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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.

With the Airy displacement integrals, the lower pressure-to-velocity ratio is $-iq\mathrm{Ai}\prime (-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha /\delta$. Hence for $\alpha =k{R}^{-1/4}$ the dispersion relation is $k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\mathrm{Im}C>0$.

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.

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.

For two incompressible inviscid layers at rest between rigid horizontal boundaries, with the heavier fluid below and no surface tension, the displayed dispersion relation follows from Laplace equation for the velocity potentials, no normal flow at the walls, equal normal velocities at the interface, and pressure continuity. The positive density contrast supplies the restoring force. In the long-wave limit the squared wave speed is $g({\rho }_2-{\rho }_1)/({\rho }_1/h_1+{\rho }_2/h_2)$.

To eliminate an unknown boundary transform by a dispersion relation symmetry, evaluate the global relation at $\nu (k)$ only where the spatial transform remains analytic. Substitute the resulting trace identity into the contour representation. An unwanted transform vanishes only if its transformed argument and exponential admit a valid contour deformation and decay estimate. For constant drift, $\nu (k)=i\alpha -k$ maps the upper domain $\mathrm{Re}(k^2-i\alpha k)<0$ into the lower transform half-plane.

The local axisymmetric density/velocity system of a pressured dust layer with linear gas drag and a fixed Keplerian shearing sheet gas flow has this cubic dispersion relation. The $\Omega /2$ azimuthal velocity coefficient includes advection of the Keplerian shear. The razor-thin disk Poisson kernel supplies the $|k|$ self-gravity term. Keeping the full amplitude determinant retains the zero-frequency branch and avoids dividing out the secular mode.

For the wave equation ${\phi }_{tt}-\nabla \cdot (c^2\nabla \phi )=0$ with slow variables and phase $e^{i\theta /\epsilon }$, define $\omega =-{\theta }_T$ and $\mathbf{k}=\nabla \theta$. The leading balance is ${\omega }^2=c^2|\mathbf{k}{|}^2$. Its positive-frequency dispersion relation is $\Omega =c|\mathbf{k}|$, and the group velocity is $c\mathbf{k}/|\mathbf{k}|$.

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

For a top-hat planar jet in stationary fluid of the same mass density, odd and even velocity potentials have the same temporal growth rate. Put $s=\tanh (|k|L)$. Their dispersion relations use $s$ and $1/s$, but $\sqrt{s}/(1+s)=\sqrt{1/s}/(1+1/s)$. Their real phase velocities are respectively $Vs/(1+s)$ and $V/(1+s)$. The common growth rate is strictly below $|kV|/2$ for finite $|k|L>0$, approaching the isolated vortex sheet value when the sheets decouple.

For two resting incompressible inviscid layers between impermeable horizontal walls, the upper layer has density ${\rho }_1$ and depth $L_1$, and the lower layer has ${\rho }_2<{\rho }_1$ and depth $L_2$. Solving the Laplace equation for the velocity potential and imposing the linearized kinematic and capillary pressure conditions gives the displayed dispersion relation. With positive surface tension $\gamma$, exponential Rayleigh-Taylor instability occurs for $0<k<k_c$, where $k_c=\sqrt{g({\rho }_1-{\rho }_2)/\gamma }$. The depths change the growth rate, but not this cutoff.

A boundary trace transformed over a finite time interval using the dispersion relation of a linear partial differential equation. For integrable trace data and a polynomial dispersion function, it is an entire function of the complex spectral parameter. Its growth depends on $\mathrm{Re}\omega (k)$; multiplying by $e^{-\omega (k)t}$ changes the integrand to the causal factor $e^{-\omega (k)(t-s)}$.

A flexural-gravity wave bends a floating elastic plate while displacing the underlying water. For a uniform plate with areal mass $m$ and bending stiffness $D$ above inviscid deep water of mass density ${\rho }_w$, put $\alpha =m/{\rho }_w$ and $\beta =D/{\rho }_w$. Combining the plate equation with the water's kinematic boundary condition gives the displayed dispersion relation. Both gravitational acceleration and plate bending restore displacement; water and plate inertia resist acceleration. It assumes small amplitude, no prestress and negligible shear deformation of the plate.

Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

At an isolated interface with continuous base velocity, a stable density drop $\Delta \rho /2$ and a jump $[U^{\prime }]$ of base shear, decaying disturbances $w\propto e^{-k|z-z_i|}$ obey the displayed dispersion relation. It follows by inserting their derivative jump $[w^{\prime }]=-2kw$ into the jump conditions for stratified inviscid shear flow. Both the density jump and the vorticity jump contribute to the intrinsic phase velocity $c-U_i$.

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.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.

Let $K^2=k^2+l^2$, $A=K^2+F$, $\mathcal{B}=K^2(K^2+2F)$ and use $e^{i(kx+ly-\sigma t)}$. The uniform upper-layer flow $V$ over a resting lower layer has the dispersion relationIts discriminant is $4F^2{\beta }^2{k}^2+\mathcal{B}{K}^2(K^2-2F)l^2{V}^2$. A negative value gives exponential baroclinic instability. This model includes advection across the interface-induced potential-vorticity gradients in a meridional two-layer current.