term in \(\omega(k)\) is retained — broadens with time according to \(\ \Delta x(t)=\Delta x_0\sqrt{1+\left(\dfrac{\beta\,t}{2\,\Delta x_0^{\,2}}\right)^{2}}\), where \(\beta=\left.\frac{d^2\omega}{dk^2}\right|_{k_0}\) is the group-velocity dispersion.

Why it matters

A single plane wave carries no information: it is infinite in extent and its energy density is uniform everywhere. Everything we can actually localize — an electron, a laser pulse, a wave group on the sea — is a superposition of many modes. The group velocity is the speed at which that localized energy, and hence any signal or particle, actually moves, and it generally differs from the phase velocity of the underlying carrier.

The same expansion that gives \(v_g\) predicts, at next order, that no packet can stay sharp forever in a dispersive medium: it spreads. This single mechanism governs the resolution limit of optical fibres, the temporal broadening of ultrashort laser pulses, the delocalization of a free quantum particle, and the dispersion of tsunami and swell trains across an ocean basin.

Assumptions

Narrow spectral band.If \(A(k)\) is not sharply peaked about \(k_0\), the Taylor expansion of \(\omega(k)\) truncated at second order is invalid; higher derivatives (third-order dispersion and beyond) distort the envelope asymmetrically and the simple group-velocity picture fails.

Smooth dispersion relation.\(\omega(k)\) must be twice differentiable at \(k_0\). Near a band edge, cutoff, or resonance where \(d\omega/dk\to0\) or a derivative diverges, the expansion coefficients are ill-defined and the packet does not propagate as a rigid group.

Linear, non-dissipative, homogeneous medium.If the medium is nonlinear the modes couple (solitons, self-phase modulation); if it absorbs, \(\omega\) acquires an imaginary part and the packet decays as well as spreads; if it is inhomogeneous, \(k\) is not a good conserved label.

Slowly varying envelope.The carrier wavelength must be short compared with the envelope scale, so that factoring \(\psi=e^{i(k_0x-\omega_0 t)}\phi\) with a smooth \(\phi\) is meaningful; otherwise "envelope" and "carrier" cannot be separated.

Real, single-valued \(\omega(k)\) with the physical branch chosen.For media with multiple branches (e.g. optical/acoustic phonons, or the two roots of a quadratic dispersion), one must expand about the correct branch; mixing branches assigns the packet an unphysical mean velocity.

Derivation

\[ \psi(x,t)=\int_{-\infty}^{\infty} A(k)\,e^{i\left(kx-\omega(k)t\right)}\,dk \]

Fourier synthesis of the field from its plane-wave modes (prior result: fourier-series-mode-decomposition, continuum limit); each mode propagates with its own phase \(e^{i(kx-\omega t)}\). A

\[ \omega(k)=\omega_0+v_g\,(k-k_0)+\tfrac{1}{2}\beta\,(k-k_0)^2+\cdots,\qquad \omega_0=\omega(k_0),\ \ v_g=\left.\frac{d\omega}{dk}\right|_{k_0},\ \ \beta=\left.\frac{d^2\omega}{dk^2}\right|_{k_0} \]

Taylor-expand the dispersion relation about the spectral centre \(k_0\); legal because \(A(k)\) is sharply peaked there (narrow-band assumption), so only small \(k-k_0\) contribute. B

\[ \psi(x,t)=e^{i(k_0x-\omega_0 t)}\underbrace{\int_{-\infty}^{\infty} A(k_0+\kappa)\,e^{\,i\left(\kappa x-\left(v_g\kappa+\frac12\beta\kappa^2\right)t\right)}\,d\kappa}_{\displaystyle \phi(x,t)} \]

Substitute the expansion, set \(\kappa=k-k_0\), and factor out the rapidly oscillating carrier \(e^{i(k_0x-\omega_0 t)}\); the remaining integral defines the slowly varying envelope \(\phi\). B

\[ \beta\to0:\quad \phi(x,t)=\int A(k_0+\kappa)\,e^{\,i\kappa\,(x-v_g t)}\,d\kappa=\phi\!\left(x-v_g t,\,0\right) \]

