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An absolute hydrodynamic instability makes the response to a localized impulse grow at a fixed spatial point in a specified frame. This differs from convective hydrodynamic instability, which amplifies a moving wave packet while its response decays at fixed positions. For analytic dispersion relations, the relevant absolute frequency is associated with an accessible zero-group velocity saddle, with the spatial-branch selection checked rather than assumed for an arbitrary dispersion relation.

An absolute wavenumber is a complex wavenumber at the selected zero-group velocity saddle of a dispersion relation. For the linear complex Ginzburg-Landau equation, $k_0=-U/[2(c_d-i)]$; the positive real part of the diffusion coefficient makes the Gaussian saddle accessible from the real Fourier contour.

For a slowly varying parallel shear flow $U(y){\mathbf{e}}_x$, the leading WKB method neglects mean-velocity gradients while retaining local advection. The continuity equation and linearized Euler equations then give the displayed acoustic dispersion relation. On the positive intrinsic-frequency branch, $\omega =U{k}_x+c_0|\mathbf{k}|$ and the group velocity is $U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$. A wavefront normal is parallel to $\mathbf{k}$; the acoustic energy-ray direction generally differs because of the advection term.

An acoustic phonon branch has $\omega (k)\to 0$ as $k\to 0$. Its long-wavelength group velocity is the speed of sound.

For ${\eta }_t+\mathbf{c}\cdot \nabla \eta =\mu \eta +\nabla \cdot (\Gamma \nabla \eta )$ with symmetric positive definite $\Gamma$, the dispersion relation is $\omega =\mathbf{c}\cdot \mathbf{k}+i(\mu -{\mathbf{k}}^T\Gamma \mathbf{k})$. Its zero-group velocity saddle is ${\mathbf{k}}_0=-i{\Gamma }^{-1}\mathbf{c}/2$. The anisotropic heat kernel has fixed-position exponential rate $\mu -{\mathbf{c}}^T{\Gamma }^{-1}\mathbf{c}/4$, so this accessible saddle gives the actual threshold. At fixed speed $q$, the threshold ranges from $q^2/(4d_{\max })$ to $q^2/(4d_{\min })$, where $d_{\min },d_{\max }$ are the eigenvalues of $\Gamma$. Absence of absolute hydrodynamic instability in every direction requires the upper bound $\mu \le q^2/(4d_{\max })$.

For the $n=1$, $m=1$ Hermite vertical mode of an isothermal disk, the local dispersion relation is $[(\omega -{\Omega }_K{)}^2-{\Omega }_K^2{]}^2=k^2{c}_s^2(\omega -{\Omega }_K{)}^2$. Its low inertial-frame frequency branches satisfy $\omega \simeq \pm k{c}_s/2$. These waves transport a changing tilt of successive annuli, or a warp, with group velocity $\pm c_s/2$. The vertical velocity is approximately independent of height. A rigid tilt is the zero-wavenumber, zero-pattern-frequency limit; the gas still executes vertical orbital motion at frequency ${\Omega }_K$ in its rotating frame.

The extraordinary wave's Poynting vector can differ in direction from its wavevector. The latter specifies wave phase propagation, while the former specifies energy flux. This angular separation is walk-off.^[ref]

In the lossless, nondispersive model above, take the gradient of $\omega =c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the group velocity, parallel to the Poynting vector. If its angle to the optic axis is $\varphi$, this gives $\tan \varphi =(n_o^2/n_e^2)\tan \theta$. Consequently the extraordinary wave generally carries energy away from its wavevector direction. An internal wavevector normal to a slab face therefore does not guarantee a normal energy path. Birefringent walk-off vanishes at $\theta =0$ and $\theta =\pi /2$; the latter direction still has unequal refractive indices. This is a useful waveplate geometry: overlapping components can acquire phase retardation without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one linear polarization can also hide a double image, for different reasons.

In a single isolated energy band, a wave packet subject to a constant force satisfies $\hslash \dot{k}=F$. Its group velocity is periodic in $k$ with reciprocal-lattice period $2\pi /a$, so its position oscillates with period $2\pi \hslash /(|F|a)$ when its mean velocity over a band is zero. For $E(k)=E_0-2J\cos ka$, integrating the sinusoidal velocity gives the oscillation explicitly. This ideal description neglects scattering and interband tunnelling.

