Dispersion relation ($\omega (k)$)

A dispersion relation specifies which angular frequencies and wavevectors a linear wave model admits. Fix the plane wave conventionHere $a$ is a constant complex amplitude, $\mathbf{k}$ is the wavevector, and $\omega$ is the angular frequency. Substitution into a constant-coefficient linear partial differential equation replaces ${\partial }_t$ by $-i\omega$ and $\nabla$ by $i\mathbf{k}$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega =\omega (\mathbf{k})$ means one chosen branch, rather than a promise that the full dispersion relation is single-valued. For real $\mathbf{k}$, $\mathrm{Im}\omega >0$ means temporal growth in this convention, while $\mathrm{Im}\omega <0$ means damping. The real-branch discussion of phase velocity and group velocity below assumes negligible loss.

For the one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$, the dispersion relation is ${\omega }^2=c^2{k}^2$. The branch $\omega =ck$ carries right-moving travelling waves for $c>0$; both its phase velocity and group velocity are $c$. One may instead use the positive-angular frequency branch $\omega =c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega )$ with $(-k,-\omega )$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a dispersion diagram.

A particularly tangible dispersive example is a small-amplitude surface gravity wave on still water of uniform depth $h$. Neglect surface tension and viscosity, assume incompressible flow and irrotational flow, and impose a flat impermeable bottom. With $k>0$ and positive angular frequency,Here $g$ is gravitational acceleration and $\tanh$ is the hyperbolic tangent. Deep water means $kh\gg 1$: $\omega \simeq \sqrt{gk}$ and longer wavelengths have larger phase speed. Shallow water means $kh\ll 1$: $\omega \simeq \sqrt{gh}k$, an approximately nondispersive wave. Thus depth relative to wavelength, rather than depth alone, determines whether crests and wave packets travel together.^[ref]

The dispersion diagram below also includes the ideal deep-water capillary wave limit $\omega =\sqrt{\gamma /\rho }{k}^{3/2}$, where $\gamma$ is surface tension and $\rho$ is mass density. This limit requires the surface tension restoring term to dominate gravity; it reverses the trend of the surface gravity wave, giving larger phase speed at shorter wavelength. The combined gravity-capillary wave has ${\omega }^2=gk+(\gamma /\rho )k^3$.^[ref]

The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single dispersion relation encodes several observable behaviours: its value gives the angular frequency, its origin-to-point slope gives the one-dimensional phase velocity, its tangent slope gives the group velocity, and variation of that tangent slope predicts distortion of a wave packet. A straight line through the origin describes a nondispersive wave; an affine branch $\omega =ck+b$ with $b\ne 0$ has constant group velocity but phase velocity $c+b/k$, so those properties must be distinguished.

On a real positive-frequency branch of ${\omega }^2=a^2+c^2{k}^2$, phase velocity is $v_P=\omega /k$ and group velocity is $v_G=d\omega /dk=c^{2}k/\omega$. Their product is $c^2$. Real propagation requires ${\omega }^2>a^2$ when solving for a nonzero real $k$; the inequality $\omega >a$ alone assumes $a$ is nonnegative. The phase-group product does not, without further assumptions on the medium, prove a general statement about signal-front propagation.

When a dispersion relation factors as $\omega P(\omega ,k)=0$, dividing by $\omega$ discards a possible time-independent Fourier mode. This can remove a necessary stationary solution from an initial value problem. For instance, a constant-coefficient temporal equation $T^{(3)}+ia{T}^{\prime \prime }+b{T}^{\prime }=0$ has a constant solution as well as the two exponential solutions from its quadratic factor. Three initial conditions generally require all three modes. Cancellation is valid only when a nonzero frequency has already been required.

For the dispersion relation $m\omega =k^2$, take two plane waves with wavenumbers $\bar{k}\pm \delta k$ and corresponding angular frequencies $\bar{\omega }\pm \delta \omega$. Adding their two relations proves the title identity, while subtracting gives $\delta \omega /\delta k=2\bar{k}/m$. Thus the exact two-tone envelope speed differs from twice the midpoint carrier phase velocity by a second-order bandwidth correction. In the narrow-band limit the derivative group velocity is exactly twice the phase velocity on the central branch.

A wave whose phase speed depends on frequency. In a one-dimensional branch, phase velocity is $\omega /k$ and the narrow-packet group velocity is $d\omega /dk$. Loss, mode conversion and broad spectral bandwidth require care before identifying group speed with energy or signal transport. Confinement can create dispersion even when the underlying unbounded material has a constant wave speed.

For the prescribed sine-mode model $\omega =k\sqrt{c_s^2+\alpha k^2}$, positive small stiffness shifts higher harmonics more strongly. Writing $\eta =\alpha k_1^2/c_s^2$ and $k_2=2k_1$ gives ${\omega }_2/{\omega }_1=2+3\eta +O({\eta }^2)$. At the original fundamental quarter-period, the fundamental displacement changes to first order, but the second-mode cosine is at an extremum and changes only to second order. If the quarter-period is redefined with the shifted fundamental frequency, that fundamental displacement is exactly zero. Comparing profiles requires specifying which clock convention is held fixed.

