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A critical pattern Fourier mode transforms under a spatial translation as $A\mapsto e^{i\varphi }A$. A forcing at three times the critical wavenumber transforms as $F\mapsto e^{3i\varphi }F$. Their lowest resonant product with phase weight one is $F{\overline{A}}^2$. This determines the form of the leading weak-forcing amplitude equation, while projection onto an adjoint eigenfunction determines its coefficient. Symmetry permits this coupling but does not ensure that it is nonzero. Travelling forcing gives a time-dependent phase to $F$.

At the marginal radial wavenumber, the forced axisymmetric density mode with vortensity has zero squared frequency. Its mass density amplitude is $C_0+C_{1}t-\Omega {\Sigma }_0^2{q}^{\prime }{t}^2$. Thus there is no exponential instability but nonzero conserved vortensity can force quadratic growth, and a homogeneous generalized mode can grow linearly. The spectral boundary $Q=1$ of the Toomre stability criterion must not be mistaken for boundedness of every initial-value solution.

For constant buoyancy frequency $N$ and $U=U_0+Sz$ with $S>0$, a stationary component of horizontal wavenumber $k$ turns at $z_t=(N/k-U_0)/S$ if it propagates at the source. Its vertical wavenumber vanishes there; WKB approximation fails and an Airy function transition connects reflection to an upper evanescent wave.

When kinematic viscosity and mass diffusivity are equal to $ϵ$, a monochromatic internal-wave beam with wavenumber $k$ and ray angle $\theta$ has leading stream-function amplitudeThe cubic dependence on $k$ makes slope-focused, short internal waves dissipate especially rapidly.

The negative-diffusion criterion for viscous disk instability is ${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$. A surface density enhancement then transports angular momentum less effectively and grows. Freezing the Keplerian viscous diffusion equation locally gives growth proportional to the square of radial wavenumber, within the thin-disc transport range; wavelengths must still be sufficiently long compared with the disk scale height for that closure. The formal short-wave limit is a backward heat equation.

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 wavelength is the spatial period of a monochromatic wave. For angular wavenumber $k$, it is $\lambda =2\pi /|k|$.

The dimensionless product of surface-wave amplitude and wavenumber, $ka$, measures orbital displacement relative to the wavelength. Small wave steepness permits a Taylor expansion of the velocity sampled by moving parcels. For a deep-water gravity wave, orbital amplitudes and the local steepness decrease exponentially with depth.