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Let $p$ denote the fluid pressure perturbation divided by the reference mass density, and let $b=-g{\rho }^{\prime }/{\rho }_0$ be the buoyancy perturbation. The buoyancy frequency satisfies $N^2=-g{\hat{\rho }}_z/{\rho }_0$. Linearization requires small boundary slope $k|{\eta }_0|\ll 1$, small displacements compared with the vertical scales of the disturbance and background, and advection small compared with the oscillatory acceleration. For a propagating internal gravity wave this includes $|m{\eta }_0|\ll 1$; boundary slope alone is insufficient when $|m|\gg k$. The Boussinesq approximation also requires small relative mass density differences over the region of interest. These conditions must hold for the resulting disturbance, including any amplification by resonance.

The inviscid Linearized Boussinesq equations and linear kinematic boundary condition areWrite the velocity field as $(u,w)=\mathrm{Re}\{(U(z),W(z))e^{i(kx-\omega t)}\}$. Incompressibility and horizontal momentum balance giveThe vertical momentum balance therefore yieldsFor a real vertical wavenumber $m$, this gives the dispersion relationWhen $0<\omega <N$, define $\ell =k\sqrt{N^2/{\omega }^2-1}$. The radiation condition selects $m=-\ell$, because energy must leave the boundary upward. The boundary-forced internal gravity wave isIts wavevector is $\mathbf{K}=(k,m)$, its phase velocity is ${\mathbf{c}}_p=\omega \mathbf{K}/(k^2+m^2)$, and its group velocity isThus the phase velocity points down and right, while the group velocity points up and right. They are perpendicular. Internal-wave polarization makes the particle motion an oscillation along the group velocity direction, with zero first-order mean transport. The constant-phase lines of an internal gravity wave are also parallel to the group velocity. Their inclination $\vartheta$ above the horizontal satisfies $\sin \vartheta =\omega /N$.

When $\omega >N$, set $\kappa =k\sqrt{1-N^2/{\omega }^2}$. Boundedness at infinity selects the evanescent waveThe horizontal and vertical velocity components are in quadrature: fluid particles describe small ellipses, and the response decays over ${\kappa }^{-1}$. There is no upward time-averaged energy flux, because $P$ and $W$ are in quadrature. A real vertical group velocity is not defined for this evanescent wave. The pattern travels horizontally with phase velocity $\omega /k$.

For $N=0$ the response is the evanescent wave with $\kappa =k$; the ratios $\omega /N$ should not be used. At the cutoff $\omega =N>0$, the bounded harmonic solution has $W=-i\omega {\eta }_0$, $U=0$: it neither decays nor has nonzero upward group velocity. It is the limiting cutoff response, rather than a localized radiating disturbance. A uniform nonzero buoyancy frequency in an infinitely deep Boussinesq approximation is itself a local idealization of the background mass density.

The arrows distinguish the phase velocity, group velocity, and oscillatory particle motion of the internal gravity wave. The right panel shows the decay envelope and particle ellipses of the evanescent wave.

Let a slowly varying wave have phase $S(\mathbf{x},t)$, wavevector $\mathbf{k}=\nabla S$, and angular frequency $\omega =-{\partial }_{t}S$. The local dispersion relation is the Hamilton-Jacobi equation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$. Equality of mixed derivatives, followed along a curve with velocity $\dot{\mathbf{x}}={\nabla }_{\mathbf{k}}\Omega$, gives the Hamiltonian ray-tracing equationsHere $d/dt={\partial }_t+\dot{\mathbf{x}}\cdot \nabla$ is the total derivative evaluated along a ray, and $\dot{\mathbf{x}}$ is the group velocity.

For $\Omega =kc(z)$, $\omega$, $k_x$, and $k_y$ are conserved. If $\psi$ is the angle between the ray and the $z$-axis, then $k\sin \psi =(k_x^2+k_y^2{)}^{1/2}$ and $k=\omega /c$. Hence Snell's law is

For the ray launched with $\mathbf{k}=k(\cos \varphi ,0,\sin \varphi )$ at $z=0$, this constant is $|\cos \varphi |/c_0$. Thereforewhich proves

For $0<\varphi <\pi /2$, write $a=\cos \varphi$. Since the ray and wavevector are parallel,Near the upper turning point, put $z=z_m-\zeta$. Expansion givesIf $x=x_m$ at the turn, integration yields the local ray pathThe ray is tangent to the line $z=z_m$, turns smoothly, crosses $z=0$, and repeats symmetrically between $-z_m$ and $z_m$. Increasing $\varphi$ decreases $z_m$, while $\varphi \downarrow 0$ sends the turning level outward.