For fixed complex frequency $\omega$, a spatial root is a complex wavenumber $k$ satisfying the dispersion relation. Analytic continuation of these branches from a causal temporal contour is essential in the Briggs-Bers criterion; their collision alone does not decide whether a spatial pinch point occurs.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

Modulational instability is growth of long-wavelength perturbations of a uniform or periodic wave due to coupling between its modulation and nonlinear dynamics. In the focusing nonlinear Schrodinger equation, the amplitude and phase perturbations can have imaginary modulation frequencies even though the unperturbed wave is an exact solution.

Linearizing a uniform Kerr-fibre background couples opposite frequency sidebands. With $i{A}_z-{\beta }_2{A}_{TT}/2+\gamma |A{|}^{2}A=0$, the displayed dispersion relation permits exponential growth precisely when ${\beta }_2\gamma <0$ and $0<{\Omega }^2<-4\gamma P/{\beta }_2$. For positive $\gamma$, this is anomalous group velocity dispersion. complex conjugation requires the paired complex-mode ansatz on the unstable branch. amplitude and power growth exponents differ by a factor of two.

Linearization around a pumped background couples the perturbation at positive frequency offset to the conjugate of the negative-offset perturbation. Their matrix eigenvalues determine amplification; assuming two unrelated scalar phase shifts misses the instability.

For the complex Ginzburg–Landau equation $A_t=A+(1+i{c}_1)A_{xx}-(1+i{c}_3)|A{|}^{2}A$, the uniform oscillation has a neutral phase mode whose growth rate is $\lambda =-(1+c_1{c}_3)k^2+O(k^4)$. Negative phase diffusion therefore causes a long-wavelength modulational instability. The Benjamin-Feir stability condition is the complementary positive-diffusion condition. On a finite domain an unstable wavenumber must actually be allowed by the boundary conditions.

For $-i{A}_t+\chi A_{xx}-\mu A+\delta A|A{|}^2=0$ with positive coefficients, linearize around $A=\sqrt{\mu /\delta }$ using real amplitude and phase perturbations. Their two-by-two dispersion relation gives the displayed equation. The modulational instability band is $0<K^2<2\mu /\chi$, with maximum growth rate $\mu$ at $K^2=\mu /\chi$.

A criterion for absolute wave-packet instability based on analytic continuation of the spatial roots of a dispersion relation. Start the temporal inversion contour above the spectrum, then lower it while deforming the spatial contour. A spatial pinch point occurs when branches originating in opposite spatial half-planes obstruct that deformation. A double root $D=D_k=0$ is only an algebraic candidate. The opposite-half-plane condition is stated in the primary study doi.org/10.1017/jfm.2016.195, section 4.2.

For $\omega (k)=i\alpha k^2-\beta k^3-i\gamma$, real-wavenumber growth is $\mathrm{Re}\alpha k^2-\mathrm{Im}\beta k^3-\mathrm{Re}\gamma$. A nonzero cubic coefficient is unbounded above on one side; after it vanishes, a positive quadratic coefficient is unbounded above on both sides. The displayed conditions are therefore necessary and sufficient for a finite temporal upper bound, which is then $-\mathrm{Re}\gamma$. The bound permits the initial causal temporal contour required by the Briggs-Bers criterion.

For the whole-line equation $A_t+a_1{A}_{xx}+a_2{A}_{xxxx}+a_{3}A=0$ with $\mathrm{Re}{a}_2>0$, the dispersion relation is $s=a_1{k}^2-a_2{k}^4-a_3$. The temporal Laplace transform of the impulse response at zero is the real-k integral of $[s+a_3-a_1{k}^2+a_2{k}^4{]}^{-1}/(2\pi )$. Factor the denominator as $a_2(k^2+{\kappa }_1^2)(k^2+{\kappa }_2^2)$ with $\mathrm{Re}{\kappa }_j>0$ for large $\mathrm{Re}s$. Partial fractions give the displayed resolvent. Square-root sheets are chosen by continuation from this causal large-s region; choosing roots independently can introduce false spatial pinch points.

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.

A collision of spatial branches which pinches the spatial inversion contour as the temporal contour is lowered from above all singularities. The branches must originate on opposite sides of the spatial contour. A collision of two upper-half-plane branches or two lower-half-plane branches can solve the double-root equations without causing absolute wave-packet instability.

For these constants, every real-wavenumber growth rate is $-k^2-1/10<0$. The real-contour impulse integral decays with envelope $e^{-t/10}{t}^{-1/2}$ at fixed position. Yet the stationary candidate $k=-2i/3$ has $\mathrm{Im}\omega =4/27-1/10>0$. It cannot be a contributing spatial pinch point; otherwise it would contradict the real-contour growth bound. This explicit example shows why solving the algebraic saddle equations is insufficient for absolute wave-packet instability.