Drop the quadratic term first. The envelope integral then depends on \(x\) and \(t\) only through the combination \(x-v_g t\), so the envelope translates rigidly at speed \(v_g\) with no change of shape: the group velocity is identified. B

\[ \frac{\partial\phi}{\partial t}=-v_g\frac{\partial\phi}{\partial x}+i\,\frac{\beta}{2}\frac{\partial^2\phi}{\partial x^2} \qquad\Longrightarrow\qquad \left.\frac{\partial\phi}{\partial t}\right|_{\xi}=i\,\frac{\beta}{2}\,\frac{\partial^2\phi}{\partial \xi^2},\quad \xi\equiv x-v_g t \]

Now keep \(\beta\). Under the integral, \(\partial_x\leftrightarrow i\kappa\) and \(\partial_t\leftrightarrow -i(v_g\kappa+\tfrac12\beta\kappa^2)\); reading off the operator identity gives the left equation. Transforming to the frame \(\xi=x-v_g t\) cancels the advection term, leaving a free Schrödinger-type dispersion equation for the envelope. C

\[ \tilde\phi(\kappa,0)\propto e^{-a^2\kappa^2/2}\ \Longrightarrow\ \tilde\phi(\kappa,t)=\tilde\phi(\kappa,0)\,e^{-i\beta\kappa^2 t/2}\propto e^{-\left(a^2+i\beta t\right)\kappa^2/2} \]

Take a Gaussian spectrum of width \(1/a\) (the minimum-uncertainty packet). In \(\kappa\)-space the dispersion equation is diagonal: each mode acquires only the phase \(e^{-i\beta\kappa^2 t/2}\). Legal because the evolution is linear and modes are independent. C

\[ \phi(\xi,t)\propto \exp\!\left(-\frac{\xi^2}{2\left(a^2+i\beta t\right)}\right),\qquad |\phi(\xi,t)|^2\propto \exp\!\left(-\frac{a^2\,\xi^2}{a^4+\beta^2 t^2}\right) \]

Inverse-transform the Gaussian back to real space (a Gaussian integral), then take the modulus squared; the real part of \(1/(a^2+i\beta t)\) sets the intensity width. C

\[ \Delta x^2(t)=\frac{a^4+\beta^2 t^2}{2a^2}=\Delta x_0^2\left(1+\frac{\beta^2 t^2}{a^4}\right),\qquad \Delta x_0=\frac{a}{\sqrt2} \]

Read the RMS width from the Gaussian intensity, \(|\phi|^2\propto e^{-\xi^2/2\Delta x^2}\), and rewrite \(a^2=2\Delta x_0^2\). This exhibits the broadening law. B

Result

\[ \boxed{\ v_g=\left.\frac{d\omega}{dk}\right|_{k_0}\ ,\qquad \Delta x(t)=\Delta x_0\sqrt{1+\left(\frac{\beta\,t}{2\,\Delta x_0^{\,2}}\right)^{2}}\ ,\qquad \beta=\left.\frac{d^2\omega}{dk^2}\right|_{k_0}\ } \]

Reading. The centre of the packet moves at the group velocity \(v_g\), the slope of the dispersion curve — not the phase velocity \(\omega/k\). The curvature \(\beta\) of that same curve makes the packet spread: high- and low-\(k\) components travel at slightly different group speeds, so an initially compact group unavoidably widens, symmetrically and quadratically in time at late times. A narrower initial packet (small \(\Delta x_0\)) spreads faster, because it needs a broader band of \(k\) to build and those components disperse more.

Units check. \([v_g]=\dfrac{\text{rad·s}^{-1}}{\text{rad·m}^{-1}}=\text{m·s}^{-1}\). \([\beta]=\dfrac{\text{rad·s}^{-1}}{(\text{rad·m}^{-1})^2}=\text{m}^2\text{·s}^{-1}\). The spreading argument \(\dfrac{\beta t}{2\Delta x_0^2}=\dfrac{(\text{m}^2\text{s}^{-1})(\text{s})}{\text{m}^2}\) is dimensionless, as it must be under the square root; \(\Delta x(t)\) carries units of metres.

