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Seismic surface-wave dispersion is the frequency dependence of Rayleigh wave or Love wave propagation caused by depth-dependent elastic properties, mode structure and possibly intrinsic anelasticity. Phase velocity is $\omega /k$; group velocity is $d\omega /dk$. Different periods and branches have different depth sensitivities, enabling surface-wave tomography.

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

Differentiating $E_k\simeq \sqrt{{\hslash }^{3}q\alpha k/2}/m$ gives the displayed group velocity when $q\alpha k>0$. Its sign is the sign of $q\alpha$, equivalently of k on that branch. Writing only a positive square root of $q\alpha /k$ would lose this sign. Exactly, the Hellmann–Feynman theorem gives $v_y=⟨\hslash k-q\alpha x^2/2⟩/m$. Evaluating this small drift from a localized wavefunction requires enough anharmonic accuracy: a leading centred Gaussian alone need not reproduce it.

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 $(Y_s-Y{)}^{-1/4}$.

MIT's documented slotted-line waveguide measurement shorts one end of a rectangular electromagnetic waveguide and excites it through a probe coupled by capacitance. A second such probe moves through a longitudinal slot, aligned with the wall electric current to reduce disturbance. Its output locates axial standing wave nulls. The book's standing-wave setup chooses a frequency above the first waveguide cutoff frequency and below higher-mode cutoffs. Lowering it below the first cutoff instead reveals spatial decay away from the source. The text supplies apparatus and an observable comparison, rather than a table of measured null positions.^[ref]

The inference can be checked without assuming that detector output is proportional to electric field amplitude. At fixed measured frequency, record several minima at coordinates $z_0,\dots ,z_N$; then estimate ${\lambda }_g=2(z_N-z_0)/N$ when the minima are consecutive. This uses positions alone. Compare the result with the guided wavelength prediction using independently measured $a$ and $f$. The $40\mathrm{mm}$ spacing above is a synthetic example, not a reading from the MIT apparatus.

Additional proposed controls are to keep probe insertion fixed, repeat the scan in both directions, recheck source frequency, and compare shallow probe settings for shifts in the minima. These test probe loading, mechanical backlash and drift. A deliberately reflecting termination supplies the standing wave; varying its position should translate the pattern without changing adjacent-null spacing in a uniform single-mode region. Source-adjacent evanescent waves, unintended reflections, additional propagating normal modes, detector noise and finite positional resolution can spoil that inference. A disappearing remote signal alone does not distinguish below-cutoff decay from loss or poor excitation, and a scan at one frequency does not measure a pulse's group velocity.

For a plane internal gravity wave in uniform buoyancy frequency, substitute $\zeta =A(\mu t,\mu z)e^{i(kx+mz-\omega t)}$ into the displacement equation. The leading order gives ${\omega }^2=N^2{k}^2/(k^2+m^2)$; the next order gives the displayed transport equation with $c_{gz}=-\omega m/(k^2+m^2)$. A boundary amplitude $a(T)$ radiates as $A=a(T-Z/c_{gz})$, choosing the sign of $m$ so the group velocity points into the fluid. The wavelength-averaged energy is $N^2|A{|}^2/2$ and its vertical flux is $c_{gz}$ times that energy. A causal ramp therefore creates a translating front, not an instantaneous monochromatic field throughout the half-space.

Slowness is the reciprocal of phase velocity, measured in time per distance. A plane wave with wavevector $\mathbf{k}$ and angular frequency $\omega$ has vector slowness $\mathbf{p}=\mathbf{k}/\omega$. Its direction is the wavefront normal, which need not be the group velocity direction in an anisotropic medium.

A perfectly transmitted spin-resolved channel contributes $e^2/h$ to two-terminal linear conductance. Two degenerate spin channels give $2e^2/h$. Flux normalization cancels group velocity against the longitudinal state density.

For a stationary internal gravity wave, the Taylor–Goldstein equation gives $m^2=N^2/U^2-U^{\prime \prime }/U-k^2$. In a propagating interval with slowly varying $m$, the WKB approximation has amplitude proportional to $m^{-1/2}$ and phase derivative $m$. The conditions $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$ fail at a regular turning level. With $U>0$, the upward branch has $m>0$ and negative intrinsic frequency; laboratory group velocity is $U(k^2,km)/(k^2+m^2)$ when the curvature term is negligible.

Define intrinsic positive frequency $\omega$ and ground frequency $\hat{\omega }=Uk+\omega$ for a nonrotating internal gravity wave. Stationarity with $U>0$ selects $k<0$ and $\sqrt{k^2+m^2}=N/U$. Differentiating the full dispersion before imposing that condition gives the displayed ground-frame group velocity. Upward energy propagation requires $m<0$ and $|k|<N/U$. Its ray inclination is ${\cos }^{-1}(U|k|/N)$. The ground-frame group velocity is opposite the normal intrinsic phase velocity; the intrinsic group velocity is perpendicular to that phase velocity. A strictly stationary ground-frame phase pattern has zero phase velocity, so mixing frames can produce a false angle claim.

For a uniform eastward current $U>0$ on a beta plane with $\beta >0$, stationary nondivergent Rossby waves have $k^2+l^2=\beta /U$. Their ground-frame group velocity is $2U(k^2,kl)/(k^2+l^2)$, tracing a circle centered at $(U,0)$ with radius $U$. Packets emitted by a localized source throughout $0<s<t$ occupy the disk shown in geometrical ray theory. This envelope is distinct from the full balanced near field. A westward current has no nontrivial stationary propagating branch in this model.