The real dispersion above is temporally stable for every real $k$, but its analytic continuation has stationary points at $k=\pm i$ with $\omega =i/2$. Writing $\omega =i\Omega$ gives $k^2=-1\pm \sqrt{1/2-\Omega }$; as $\Omega$ decreases to $1/2$, both branches at $+i$ approach from the upper half-plane and both at $-i$ from the lower. These are not spatial pinch points. This provides a concrete counterexample to treating growing algebraic double roots as sufficient for absolute wave-packet instability.

A localized disturbance grows along some moving rays but decays at its original fixed location. This wave-packet meaning differs from buoyancy-driven stellar convection and is therefore given a distinct canonical title. Changing the observation frame can change the absolute/convective classification. The Briggs-Bers criterion uses the laboratory-frame impulse response to distinguish this case from absolute wave-packet instability.

A localized perturbation has positive asymptotic exponential growth at a fixed observation point in the chosen reference frame. The relevant spatial pinch point of the Briggs-Bers criterion has positive imaginary frequency. Merely finding a growing temporal mode or a formal complex saddle is not sufficient; the saddle must contribute to the localized impulse response.

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.

Wave dispersion means that phase velocity depends on wavenumber or, equivalently, wavelength. Wave packets generally travel with group velocity, which need not equal phase velocity. The linear rotating shallow-water dispersion relation ${\omega }^2=f^2+c^2{k}^2$ has faster long-wave phase motion but slower long-wave packet motion when $f\ne 0$.

The change of an optical propagation angle with wavelength. A diffraction grating at normal incidence has $d\sin \theta =m\lambda$, so differentiating gives $d\theta /d\lambda =m/(d\cos \theta )$. This converts a small spectral-line separation into angular separation, but resolving the two images also requires sufficient aperture and a narrow source image.

Intermodal dispersion is the temporal separation of components in different propagating normal modes because their group velocities differ. A pulse exciting several modes can therefore broaden even when its individual modal components have negligible bandwidth. It is distinct from group velocity dispersion within a single mode.

A nondispersive wave has constant phase velocity on a propagation branch. For $\omega =ck$, its group velocity is also $c$. A frequency relation with a nonzero offset, such as $\omega =ck+b$, has constant group velocity but a phase velocity $c+b/k$ depending on wavenumber; constant group velocity alone is therefore not sufficient for nondispersion.

A dispersion diagram plots the branches of a dispersion relation, usually with wavenumber on the horizontal axis and angular frequency on the vertical axis. Its slopes, intersections, and gaps display group velocities, mode conversion, and forbidden frequency ranges.

Ray tracing follows the paths along which a slowly modulated wave packet propagates. For a local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, the ray velocity is the group velocity ${\nabla }_{\mathbf{k}}\Omega$.

A path followed by a slowly modulated wave packet under a local dispersion relation. Its velocity is the group velocity, while its wavevector evolves as $d\mathbf{k}/dt=-{\nabla }_{\mathbf{x}}\omega$. In a uniform stationary medium both are constant, so the ray is straight even when phase velocity and group velocity differ.

A wave's travel time along a elastic ray is the integral of local slowness against arc length. Its variation with source and receiver position describes the arrival branches, including transmitted and reflected paths. Different paths can have different travel times between the same endpoints.

In a smooth isotropic medium, an elastic ray is the path followed by a local wavefront normal, with the appropriate P or S wave speed $c$. Arclength $s$ gives the displayed trajectory and travel-time equations. Its neighboring ray paths determine geometrical spreading, while the eikonal equation determines the normals. A caustic occurs where the neighboring paths focus and the elementary ray wave amplitude description ceases to be uniform.

For an isotropic radial wave speed $\alpha (r)$, the elastic ray equations imply conservation of $\mathbf{x}\times \mathbf{y}$, where $\mathbf{y}$ is the slowness vector. Indeed, its derivative is $\alpha \mathbf{y}\times \mathbf{y}+\mathbf{x}\times \nabla (1/\alpha )=0$. Thus each nonradial elastic ray lies in a fixed plane through the centre and $p=|\mathbf{x}\times \mathbf{y}|$ is constant. The angle $i$ is measured from the radial direction; tangential slowness continuity preserves $p$ across spherical interfaces.

For a spherically symmetric speed with strictly increasing $\eta (r)=r/v(r)$, a turning ray of parameter $p$ has $\Delta (p)=2p{\int }_p^{{\eta }_a}(d\log r/d\eta )({\eta }^2-p^2{)}^{-1/2}d\eta$. Integrate this against $(p^2-{\mu }^2{)}^{-1/2}$ and interchange the triangular integration region. The inner kernel integral is $\pi /2$, giving the displayed radius and then $v(r(\mu ))=r(\mu )/\mu$. Complete turning-ray branch data are required.

Where $\eta =r/v$ decreases outwards, it increases as a surface ray travels inward, so no surface ray turns inside that interval. Penetrating rays may traverse it and turn deeper. The radius is then not a single-valued function of $\eta$, and angular ray data constrain combined branch contributions rather than the single function inverted by Herglotz–Wiechert inversion. Additional structural or observational information is needed to locate the hidden profile.