For unit plane wave incidence ${\psi }_i=e^{i{k}_0{\hat{\mathbf{r}}}_0\cdot \mathbf{r}}$, define the scattering wavevector difference $\mathbf{q}=k_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})$ andThe outgoing Green function for the three-dimensional Helmholtz equation givesThe Born far-field pattern is $A$, a Fourier transform sample of the potential. Expanding the far-field Rytov exponential in $1/r$ gives the same leading outgoing term, while the logarithmic representation preserves its accumulated phase and amplitude correction.

Take $U_0,N_0,k>0$ and assume an initially propagating mode, $k<N_0/U_0$. Put $s=N_0/U_0$ and $\delta (z)=1+\gamma z$. Neglecting $U^{\prime \prime }/U$ as stipulated, the Scorer parameter and stationary internal-wave WKB solution areThe positive root $m>0$ selects upward group velocity for the negative intrinsic frequency branch $\hat{\omega }=-kU$. The WKB approximation needs a slowly varying background relative to the local vertical wavelength, in particular $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$, away from a turning level or singular mean profile.

The local dispersion relation is $\omega =kU-Nk/(k^2+m^2{)}^{1/2}=0$. Its intrinsic and laboratory group velocities are, with $K=(k^2+m^2{)}^{1/2}$,Thus intrinsic energy propagation is upstream and upward, while laboratory energy propagation is downstream and upward, parallel to $(k,m)$. Constant-phase lines of an internal gravity wave have slope $dz/dx=-k/m$ and normal spacing $2\pi /K$; the horizontal spacing remains $2\pi /k$.

If $\gamma <0$, $m$ grows with height. The wavevector and laboratory energy ray become more vertical; phase lines become flatter and closer together. For finite nonzero $N$, the heightis a critical level of an internal gravity wave: $U\to 0$ and the vertical wavenumber diverges. The laboratory group velocity tends to zero. Importantly, $|m^{\prime }|/m^2\to |\gamma |/s$, so the formal WKB condition need not deteriorate before this level if that ratio is small. Nevertheless the stationary equation is singular at the level, the phase has no finite limit, and the horizontal perturbation velocity grows as $m^{1/2}$; the inviscid linear wave cannot be continued uniformly through it without additional physics.

If $\gamma >0$, $m$ decreases to zero atThe wavevector and laboratory ray become horizontal, the phase lines become vertical, and the normal spacing increases. Above $z_t$, $m^2<0$ and the disturbance is an evanescent wave. Here $|m^{\prime }|/m^2$ diverges, so the WKB approximation fails in a turning region and must be replaced by a local connection solution.

The physical rapid phase isDefine the local wavevector, angular frequency, and group velocity byThe equality of mixed partial derivatives gives the phase-compatibility equationsA ray is defined to move with the local group velocity:

Differentiate the local dispersion relationwith respect to $x_i$. Along a ray,The last two terms cancel because $\mathbf{k}$ is a gradient. Similarly,Finally,Thus the Hamiltonian ray-tracing equations are

Surfactants accumulate at an interface and orient their two chemically distinct ends toward their preferred fluid components. The relaxation found in part c lowers the coefficient of the gradient energy, and hence generally lowers the surface tension. If the reduction makes $\tilde{\kappa }<0$, the positive fourth-gradient term proportional to $\gamma$ can stabilize structure at a nonzero wavevector, producing the modulated correlations characteristic of a microemulsion.

Let the group velocity make an angle $\theta$ above the horizontal. A convenient right-handed choice of rotated unit vectors isThe first vector is parallel to the wavevector, the second is parallel to the group velocity, and their perpendicularity is the defining geometry of an internal-wave phase and group velocity. ThusWriting ${\Delta }_{\xi \zeta }={\partial }_{\xi }^2+{\partial }_{\zeta }^2$, the equation becomesFor an inviscid plane wave proportional to $e^{i(k\xi -\omega t)}$, its dispersion relation is $\omega =N\sin \theta$.

