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For this metric, $\sqrt{-g}=a^2$, while $g^{\eta \eta }=-a^{-2}$, $g^{zz}=a^{-2}$ and $g^{ij}={\delta }^{ij}$. The massless Klein-Gordon equation is consequentlyInsert the spatial Fourier transform given in the question. Each wavenumber mode then satisfiesThus every field mode is a simple harmonic oscillator with a time-dependent angular frequency.

A resonant interaction requires matching both wavenumber and angular frequency: $k_3=k_1+k_2$ and ${\omega }_3={\omega }_1+{\omega }_2$ for a sum interaction, with corresponding sign changes for differences. A common phase velocity alone does not guarantee that a generated harmonic lies on the dispersion relation.

For two fluids with dynamic viscosities ${\mu }_1<{\mu }_2$, uniform permeability of a porous medium $k$, porosity $\varphi$, and imposed Darcy velocity $U$, the Saffman–Taylor instability of a planar interface has growth rate $\sigma =\alpha U({\mu }_2-{\mu }_1)/[\varphi ({\mu }_2+{\mu }_1)]$ without capillarity. With constant interfacial tension $\gamma$ and the stabilizing pressure-jump convention, a perturbation of transverse wavenumber $\alpha >0$ instead hasThe fastest-growing mode satisfies ${\alpha }_m^2=({\mu }_2-{\mu }_1)U/(3k\gamma )$.

For a frozen real parallel profile and modes proportional to $e^{i\alpha (x-ct)}$, continuity eliminates tangential velocity and gives this third-order equation. Wall conditions are $v=v^{\prime }=0$; outer tangential matching gives $v^{\prime }\to 0$, with bounded $v$. Real coefficients give the spectral symmetry $(\alpha ,c)\mapsto (-\alpha ,c^{\ast })$. For positive real wavenumber, growth means $\mathrm{Im}c>0$.

A quadratic product of plane waves generates sums and differences of their wavenumbers and angular frequencies. A component drives a secular term when its generated pair also satisfies the linear dispersion relation. A solvability condition in the method of multiple scales then determines the coupled amplitude equations on a slow time.

For a horizontal Fourier mode of wavenumber $k>0$, the midplane gravitational potential of a razor-thin disk is

A lower ray reaches $H$ after ${\tau }_1=H/[U_1\sin {\theta }_1\cos {\theta }_1]$, at $x_H=H\cot {\theta }_1$. Conserved horizontal wavenumber gives $\cos {\theta }_2=(U_2{N}_1/U_1{N}_2)\cos {\theta }_1$. The transmitted front is $(x_H,H)+(t-{\tau }_1){\mathbf{c}}_{g2}$ on propagating branches; reflection reverses the vertical group velocity.

For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal wavenumber $k$ and vertical index $n$, put $a=k^2+n^2{\pi }^2$ and $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.

Combining the Exner equation, sediment transport saturation, bed shear response, and inclined-bed sediment threshold gives a competition between upstream forcing and downstream relaxation. With $a={\tau }_{0}A$, $b={\tau }_{0}B-{\tau }_{\mathrm{th},0}/{\mu }_s$, $C=\chi \gamma ({\tau }_0-{\tau }_{\mathrm{th},0}{)}^{\gamma -1}$, and positive wavenumber $k$, the growth rate is $\sigma =C{k}^2(b-ak{L}_{\mathrm{sat}})/(1+k^2{L}_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite saturation length stabilizes shorter waves.

A sideband instability is growth of disturbances with wavenumbers slightly displaced from a carrier pattern's wavenumber. A real physical perturbation uses the displaced mode together with its complex conjugate; a complex amplitude equation can couple those two sidebands through its cubic term. The Eckhaus instability is a longitudinal phase-modulation example.

Linear axisymmetric compressive modes of a barotropic closure of a razor-thin disk combine radial epicyclic restoration, softened self-gravity and pressure. The off-plane razor-thin Poisson kernel supplies the exponential gravity reduction. Here $k$ is the nonnegative radial wavenumber magnitude. The full linear system also has an axisymmetric geostrophic mode.

A steady spatial forcing at twice the critical wavenumber permits coupling of the negative critical Fourier mode to the positive critical mode. A weakly nonlinear expansion then permits a term $\beta \overline{A}$ in the Landau amplitude equation. Its coefficient is found by projecting the resonant forcing-advection terms onto the adjoint eigenfunction. Reflection-symmetric forcing permits real coefficients. The symmetry permission does not prove a nonzero coefficient: it can vanish for a particular model or mode structure.

With the time convention $e^{-i\omega t}$, pressure amplitude $p$ and background density ${\rho }_0$, the acoustic energy flux is $I_x=(2{\rho }_0\omega {)}^{-1}\mathrm{Im}(\overline{p}{\partial }_{x}p)$. A transverse plane wave mode with positive axial wavenumber $\kappa$ carries flux $\kappa |p{|}^2/(2{\rho }_0\omega )$. For the reduced pressure $p=e^{ikx}E$, the parabolic wave equation gives $\kappa =k-{\nu }^2/(2k)$, approximated by $k$ at leading paraxial order. Stationary illumination uses the infinite-aperture spectral normalization: replace $|p{|}^2$ by the corresponding spectral power per unit transverse length. Other acoustic amplitude conventions change the dimensional prefactor.

A primordial bispectrum is squeezed when one external Fourier mode has much smaller wavenumber than the other two. Momentum conservation then makes the two short wavevectors almost opposite. This configuration measures the response of short-scale correlations to a long perturbation.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.

For a fixed-temperature layer with stress-free boundary conditions, a vertical Fourier sine series has $a^2={\lambda }^2+n^2{\pi }^2$, where $\lambda$ is horizontal wavenumber. The coupled vertical velocity and temperature amplitudes satisfy the displayed quadratic for their growth rate $\sigma$. Its roots are real for positive Rayleigh number, while both roots have negative real parts for negative Rayleigh number. Setting $\sigma =0$ gives the free-slip convection neutral curve.

Suppose the apparent capillary pressure is $p_1-p_2=\gamma (1+\beta x)({\kappa }_0+\nabla \cdot \mathbf{n})$, with normal towards the displaced fluid and positive apparent tension at the flat front $x=X$. Linearizing the Darcy law interface problem gives the local growth rateThe pressure increase experienced by a forward protrusion is stabilizing when ${\kappa }_0\beta >0$. It suppresses every positive wavenumber when $U\le k\gamma {\kappa }_0\beta /({\mu }_2-{\mu }_1)$. For a fixed spatial gradient, $X=Ut/\varphi$ makes the rate time dependent; constant exponential growth is a frozen-position approximation.

The Taylor–Goldstein equation governs linear two-dimensional disturbances of an inviscid, vertically sheared, stably stratified parallel flow. For horizontal phase velocity $c$, base velocity $U(z)$, buoyancy frequency $N(z)$, and horizontal wavenumber $k$,

With thermal diffusivity $\xi$, a Fourier mode of wavenumber $k$ decays by diffusion at this rate.