In a constant-speed spherical shell with outer radius $R$ and inner radius $a$, a symmetric single reflection from the inner boundary has the displayed travel time. The boundary lies on the angular bisector of source and receiver. The branch exists only for $0\le \Delta \le 2\arccos (a/R)$, since the incident line must remain in the shell. At the endpoint it joins the mantle-only chord branch with the same slope $a/c$. Continuing the formula beyond that endpoint would make the line intersect the inner boundary before its supposed reflection point.

A solid-to-fluid interface can turn spherical source P waves into parallel vertical rays if its height $Z(R)$ satisfies the displayed travel-time identity, with $r=\sqrt{R^2+Z^2}$. The positive branch through $Z(0)=z_0$ is a hyperboloid when $\alpha >{\alpha }_f$. Differentiating the travel-time identity enforces Snell law for elastic and acoustic waves. Travel time above the interface is then $z_0/\alpha +(z-z_0)/{\alpha }_f$, independent of cylindrical radius. The leading wavefront is planar although its wave amplitude need not be uniform across that plane.

A local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{X},T)$ yieldsThe spatial and temporal derivatives on the right hold the wavevector fixed. Translation symmetries conserve the corresponding wavevector components, while time independence conserves frequency.

A simple turning point for a stratified wavespeed occurs at $c(Y_s)=\omega /k$, where $Q={\omega }^2/c^2-k^2$ has a nonzero derivative. If $c_Y(Y_s)>0$, write $q=-Q^{\prime }(Y_s)>0$; then $\ell \sim [q(Y_s-Y){]}^{1/2}$ below it. The vertical group velocity vanishes and the stationary wave amplitude in a stratified wavespeed diverges as $(Y_s-Y{)}^{-1/4}$.

In ${\epsilon }^2{\Psi }^{\prime \prime }+2{\epsilon }^2(c^{\prime }/c){\Psi }^{\prime }+Q(Y)\Psi =0$, with $Q\sim -q(Y-Y_s)$ and $q>0$, set $Y-Y_s=({\epsilon }^2/q{)}^{1/3}\eta$. The leading equation is ${\Psi }_{\eta \eta }-\eta \Psi =0$, the Airy ordinary differential equation. The first derivative and nonlinear coefficient corrections are smaller by $O({\epsilon }^{2/3})$.

The local inner solution is a combination of Airy functions $\mathrm{Ai},\mathrm{Bi}$. Decay in the forbidden region removes $\mathrm{Bi}$. The remaining Airy turning-point connection formula is oscillatory on the allowed side and represents an incident/reflected pair, not a single upward branch. Without the forbidden-side condition the local differential equation alone leaves both coefficients free.

A simple ray turning point has outer amplitude proportional to $|Y-Y_s{|}^{-1/4}$. At the Airy scaling at a variable-speed wave turning point, $|Y-Y_s|=O({\epsilon }^{2/3})$, this gives an inner wavefield of size $O({\epsilon }^{-1/6})$ when the incident action normalization is order one. The enhancement is finite for nonzero $\epsilon$ and is resolved by the Airy function.

For the zero-frequency dispersion relationwrite $\mathbf{k}=\kappa (\cos \varphi ,\sin \varphi )$. Thenwhere $\psi$ is the direction of the group velocity. Maximizing the angular slope gives a downstream wake wedge of semi-angle

For a normal mode proportional to $e^{\sigma t}$, the real part $\mathrm{Re}\sigma$ is its exponential growth rate. A positive growth rate signals linear instability.

For a dispersion relation $\omega (k)$, the phase velocity is $c_p=\omega /k$ and the group velocity is $c_g=d\omega /dk$. For the Klein-Gordon relation at $k>0$,

For a differentiable dispersion relation and nonzero wavenumber $k$, group velocity equals phase velocity precisely where $d{v}_p/dk=0$. Equality at an isolated stationary point of $v_p$ does not establish a nondispersive branch. For a gravity-capillary wave, the equality occurs at $k=\sqrt{\rho g/T}$ and the phase-speed minimum, even though the relation is dispersive elsewhere.

A useful stipulated dispersive medium obeys the Klein-Gordon equation $u_{tt}-c^2{u}_{xx}+{\omega }_0^{2}u=0$, with $c>0$ and ${\omega }_0>0$. Its positive-angular frequency dispersion relation isHere ${\omega }_0$ is the gap at zero wavenumber, not the central angular frequency ${\omega }_c=\omega (k_0)$ of a wave packet; the parameter in the preceding formula is $A={\omega }_0/c$. For $k>0$, the phase velocity satisfies $v_p>c$, the group velocity satisfies $0<v_g<c$, and $v_p{v}_g=c^2$. In this stipulated medium $c$ is the limiting propagation speed of the model; it need not mean the vacuum speed of light. The same square-root dispersion relation appears in waveguide phase and group velocities, with a waveguide cutoff frequency playing the role of ${\omega }_0$.^[ref]

For the original dimensionless choice $c={\omega }_0=k_0=1$, a carrier wave has phase velocity $v_p=\sqrt{2}$, while a narrow wave packet has leading group velocity $v_g=1/\sqrt{2}$. After time $t=8$, the carrier's zero-wave phase reference has moved to $x=8\sqrt{2}\simeq 11.31$, whereas the leading wave envelope centre is at $x=8/\sqrt{2}\simeq 5.66$. A carrier wave crest therefore moves through the wave envelope; it does not carry an identifiable piece of the envelope with it.