Limiting cases

Non-dispersive medium (\(\beta=0\), e.g. light in vacuum, ideal string): \(\Delta x(t)=\Delta x_0\) for all time and \(v_g=v_p=\omega/k\) constant — the packet is a rigid, non-spreading pulse.
Early times \(t\ll 2\Delta x_0^2/\beta\): \(\Delta x(t)\approx\Delta x_0\big[1+\tfrac12(\beta t/2\Delta x_0^2)^2\big]\), spreading is negligible and quadratic in \(t\).
Late times \(t\gg 2\Delta x_0^2/\beta\): \(\Delta x(t)\approx \dfrac{\beta\,t}{2\Delta x_0}\), width grows linearly and the spread velocity \(\beta/2\Delta x_0\) equals the group-velocity range across the initial band.
Free quantum particle \(\omega=\hbar k^2/2m\): \(v_g=\hbar k_0/m=p/m\) (the classical velocity) and \(\beta=\hbar/m\), giving \(\Delta x(t)=\Delta x_0\sqrt{1+(\hbar t/2m\Delta x_0^2)^2}\).
Deep-water gravity waves \(\omega=\sqrt{gk}\): \(v_g=\tfrac12\sqrt{g/k}=\tfrac12 v_p\), the group runs at half the crest speed, so crests appear to overtake and vanish through the group.

Breaks when

Broadband packets. If the spectral width is comparable to \(k_0\), the second-order truncation of \(\omega(k)\) is inadequate; third-order dispersion skews the envelope and produces oscillatory tails, so a single \(v_g\) and a symmetric spreading law no longer describe the motion.
Near a band edge or resonance. Where \(d\omega/dk=0\) (photonic/electronic band edge, roton minimum) the group velocity vanishes and \(\beta\) dominates; where a derivative diverges the expansion is meaningless. The packet stalls or breaks up rather than translating rigidly.
Nonlinear media. At high amplitude the modes couple: self-focusing and modulation instability can cancel dispersion (solitons) or catastrophically compress the packet, contradicting the linear spreading law entirely.
Absorbing/gain media (anomalous "superluminal" regimes). When \(\omega(k)\) is complex, \(v_g\) can exceed \(c\) or become negative; it then no longer represents the signal (front) velocity, and the boxed formulas describe pulse reshaping, not genuine energy transport.

Failure modes

Confusing phase and group velocity. Using \(v_p=\omega/k\) as the packet speed. For deep-water waves this is off by a factor of two; for a free electron the phase velocity is again half the group velocity, in the opposite sense.
Sign/curvature error in \(\beta\). Writing \(\beta=d^2\omega/dk^2\) but using \(d^2k/d\omega^2\) (the fibre GVD parameter \(\beta_2\)); these are reciprocal-related, not equal, and mixing them mis-scales the spreading time.
Thinking a narrower packet spreads more slowly. The intuition is backwards: small \(\Delta x_0\) means large \(\Delta k\), hence faster spreading — the \(\Delta x_0^{-2}\) in the argument is easy to miss.
Assuming the peak amplitude is conserved. Students hold \(\Delta x\) fixed and let height grow; in fact the norm (area) is conserved, so as \(\Delta x\) grows the peak intensity falls as \(1/\Delta x\).
Expanding about \(k=0\) instead of \(k_0\). Setting \(v_g=\omega/k\) evaluated at the origin rather than the slope at the carrier, valid only for a linear (non-dispersive) relation.
Forgetting the moving frame. Looking for spreading in the lab frame and being confused by the simultaneous translation; the clean broadening law lives in the \(\xi=x-v_g t\) frame.

Discussion

The deep content of this derivation is that information and energy travel with the group, not the phase. A pure sinusoid \(e^{i(kx-\omega t)}\) has crests moving at \(v_p=\omega/k\), but those crests carry no marker — to send a signal you must impose a modulation, and any modulation is a superposition of nearby \(k\), whose common motion is set by the slope \(d\omega/dk\). This is why \(v_p>c\) is permitted in a plasma or waveguide while \(v_g<c\) still protects causality: the phase velocity is a bookkeeping speed of unmarked crests, whereas the group velocity is (in transparent media) the speed of the energy flux.