Setwhere $r$ is the magnitude of the internal-wave ray slope and $s$ is the bottom slope. A rightward ray remains in the triangular basin precisely when its bottom reflection is subcritical internal-wave reflection, namely $r>s$. Therefore

Conservation of frequency and of the component of the wavevector tangent to the slope gives the focusing power of internal-wave reflectionwhere $\tan \alpha =s$. In this range $\gamma >1$, so the reflected wavelength is shorter by the factor $\gamma$. The reflected normal group velocity is smaller by ${\gamma }^2$; conservation of normal energy flux therefore increases the wavelength-averaged energy density by

In the razor-thin disk approximation, the three-dimensional Poisson equation isFor a horizontal Fourier mode with wavevector $\mathbf{k}$ and $k=|\mathbf{k}|>0$, the Fourier transform of this equation isThe solution that decays away from the disk isConsequently the required razor-thin disk Poisson kernel in the midplane isThe spatially uniform $k=0$ background is excluded from this local perturbation formula.

For zero forcing, the given equation for $v_y$ isUse the shearing-wave ansatzThe explicit $x$ dependence cancels from the material derivative whenThe amplitude then obeys the time-dependent harmonic oscillator equationThis evolving wavevector is the characteristic signature of a shearing wave.

Let $k_S=\omega /c_S$ and $k_P=\omega /c_P$. Snell law for elastic waves fixes the reflected waves' common tangential wavenumber:Choose the reflected SV-wave and converted reflected P-wave asThe polarizations are respectively perpendicular and parallel to their wavevectors. Phase matching makes all three exponential factors equal at $z=0$. The rigid boundary condition for an elastic wave ${\mathbf{u}}_I+{\mathbf{u}}_{RS}+{\mathbf{u}}_{RP}=0$ there givesSolving this two-by-two linear system yieldsThese fields give the Reflection of an SV-wave from a rigid plane.

Use periodic boundary conditions on the square crystal. Each allowed wavevector occupies area $(2\pi {)}^2/A$ in two-dimensional $k$-space. Including the two phonon polarizations, the number of modes in the disk $|k|\le K$ isThe phonon dispersion relation $\omega =C|k{|}^{\alpha }$ gives $K=(\omega /C{)}^{1/\alpha }$, soDifferentiating the cumulative mode count produces the density of states for a two-dimensional power-law phonon dispersion:

Use the Fourier transform conventionThe zero mode vanishes because the compositional order parameter has zero spatial average. Parseval identity and the Fourier transform of a derivative turn the quadratic part of the dimensionless free energy intoHere ${\sum }_{\mathbf{q}}^{+}$ is the positive-wavevector sum for a real field. Each independent complex amplitude has density proportional to $e^{-G(q)|{\varphi }_{\mathbf{q}}{|}^2}$, so its elementary Gaussian integral gives the static structure factorwhenever $G(q)>0$.

The stationary points of the Brazovskii model kernel obeyBecause $\kappa <0<\gamma$, the nonzero minimum is the nonzero-wavevector soft-mode sphereAt this wavevector,The first Gaussian field theory divergence therefore occurs at

A phonon dispersion relation gives its angular frequency $\omega$ as a function of its wavevector $k$. For an isotropic power law, $\omega =C|k{|}^{\alpha }$.

The wavevectors of a nonzero-frequency Rossby wave in a resting shallow layer satisfyThis follows by completing the square in the shallow-water Rossby-wave dispersion relation. A real circle requires $|\omega |\le |\beta |/(2|a|)$. Its two intersections with a given tangential wavenumber provide the incident and reflected normal wavenumbers at a meridional wall.

A space-time average over periodic modes integrates over the spatial periodic cell and one temporal period. For real spatial Fourier modes sharing a wavevector, $⟨\mathrm{Re}(\mathbf{v}{e}^{i\mathbf{k}\cdot \mathbf{x}})\times \mathrm{Re}(\mathbf{w}{e}^{i\mathbf{k}\cdot \mathbf{x}}){⟩}_x=\mathrm{Re}({\mathbf{v}}^{\ast }\times \mathbf{w})/2$. A matching sine or cosine temporal factor supplies another $1/2$. Opposite wavevectors can have nonzero spatial cross averages, so their temporal coefficients must be checked rather than discarded merely because the wavevectors are distinct.

Vertical shear instability destabilizes differential rotation that varies with height, when buoyancy restoration is absent or sufficiently weakened. In the homogeneous incompressible flow model with background $U_y=-3\Omega x/2+q\Omega z$, an axisymmetric perturbation with wavevector $(k_x,0,k_z)\ne 0$ hasThe pressure-free radial/vertical momentum equation and the azimuthal equation give this result after using incompressibility. Growth requires $2q{k}_x{k}_z>k_z^2$, so a small vertical shear favors nearly radial wavevectors. Here $q$ is the vertical shear amplitude, distinct from the orbital shear parameter $3/2$. Stratification, cooling and boundaries can change this simplified criterion.