The origin-to-curve secant on the dispersion diagram gives phase velocity, while its local tangent gives group velocity. As $k\to 0^{+}$, $v_p\to \infty$ and $v_g\to 0$; at $k=0$ the phase velocity itself is undefined. As $k\to \infty$, both approach $c$, from opposite sides. On the same positive-angular frequency branch with $k<0$, both signed velocities are negative and their product remains $c^2$. These conventions distinguish direction from phase speed. Reversing both $k$ and $\omega$ represents the same real plane wave; changing just one sign changes its direction.

For an original finite-bandwidth check, take equal-amplitude plane waves at $k_1=0.8$ and $k_2=1.2$, still with $c={\omega }_0=1$. The exact two-tone wave envelope speed iswhile its carrier wave has phase velocity $(\sqrt{1+0.8^2}+\sqrt{1+1.2^2})/2\simeq 1.421337$. Their product is exactly $1$: rationalizing the difference gives $v_{\mathrm{beat}}=2/({\omega }_1+{\omega }_2)$. Nevertheless $v_{\mathrm{beat}}\ne {\omega }^{\prime }(1)$; the derivative describes the limit of vanishing separation, not an arbitrary pair of tones. The two-tone wave envelope repeats indefinitely, so it has no isolated first arrival and cannot measure front velocity.

If a differentiable dispersion relation $\omega (\mathbf{k})$ is homogeneous of degree zero at nonzero $\mathbf{k}$, the Euler homogeneous function theorem gives $\mathbf{k}\cdot {\nabla }_{\mathbf{k}}\omega =0$. Therefore phase velocity $\omega \mathbf{k}/k^2$ is orthogonal to group velocity ${\nabla }_{\mathbf{k}}\omega$. The result applies exactly to limiting inertial waves; finite compressibility can alter the dispersion.

The phase velocity is the motion of a constant-wave phase surface. For a one-dimensional plane wave $u=a\cos (kx-\omega t+{\varphi }_0)$ with real $k\ne 0$ and real $\omega$, fixing $kx-\omega t+{\varphi }_0$ givesEach wave crest of this monochromatic travelling wave advances at that signed phase velocity. Its magnitude is the phase speed $|c_p|=\lambda /T$, where $\lambda =2\pi /|k|$ is the wavelength and $T=2\pi /|\omega |$ is the temporal period when $\omega \ne 0$. Choosing $\omega >0$ makes positive $k$ mean rightward phase velocity and negative $k$ mean leftward phase velocity. Reversing both signs leaves the real plane wave and its phase velocity unchanged. At $k=0$ there are no spatial crests to track, so $\omega /k$ is undefined; a finite limit as $k\to 0$ is a limiting phase speed, not the speed of a spatially uniform oscillation.

In several dimensions, a plane wave has wave phase $\mathbf{k}\cdot \mathbf{x}-\omega t+{\varphi }_0$. Constant-wave phase planes obey $\mathbf{k}\cdot \mathbf{v}=\omega$. Their normal phase velocity isTangential motion within a plane does not move that wavefront, which is why the normal direction fixes the vector convention. In an anisotropic medium the group velocity need not point along this wavevector. Likewise the horizontal phase velocity of a tilted wavefront is its crossing speed along a horizontal line, rather than a component of the normal phase velocity.

For the nondispersive wave branch $\omega =ck$, a wave crest and a wave envelope both move at $c$. For deep-water surface gravity waves, $\omega =\sqrt{gk}$ gives $c_p=\sqrt{g/k}$ and group velocity $c_g=c_p/2$. With the illustrative choices $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $\lambda =1\mathrm{m}$, the model predicts $c_p\simeq 1.25\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.625\mathrm{m}{\mathrm{s}}^{-1}$. These are calculated values for negligible surface tension and sufficiently large $kh$, not measurements. A wave crest moves forward through the wave packet, so timing a crest does not time the packet's journey.

