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The absolute frequency is the generally complex angular frequency at the physically selected absolute wavenumber. Its imaginary part is the fixed-position exponential growth rate under the convention $e^{-i\omega t}$. Marginal zero exponential growth may still carry an algebraic prefactor.

The absolute growth rate is the exponential growth rate of a localized impulse at a fixed position in the chosen observation frame. Under the convention $e^{i(kx-\omega t)}$, it is the imaginary part of the physically selected absolute frequency. Positive values give absolute hydrodynamic instability; negative values can still coexist with convective hydrodynamic instability.

Let fluid 1 lie above fluid 2, with mass densities ${\rho }_1<{\rho }_2$ and dynamic viscosities ${\mu }_1<{\mu }_2$. Define $\Delta \rho ={\rho }_2-{\rho }_1>0$, $\Delta \mu ={\mu }_2-{\mu }_1>0$, porosity $\varphi$, permeability of a porous medium $k$, and downward interface speed $W$. With Darcy flux $\varphi W$, no capillarity, and two unbounded layers, Darcy's law givesFor a Fourier mode of interface displacement, the two perturbation pressures decay exponentially into the respective layers. The kinematic boundary condition makes their amplitudes proportional to $\varphi {\mu }_1\sigma /(k|\alpha |)$ and $-\varphi {\mu }_2\sigma /(k|\alpha |)$. Linearized pressure continuity adds the base-gradient difference $\varphi W\Delta \mu /k-g\Delta \rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta \rho /(\varphi \Delta \mu )$; the critical Darcy velocity is $kg\Delta \rho /\Delta \mu$. Without surface tension this model has unbounded short-wave growth rate on its unstable side.

The complex Ginzburg–Landau equation is a nonlinear complex-amplitude evolution equation with linear and cubic terms and generally complex coefficients. The driven-condensate form ${\Psi }_t=(\alpha -\beta |\Psi {|}^2)\Psi +i({\nabla }^2-g|\Psi {|}^2+s)\Psi$ has gain/loss coefficients $\alpha ,\beta$ and conservative interaction/frequency coefficients $g,s$. A uniform nonzero state has number density $\alpha /\beta$ when $\alpha ,\beta >0$.

A complex Ginzburg–Landau equation describes a complex number disturbance amplitude through linear amplification, advection, complex diffusion equation terms, and often cubic amplitude saturation. A constant-coefficient example is ${\psi }_t+U{\psi }_x=\mu \psi +(1+i{c}_d){\psi }_{xx}-|\psi {|}^2\psi$. Here real $U,\mu ,c_d$ describe transport, linear growth rate and dispersive diffusion equation terms. Omitting the cubic term gives the linear complex Ginzburg-Landau equation.

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 a planar vortex sheet separating equal-density streams of velocities $U_1$ and $U_2$, a mode of nonzero wavenumber $k$ has complex growth exponentOne sign has positive growth rate, so every nonzero wavenumber is unstable in the inviscid zero-thickness model.

Linear instability is growth of small disturbances in the linearized dynamics. An eigenmode with positive temporal growth rate is sufficient to establish it. In a non-normal system, finite transient amplification is a different issue from the existence of an exponentially growing eigenmode. For normal modes written as $e^{i\alpha (x-ct)}$, the temporal rate is $\alpha \mathrm{Im}c$.

A planar advection-driven melting front in a porous matrix with permeabilities $k$ ahead and $k+\Delta k$ behind has the ideal sharp-front growth rateThis follows by matching harmonic pressure perturbations and normal Darcy flux, then converting flux to front speed by the enthalpy balance. It predicts no finite fastest-growing wavelength when $\Delta k>0$.

A perturbation proportional to $\sin (n\pi z)e^{st+ikx}$ has growth rateMinimizing the neutral curve $(k^2+n^2{\pi }^2{)}^2/k^2$ gives $n=1$, $k=\pi$ and $R_c=4{\pi }^2$. Thus the first convection rolls have horizontal wavelength twice the layer depth.