Spreading is the price of localization in any dispersive system, and it is the classical wave root of quantum spreading in its guise as time evolution. For a free particle the envelope equation of Step 5 is literally the Schrödinger equation, with \(\beta=\hbar/m\) playing the role of the inverse mass. The \(\Delta x_0^{-2}\) scaling then expresses the uncertainty principle dynamically: pin a particle tightly and its momentum spread — hence its range of group velocities — grows, guaranteeing rapid delocalization. A macroscopic mass has \(\beta=\hbar/m\) so minuscule that spreading is unobservable over the age of the universe, which is why classical particles keep sharp trajectories.

The generating structure is stationary phase. The synthesis integral \(\int A(k)e^{i\Phi(k)}\,dk\) with \(\Phi=kx-\omega(k)t\) is dominated, for large \(x,t\), by wavenumbers where \(\partial\Phi/\partial k=0\), i.e. \(x=\omega'(k)t\): at position \(x\) and time \(t\) one selectively sees the wavenumber whose group velocity points there. The second derivative \(\Phi''=-\omega''(k)t=-\beta t\) controls the width of that stationary region, and its growth \(\propto|\beta t|^{-1/2}\) in amplitude (with the corresponding \(\propto|\beta t|\) spatial spread) is exactly the spreading law seen from the saddle-point side. Group velocity and dispersion are thus the first and second derivatives of a single phase function, unified by the method of stationary phase.

Common misconceptions. (i) "The wave packet spreads because the medium absorbs part of it" — no; spreading is unitary and lossless, the norm is exactly conserved while the width grows and the height falls. (ii) "Group velocity is always the signal velocity" — only in transparent regions; near strong absorption \(v_g\) loses that meaning and the true signal front still moves at \(c\). (iii) "A wave packet eventually settles to a fixed size" — in a purely dispersive linear medium it never stops spreading; only nonlinearity (solitons) or confinement can arrest it.

Worked examples

Example 1 — Spreading of a free-electron wave packet.

\[ \omega(k)=\frac{\hbar k^2}{2m}\ \Rightarrow\ v_g=\frac{d\omega}{dk}=\frac{\hbar k_0}{m},\qquad \beta=\frac{d^2\omega}{dk^2}=\frac{\hbar}{m} \]

Free-particle dispersion; group velocity is the classical velocity \(p/m\). A

\[ \beta=\frac{\hbar}{m}=\frac{1.055\times10^{-34}\ \text{J·s}}{9.11\times10^{-31}\ \text{kg}}=1.16\times10^{-4}\ \text{m}^2\text{s}^{-1} \]

Insert \(\hbar\) and the electron mass; units \(\text{J·s/kg}=\text{m}^2\text{s}^{-1}\). A

\[ \Delta x(t)=2\Delta x_0\ \Rightarrow\ \left(\frac{\beta t}{2\Delta x_0^2}\right)^2=3\ \Rightarrow\ t=\frac{2\sqrt3\,\Delta x_0^2}{\beta} \]

Solve the width law for the time to double the RMS width. B

\[ \Delta x_0=1\ \text{nm}:\quad t=\frac{2\sqrt3\,(10^{-9})^2}{1.16\times10^{-4}}=\frac{3.46\times10^{-18}}{1.16\times10^{-4}}\approx 3.0\times10^{-14}\ \text{s} \]

Numbers in; \(\text{m}^2/(\text{m}^2\text{s}^{-1})=\text{s}\). A

\[ t_{\times2}\approx 30\ \text{fs} \]

Reading. A nanometre-localized electron doubles in width in about thirty femtoseconds — spreading is dramatic at atomic scales. The same formula with the mass of a \(1\ \mu\text{g}\) grain gives \(t_{\times2}\sim10^{18}\,\text{s}\), longer than the age of the universe: dispersion is why small things smear and large things do not.