There is a direct laboratory meaning to this distinction. MIT's 2005 surface-wave lab uses a long water tank, a controllable hydraulic paddle, and an absorbing beach. Resistance probes convert submerged length into voltage; calibration converts their readouts into free surface elevation versus time. Two probes a known distance apart, backed by a ruler and stopwatch, let students time a wave crest and separately time a feature of the wave packet. The lab records depth and varies frequency and wave amplitude.^[ref]

For a clean measurement of phase velocity, fit the periodic portion of both records or follow one identifiable wave crest. A phase difference is only known modulo $2\pi$: probe spacing below half the shortest relevant wavelength, or continuity while sweeping frequency, avoids assigning the wrong whole-cycle delay. Then compare the timed phase velocity with $\lambda /T$. Tracking pepper grains instead measures fluid-particle motion, which has a different meaning from either phase velocity or group velocity.

The Klein-Gordon packet velocities example makes the distinction especially sharp: its phase velocity exceeds $c$, yet its narrow wave envelope moves below $c$. A carrier wave labels successive oscillations; its speed alone does not establish information velocity. The front velocity depends on the full causal model, including angular frequencies far outside the carrier's narrow range.

For phase $kx+mz-\omega t$ with $k\ne 0$, hold $z$ fixed: a contour of constant wave phase advances in $x$ at $\omega /k$. This is the speed at which a tilted wavefront crosses a horizontal line. It differs from the horizontal component of the vector phase velocity, $\omega k/(k^2+m^2)$, which is defined along the wavevector.

The magnitude of the phase velocity of a real-frequency plane Fourier mode is $|\omega |/|\mathbf{k}|$. In an anisotropic medium it depends on the direction of the wave vector and need not equal the group velocity magnitude.

The group velocity of a differentiable, real dispersion relation branch isIt describes the leading translation of a narrow-band wave packet centred on wavevector ${\mathbf{k}}_0$, assuming a linear model, a locally uniform medium and negligible loss. Narrow-band means that the appreciably excited wavevectors occupy a small neighbourhood on which one smooth branch can be linearized. Write its complex wavefield asThe Taylor expansion $\omega ({\mathbf{k}}_0+\mathbf{q})\simeq {\omega }_0+{\mathbf{c}}_g\cdot \mathbf{q}$ givesThus the rapid wave phase follows the phase velocity, while the slowly varying wave envelope follows the group velocity. This Fourier analysis argument is an outline of the approximation, rather than a statement that every wave packet translates rigidly forever. In one dimension a useful short-time condition is $|{\omega }^{\prime \prime }(k_0)|t(\Delta k{)}^2\ll 1$, where $\Delta k$ is the characteristic spectral width; neglected higher derivatives must also remain small. Curvature of the dispersion relation eventually changes the wave envelope.^[ref]

Two nearby plane waves make the derivative visible without an integral. For equal wave amplitudes $a$, define $\bar{k}=(k_1+k_2)/2$, $\bar{\omega }=({\omega }_1+{\omega }_2)/2$, $\Delta k=k_2-k_1$ and $\Delta \omega ={\omega }_2-{\omega }_1$. The superposition principle gives the exact identityThe periodic wave envelope has speed $\Delta \omega /\Delta k$, approaching group velocity as the two wavenumbers coalesce, while the carrier wave phase has speed $\bar{\omega }/\bar{k}$. Two tones alone form an infinite beat train, not a localized wave packet. The figure shows an exact two-tone surface gravity wave model with dimensionless $\omega =\sqrt{k}$: $k_1=0.9$, $k_2=1.1$. Its carrier phase velocity is approximately $0.99875$, its finite-difference envelope velocity approximately $0.50063$, and ${\omega }^{\prime }(1)=0.5$ is the limiting group velocity. The marked carrier phase line need not coincide exactly with a maximum of the full modulated wave.

The deep-water surface gravity wave therefore has $c_g=c_p/2$, whereas the deep-water capillary wave limit has $c_g=3c_p/2$: crests can drift either forward or backward relative to a wave envelope. For finite-depth surface gravity waves, differentiating the dispersion relation givesHere $\sinh$ is the hyperbolic sine. The shallow-water limit has $c_g\simeq c_p\simeq \sqrt{gh}$; the deep-water limit has $c_g\simeq c_p/2$.^[ref]

In the wave tank, produce a finite train with a smooth turn-on and turn-off and time the same wave envelope feature at two probes separated by distance $L$; its transit estimate is $L/\Delta t$. Independently follow wave crests through that train. Techet's 2003 lab asks students to observe a deep-water train's foremost crest disappearing as the group moves forward, and to compare deep- and shallow-water cases. The interpretation is that a wave crest overtakes the slower wave envelope, dwindling at its front while other crests become visible at its rear. This is a qualitative expected observation in the teaching protocol, not a published set of measured curves.^[ref]

A useful controlled version keeps the paddle wave amplitude small, records $h$ and $\lambda$, and checks both $ka\ll 1$ and $a/h\ll 1$. Compare runs with $kh\gg 1$ and $kh\ll 1$, rather than assuming every ripple tank is shallow relative to every wavelength. Increase train duration at fixed central frequency to narrow its spectrum, and check that the envelope transit estimate stabilizes. Calibrate both probes, sample fast enough to resolve the carrier, and compare records before reflected waves return; an absorbing beach reduces this contamination. A current changes laboratory velocities, large wave amplitudes introduce nonlinear effects, and very short wavelengths require surface tension. Strong damping or noticeable reshaping makes the chosen threshold or peak an unreliable marker of simple group velocity. These are proposed controls and model limits, not claims that the cited lab measured every correction.