For real $\alpha ,V>0$, use the zero-frequency dispersion relationwrite $\mathbf{k}=\kappa (\cos \varphi ,\sin \varphi )$ with $\kappa >0$ and $\cos \varphi >0$. Squaring alone would also admit a spurious negative-cosine branch. Thenwhere $\psi$ is the direction of the group velocity. Maximizing the angular slope gives a downstream wake wedge of semi-angle

A linear deep-water gravity wave of surface-displacement amplitude $a$ has equal mean kinetic energy and potential energy, and total mean energy per horizontal area $\mathcal{E}={\rho }_{w}g{a}^2/2$. Its energy transport speed is the group velocity. This amplitude is half the crest-to-trough height.

For a regular real mode between rigid boundaries, multiply the Taylor–Goldstein equation by its vertical structure and integrate to get the stationary quotient $k^2=\int ({\ell }^2{\varphi }^2-{\varphi }_z^2)dz/\int {\varphi }^{2}dz$. Along a differentiable dispersion branch, stationarity removes the variation of the eigenfunction, giving $2k=(\partial I/\partial c)dc/dk$. Since ${\partial }_c{\ell }^2=N^2/(U-c{)}^3+{\ell }^2/(U-c)$, the displayed group velocity follows from $d(kc)/dk$. Critical levels and vanishing branch derivatives require a separate limiting analysis.

Consider nonrotating internal gravity waves in constant buoyancy frequency $N>0$ and the imposed planar strain-rate tensor corresponding to $\mathbf{U}=\gamma (x,0,-z)$, with $\gamma >0$. The internal-wave ray tracing equations advect the horizontal and vertical wavenumbers asFor $k_0>0$, $m_0<0$ and initial height $z_0>0$, the intrinsic vertical group velocity is $c_{gz}=-Nkm/(k^2+m^2{)}^{3/2}>0$. Including mean advection givesEvery term in the bracket is positive. Moreover, $c_{gz}=O(e^{-3\gamma t})$, so the bracket tends to a positive finite limit. Thus the ray remains above $z=0$ at finite time and approaches it as $C{e}^{-\gamma t}$, despite its upward intrinsic group velocity. This is a ray prediction within the imposed flow with density stratification, rather than a statement that a finite-amplitude beam can contract indefinitely without viscous attenuation of an internal-wave beam.

With weak kinematic viscosity $\nu$, zero buoyancy diffusion and locally unchanged plane internal gravity wave polarization, the displacement pseudomomentum of an internal gravity wave obeys $P_t+{\partial }_z(c_{gz}P)=-\nu K^{2}P$. For constant wavenumbers and a stationary wave envelope, $P\propto |\hat{\sigma }{|}^2$ proves the displayed amplitude decay. Decay is along the group velocity, so $c_{gz}<0$ describes a source above the receiving level. In a slowly varying shear flow, use wave action and the changing intrinsic frequency rather than treating $P/|\hat{\sigma }{|}^2$ as constant.

The Whitham modulation equation gives $W_T+J_X=0$ with $W={\overline{L}}_{\omega }$ and $J=-{\overline{L}}_k$. For a linear wave family with $\overline{L}=A^{2}D(\omega ,k)/4$ and simple dispersion relation $D(\omega ,k)=0$, assume $D_{\omega }\ne 0$. Implicit differentiation gives the group velocityHence $W_T+{\partial }_X(c_{g}W)=0$. In several space dimensions this becomes $W_T+{\nabla }_X\cdot ({\mathbf{c}}_{g}W)=0$ for the corresponding slowly varying conservative linear-wave branch. A ray tube can therefore change local wave-action density by spreading or compression while conserving its transported integral when boundary fluxes vanish.

The relation between flux and group velocity uses the linear dispersion branch. General nonlinear averaged systems still have $J=-{\overline{L}}_k$, but this need not equal one scalar group speed times $W$. Dissipation, external forcing or interactions can also add source terms, so the homogeneous law requires its conservative, scale-separated setting.

For a small-amplitude conservative wave packet in a slowly varying fluid flow, its intrinsic energy divided by intrinsic frequency is transported along its group velocity, with the basic advection added in the laboratory frame. A stationary background conserves laboratory angular frequency, but a changing mean velocity changes the intrinsic one through the Doppler effect; intrinsic energy alone need not be conserved. Geometrical spreading and dissipation enter the action-flux balance.

The first-order imaginary balance of the wave equation in geometrical optics gives ${\partial }_T(\omega A^2)+\nabla \cdot (c^2\mathbf{k}{A}^2)=0$ for a real leading amplitude. The action density $\omega A^2$ travels with the group velocity $c^2\mathbf{k}/\omega$. For a complex amplitude replace $A^2$ by $|A{|}^2$.

For a smooth phase $\Theta$, define local wavenumber $k={\Theta }_x$ and local angular frequency $\omega =-{\Theta }_t$. Equality of mixed derivatives gives $k_t+{\omega }_x=0$. Dividing by $2\pi$ expresses conservation of signed crest count with density $k/(2\pi )$ and flux $\omega /(2\pi )$. When a homogeneous local dispersion relation gives $\omega =\omega (k)$, the equation becomes $k_t+v_g(k)k_x=0$; its characteristic curves move at the group velocity. This phase compatibility is distinct from energy or wave-action conservation.