Example 2 — Group velocity and spreading of a deep-water swell train.

\[ \omega(k)=\sqrt{gk}\ \Rightarrow\ v_g=\frac{d\omega}{dk}=\frac12\sqrt{\frac{g}{k}},\qquad \beta=\frac{d^2\omega}{dk^2}=-\frac14\sqrt{g}\,k^{-3/2} \]

Deep-water gravity-wave dispersion; \(v_g=\tfrac12 v_p\). B

\[ \lambda_0=10\ \text{m}\Rightarrow k_0=\frac{2\pi}{\lambda_0}=0.628\ \text{m}^{-1},\quad v_g=\tfrac12\sqrt{\tfrac{9.81}{0.628}}=1.98\ \text{m·s}^{-1} \]

Evaluate the group velocity at the carrier wavenumber; \(v_p=3.95\,\text{m·s}^{-1}\), twice \(v_g\). A

\[ |\beta|=\frac14\sqrt{9.81}\,(0.628)^{-3/2}=\frac14(3.13)(2.01)=1.57\ \text{m}^2\text{s}^{-1} \]

Magnitude of the group-velocity dispersion at \(k_0\). A

\[ \Delta x_0=20\ \text{m},\quad \Delta x=2\Delta x_0:\ \ t=\frac{2\sqrt3\,\Delta x_0^2}{|\beta|}=\frac{2\sqrt3\,(400)}{1.57}\approx 8.8\times10^{2}\ \text{s} \]

Same doubling condition as Example 1, now with water-wave \(\beta\). B

\[ v_g\approx 2.0\ \text{m·s}^{-1},\qquad t_{\times2}\approx 15\ \text{min} \]

Reading. A 20 m swell group of 10 m waves drifts at about \(2\ \text{m·s}^{-1}\) while individual crests race through it at \(4\ \text{m·s}^{-1}\), and the group visibly broadens over a quarter hour — the mechanism that sorts ocean swell into long, clean sets by the time it reaches a distant shore.