Three velocities should consequently be kept separate. The group velocity predicts narrow-band wave envelope motion. An energy-transport velocity is energy flux divided by energy density, using the appropriate time average; it agrees with group velocity in standard lossless linear wave models when all stored energy is included.^[ref] A signal-front velocity concerns the earliest response to a new disturbance and requires a causal model over its full spectrum. A strictly band-limited wave packet has no sharply bounded onset, and a mathematical two-tone envelope cannot test front arrival. In strongly absorbing or broadband propagation, a single group velocity need not describe the observed peak, energy transport, or first detectable response. A complex $\omega$ or $k$ also prevents interpreting its derivative as an ordinary real transport velocity without further analysis.

In particular, an unusually large or negative group velocity from pulse reshaping does not establish superluminal information transmission. For the causal dielectric model analysed by Fitzpatrick, the field ahead of the front vanishes for $t<x/c$, even when its phase velocity exceeds the vacuum speed of light $c$. That result concerns the electromagnetic model's causality; the ideal water-wave diagrams above contain no relativistic signal-front claim.^[ref]

Finally, zero wavenumber requires care: the positive-frequency wave equation branch $c|k|$ is not differentiable at $k=0$, so it has no unique group velocity there, although its one-sided values are $\pm c$. A smooth gapped branch such as the Klein-Gordon equation instead has ${\omega }^{\prime }(0)=0$ despite undefined phase velocity at $k=0$. The same signed branch must be used in both derivatives and ratios; a negative group velocity means an envelope moving in the negative coordinate direction, not a negative elapsed time. The practical value of group velocity is precisely this translation between a local slope on a dispersion diagram and a measurable packet transit, with its assumptions made explicit.

The Gaussian envelope in a Klein-Gordon medium supplies a quantitative next step: the first derivative translates the wave envelope, the second derivative spreads it, and higher derivatives limit that approximation. At finite spectral width, a peak's measured speed need not equal the group velocity at the central wavenumber. Neither a peak nor a fixed fraction of its height generally identifies front velocity.

Since $k=\omega /c(\omega )$, differentiation gives $u=(dk/d\omega {)}^{-1}=c/[1-(\omega /c)dc/d\omega ]$. Using the period $P=2\pi /\omega$ gives the displayed formula. A positive period slope makes the group velocity smaller than the phase velocity on this branch. A slope read from a graph produces an approximate local result, not a globally fitted dispersion law.

The relative envelope delay accumulated by waves with different group velocities over length $L$. In nonlinear pulse gating this can blur the temporal overlap and limits the useful crystal thickness.

Writing the material wavelength as ${\lambda }_m=2\pi /k$ and its phase velocity as $v_p({\lambda }_m)$ gives $v_g=v_p-{\lambda }_{m}d{v}_p/d{\lambda }_m$. This follows by differentiating $\omega =k{v}_p$ and using $kd{\lambda }_m/dk=-{\lambda }_m$. If instead the independent variable is vacuum wavelength ${\lambda }_0=2\pi c/\omega$, then $k=\omega /v_p({\lambda }_0)$ givesSubstituting vacuum wavelengths into the material-wavelength derivative formula gives an erroneous result beyond first order in dispersion. The two coordinates describe the same dispersion relation, but their derivatives differ by the chain rule.

Variation of inverse group velocity with angular frequency; for a uniform medium of length $L$, the accumulated group delay dispersion is $L{d}^{2}k/d{\omega }^2$.

A pulse's frequency components accumulate a differential group delay $\Delta t=L|{\partial }_f(1/v_g)|\Delta f$. If that spread approaches the pulse repetition period $T_r$, neighbouring symbols overlap. The resulting length estimate depends on bandwidth, initial pulse width, chirp and the receiver's allowable intersymbol interference. For $n(f)=n_0+\tau (f-f_0)$, the group index is $n_0+\tau (2f-f_0)$ and ${\partial }_f(1/v_g)=2\tau /c$: differentiating the phase index alone loses a factor of two.

A digital repeater can recover the symbol sequence and generate new pulses with restored amplitude, shape and timing. This resets waveform degradation between spans, provided decisions remain reliable. An amplifier alone increases the broadened pulse amplitude and also adds noise; it does not undo group velocity dispersion. optical dispersion compensation can instead reverse an accumulated frequency-dependent delay within its design bandwidth.

The distance over which second-order group velocity dispersion substantially changes a pulse of characteristic duration $T_0$. Width conventions and initial chirp affect numerical broadening factors; the displayed definition is the standard envelope-equation scale.