A moving boundary excites an internal gravity wave through its kinematic boundary condition. For wavenumber $k$ and angular frequency $\omega$, the vertical velocity field satisfies $W^{\prime \prime }+k^2(N^2/{\omega }^2-1)W=0$. The radiation condition selects outward group velocity; when $\omega >N$, the bounded response is an evanescent wave.

The capillary length measures when gravity and surface tension are comparably important in shaping a liquid interface. For a liquid of uniform mass density $\rho$, beneath a gas whose mass density is negligible, with uniform gravitational acceleration $g>0$ and constant surface tension $\gamma >0$,A deformation of height $a$ and horizontal length scale $L$ has hydrostatic pressure variation of order $\rho ga$ and small-slope capillary pressure of order $\gamma a/L^2$. Equating these gives $L\sim {\ell }_c$: surface tension smooths small features, whereas gravity matters for broad ones. For two resting fluids, replace $\rho$ by their positive mass density difference $\Delta \rho$ in this static balance. This change alone does not give the two-fluid wave dynamics, because both fluids can have inertia. ^[ref]

With the stipulated illustrative parameters $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, the capillary length is ${\ell }_c=2.71\mathrm{mm}$. These parameters define the ideal calculations below; they are not a measurement of a particular sample.

For a sinusoid with wavenumber $k>0$, the relevant horizontal scale is $L=1/k$, while its wavelength is $\lambda =2\pi /k$. Thus the Bond number for this scale and the reciprocal restoring-term ratio areEquality is at $k_c=1/{\ell }_c$, hence ${\lambda }_c=2\pi {\ell }_c=17.02\mathrm{mm}$ for these parameters. Using $L=\lambda$ instead produces a different numerical Bond number; state the length convention. The capillary length itself is not the equality wavelength. This distinction helps choose whether a measured ripple should be modeled as a surface gravity wave, a nearly pure capillary wave, or a gravity-capillary wave. ^[ref]

In uniform positive flow $U$, upward stationary internal gravity waves have laboratory group velocity $U({\cos }^2\theta ,\sin \theta \cos \theta )$. Packets emitted by a localized source at startup lie on $(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$, with later emissions inside. This envelope belongs to the permanent stationary component, not every possible startup transient.

A weak horizontal force $f^{\prime }(x)e^{imz}$, with $m>0$ and $\int f^{\prime }(x)dx=\epsilon$, admits many steady pressures differing by an arbitrary function of height. Switching the force on in initially motionless, unbounded, uniformly stratified fluid selects $p_{\infty }=[f(x)-\epsilon /2]e^{imz}$, up to a spatially constant pressure gauge. For horizontal wavenumber $q$, the transient frequency is $\Omega (q)=Nq/\sqrt{q^2+m^2}$. Its long-wave group velocity is $N/m$, and oppositely propagating outgoing fronts leave pressure limits $-\epsilon /2$ and $+\epsilon /2$. The limit is at fixed $x$ after the fronts have passed; dispersive tails need not vanish at any finite time.

For a stationary internal gravity wave in constant buoyancy frequency $N$ and wind $U=U_0(1+z/H)$, set $q_0=k{U}_0/N<1$. Its upward group velocity has ratio $dx/dz=k/m=q/\sqrt{1-q^2}$, where $q=q_0(1+z/H)$. Integrating from the origin gives a circular ray with center $(H\sqrt{1-q_0^2}/q_0,-H)$ and radius $H/q_0$. The ascending ray becomes horizontal at the turning level of a stationary internal wave; its reflected branch continues downwind along the same circle.

The conduit equation evolves the positive cross-sectional area $A$ of a viscous buoyant conduit. Its scaling uses axial length $a_0/\sqrt{8\lambda }$ and velocity $\Delta \rho g{a}_0^2/(8\lambda {\mu }_o)$, where $\lambda ={\mu }_i/{\mu }_o$. Linearization at $A=1$ gives $\omega =2k/(1+k^2)$: phase velocity is upward but group velocity changes sign at $|k|=1$. Nonlinear elevation waves obey the solitary-wave amplitude-speed relation for the conduit equation.

For $S<0$ and constant positive buoyancy frequency, the critical height is $U_0/|S|$. A stationary upward ray approaches it with vertical group velocity proportional to $U^2$, requiring infinite ideal ray travel time. Large gradient Richardson number can preserve the short-wavelength WKB approximation, but parcel-displacement amplitude grows and eventually invalidates linearity.

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.

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.