Problems

(A) Two-wave beat. Add two equal-amplitude waves \(\cos(k_1x-\omega_1 t)+\cos(k_2x-\omega_2 t)\) with \(k_2=k_1+\Delta k\), \(\omega_2=\omega_1+\Delta\omega\). Show the result is a carrier modulated by an envelope, and identify the envelope speed.
Solution
Using \(\cos A+\cos B=2\cos\tfrac{A-B}{2}\cos\tfrac{A+B}{2}\): the sum is \(2\cos\!\big(\tfrac{\Delta k}{2}x-\tfrac{\Delta\omega}{2}t\big)\cos\!\big(\bar k x-\bar\omega t\big)\) with \(\bar k=k_1+\tfrac{\Delta k}{2}\), \(\bar\omega=\omega_1+\tfrac{\Delta\omega}{2}\). The slow factor is the envelope: it is constant along \(\tfrac{\Delta k}{2}x-\tfrac{\Delta\omega}{2}t=\text{const}\), i.e. moving at \(x/t=\Delta\omega/\Delta k\). The carrier moves at \(\bar\omega/\bar k\). In the limit \(\Delta k\to0\) the envelope speed \(\Delta\omega/\Delta k\to d\omega/dk=v_g\).
(B) Group velocity of a de Broglie wave. For a relativistic free particle \(E=\sqrt{(pc)^2+(mc^2)^2}\), with \(\omega=E/\hbar\) and \(k=p/\hbar\), show that \(v_g\) equals the particle's mechanical velocity \(v=pc^2/E\).
Solution
\(v_g=\dfrac{d\omega}{dk}=\dfrac{dE}{dp}\). Differentiate \(E^2=p^2c^2+m^2c^4\): \(2E\,dE=2pc^2\,dp\Rightarrow \dfrac{dE}{dp}=\dfrac{pc^2}{E}\). Since \(E=\gamma mc^2\) and \(p=\gamma mv\), \(\dfrac{pc^2}{E}=\dfrac{\gamma m v c^2}{\gamma mc^2}=v\). So \(v_g=v\), whereas the phase velocity \(v_p=\omega/k=E/p=c^2/v>c\): the phase runs faster than light, the group at the particle speed.
(C) Electron spreading time. An electron is localized to \(\Delta x_0=0.10\ \text{nm}\) (atomic scale). Using \(\beta=\hbar/m\), find the time for its RMS width to grow to \(\sqrt2\,\Delta x_0\).
Solution
The \(\sqrt2\) condition means \(\big(\tfrac{\beta t}{2\Delta x_0^2}\big)^2=1\Rightarrow t=\dfrac{2\Delta x_0^2}{\beta}\). With \(\beta=1.16\times10^{-4}\,\text{m}^2\text{s}^{-1}\) and \(\Delta x_0=1.0\times10^{-10}\,\text{m}\): \(t=\dfrac{2(1.0\times10^{-10})^2}{1.16\times10^{-4}}=\dfrac{2.0\times10^{-20}}{1.16\times10^{-4}}\approx1.7\times10^{-16}\,\text{s}=0.17\ \text{fs}\). Ten times tighter localization than Example 1 gives a hundredfold shorter spreading time, confirming the \(\Delta x_0^2\) scaling.
(B) Half-speed crests. For deep-water waves \(\omega=\sqrt{gk}\), a group is centred on \(\lambda_0=100\ \text{m}\). Find \(v_p\), \(v_g\), and the time for a crest to traverse a group of length \(L=300\ \text{m}\) relative to the group.
Solution
\(k_0=2\pi/100=0.0628\,\text{m}^{-1}\), \(\omega_0=\sqrt{9.81\times0.0628}=0.785\,\text{s}^{-1}\). \(v_p=\omega_0/k_0=12.5\,\text{m·s}^{-1}\); \(v_g=\tfrac12 v_p=6.25\,\text{m·s}^{-1}\). A crest overtakes the group at \(v_p-v_g=6.25\,\text{m·s}^{-1}\); to cross \(L=300\) m it takes \(t=L/(v_p-v_g)=300/6.25=48\ \text{s}\). Crests are continually born at the back of the group, sweep forward, and die at the front — the signature of \(v_g=\tfrac12 v_p\).
(C) Envelope obeys the dispersion equation. Starting from \(\phi(x,t)=\int A(\kappa)\,e^{i(\kappa x-(v_g\kappa+\frac12\beta\kappa^2)t)}\,d\kappa\), prove directly that \(\partial_t\phi+v_g\partial_x\phi=i\tfrac{\beta}{2}\partial_x^2\phi\), and hence that in the frame \(\xi=x-v_g t\) the envelope satisfies a free Schrödinger equation.
Solution
Differentiate under the integral: \(\partial_x\phi=\int A\,(i\kappa)e^{(\cdots)}d\kappa\), \(\partial_x^2\phi=\int A\,(-\kappa^2)e^{(\cdots)}d\kappa\), and \(\partial_t\phi=\int A\,\big(-i(v_g\kappa+\tfrac12\beta\kappa^2)\big)e^{(\cdots)}d\kappa\). Then \(\partial_t\phi+v_g\partial_x\phi=\int A\,\big(-i v_g\kappa-\tfrac{i}{2}\beta\kappa^2+i v_g\kappa\big)e^{(\cdots)}d\kappa=\int A\,\big(-\tfrac{i}{2}\beta\kappa^2\big)e^{(\cdots)}d\kappa\). Compare with \(i\tfrac{\beta}{2}\partial_x^2\phi=i\tfrac{\beta}{2}\int A(-\kappa^2)e^{(\cdots)}d\kappa=\int A(-\tfrac{i}{2}\beta\kappa^2)e^{(\cdots)}d\kappa\). The two agree, so \(\partial_t\phi+v_g\partial_x\phi=i\tfrac{\beta}{2}\partial_x^2\phi\). Changing to \(\xi=x-v_g t\), \(\tau=t\): \(\partial_t|_x=\partial_\tau-v_g\partial_\xi\) and \(\partial_x=\partial_\xi\), so the advection terms cancel and \(\partial_\tau\phi=i\tfrac{\beta}{2}\partial_\xi^2\phi\) — the free Schrödinger equation with \(\hbar/m\to\beta\).