The front velocity is the propagation speed of the earliest boundary of a newly introduced disturbance's support. It asks when an initially unaffected observer can first respond, whereas group velocity asks how a narrow wave envelope translates. A detector threshold defines a practical arrival time, but can miss arbitrarily small earlier responses; a packet peak is another distinct timing convention. In a causal dielectric model, a high-angular frequency contour argument gives exactly zero field for $t<x/c$, even when phase velocity exceeds $c$. This is a statement about that model's relativistic causality, not a universal identification of group velocity with front velocity.^[ref]

For the stipulated Klein-Gordon equation with compactly supported real Cauchy data, front velocity is bounded by $c$, and $c$ is the model's maximal speed of influence. A short energy method outline usesIntegration outside an expanding interval shows that zero initial energy there remains zero. Thus initial support in $[-a,a]$ cannot influence $x>a$ before $t=(x-a)/c$. The positive energy density includes the restoring term ${\omega }_0^2{u}^2/2$: this adds oscillation behind the front without extending its domain of dependence. A particular disturbance may fail to excite the leading boundary, so the maximal speed is not a promise of a detectable response for every source.

A Gaussian function is nonzero arbitrarily far away, so the Gaussian envelope in a Klein-Gordon medium has no sharp initial support boundary. Its small distant tails are already in the initial data. Likewise, keeping only a narrow interval of the frequency spectrum cannot reproduce an exact abrupt onset. Cutting off a Gaussian tail changes its Fourier transform and the accompanying initial velocity; it is a different initial-value problem. The exponential initial-velocity tail for the Klein-Gordon equation is another concrete example of a pre-existing tail outside the main disturbance.

The water-tank measurement in phase velocity records free surface elevation with calibrated resistance probes and uses a ruler and stopwatch for an independent check. The inspected protocol calls the front or back of a group a timing feature. That visible edge establishes an envelope transit estimate, not the first nonzero disturbance in the sense of front velocity.^[ref] A proposed additional control would repeat the transit fit at several detection thresholds and train durations, using matched probe calibration and the same sampling rate. Dependence on the threshold diagnoses wave envelope reshaping or limited sensitivity; records should be compared before reflections return. Such a test could expose limitations of a group velocity estimate, but would not establish an exact causal front or validate the stipulated Klein-Gordon equation for water waves.

An information velocity must specify what counts as a newly distinguishable message and how it is detected. In a causal model, receivers cannot distinguish two otherwise identical preparations before their difference can arrive. Front velocity therefore bounds the arrival of genuinely new changes when the model has a finite domain of dependence. Group velocity can estimate the delay of a weakly distorted narrow wave packet, but it is not a universal speed of information: damping, amplification and reshaping can move the peak without advancing the first possible response. A Gaussian peak shift alone does not test relativistic causality.^[ref]

The wave crest and wave trough are respectively the highest and lowest displacements in a wave.

A wave crest is a local maximum of the wave's displacement. In a monochromatic travelling wave it lies at a fixed phase and propagates at the phase velocity.

A wave trough is a local minimum of the wave's displacement.

A wave packet is a localized superposition of Fourier modes. In a narrow packet, nearby wavenumbers interfere to produce a slowly varying wave envelope around a rapid carrier wave. Its leading translation is governed by group velocity; the carrier's wave phase follows phase velocity. Localized does not necessarily mean compact support: a Gaussian function decays rapidly but extends over the whole line. The Gaussian envelope in a Klein-Gordon medium gives an original example with both translation and spreading.

For a nearly monochromatic packet of length $L$ travelling past a stationary observer, the snapshot count is $L/\lambda$ but the crossing count is $fL/v_g=(L/\lambda )(v_p/v_g)$. In the envelope frame the observer moves at $-v_g$ and the crests at $v_p-v_g$. For deep-water gravity waves, $v_p=2v_g$, so the observer counts twice as many crests as appear in an instantaneous snapshot. A sharply bounded monochromatic packet is an idealization; the count describes the approximately uniform central train.

A wave envelope is a slowly varying wave amplitude multiplying a rapidly oscillating carrier wave. For a complex wavefield $\psi =A(x,t)e^{i(k_{0}x-{\omega }_{c}t)}$, its nonnegative amplitude envelope is $|A|=|\psi |$; the complex $A$ can also encode a slowly varying wave phase. This choice distinguishes the amplitude envelope from a signed modulation factor, such as the cosine factor in a two-tone superposition. A slowly varying wave envelope is useful when the carrier's wavelength and period are short compared with its variation scales. Strong overlap between distinct packets or a broad frequency spectrum can spoil a unique carrier-envelope interpretation.

A carrier wave is the rapid reference oscillation in a modulated wave. In the narrow-packet representation $\psi =A(x,t)e^{i(k_{0}x-{\omega }_{c}t)}$, choose $k_0$ near the centre of the frequency spectrum and ${\omega }_c=\omega (k_0)$ on its dispersion relation. Its constant-wave phase reference moves at phase velocity ${\omega }_c/k_0$. This reference need not coincide with a local maximum of the complete real wavefield when the wave envelope varies. Moving the reference $k_0$ shifts some slow wave phase into $A$ without changing $\psi$.