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For a single phase $\varphi =kx+mz-\omega t$, every disturbance field is a function of $\varphi$ alone. Its velocity is tangent to the phase surfaces:Consequently both nonlinear Jacobians vanish exactly, irrespective of amplitude. Substitution givesA nonzero amplitude therefore requires the internal gravity wave dispersion relationThe wave is an exact nonlinear solution of the inviscid Boussinesq equations. The parameter $ϵ$ merely sets its amplitude; it does not change its frequency or waveform. Very large $ϵ$ can nevertheless invalidate the Boussinesq model or make the total stratification locally overturn.

The shearing sheet is a local Cartesian expansion about a reference circular orbit in a differentially rotating disk. One works in a frame rotating at $\Omega$, identifies $x,y,z$ with radial, azimuthal, and vertical directions, retains the linear background shear and radial tidal acceleration, and neglects curvature and background variation across a patch much smaller than the orbital radius. The incompressible version also takes constant reference density and imposes $\nabla \cdot \mathbf{u}=0$, filtering sound waves.

It represents local three-dimensional vortical motion, epicyclic motion, shear dynamics, inertial waves, and, after adding buoyancy, internal gravity waves and convection. It cannot represent global geometry or boundaries, order-radius structures, density-changing compressible motion, or acoustic waves.

Let $u=ps{i}_z$ and $w=-ps{i}_x$, so mass conservation for the two-dimensional incompressible flow is automatic. Write $b$ for buoyancy and introduce the diffusive operatorsThe Linearized Boussinesq equations areTaking the curl of the momentum conservation equations eliminates the pressure and givesApply $D_{\kappa }$ and use $D_{\kappa }b=N^2{\psi }_x$. Since the constant-coefficient linear partial differential operators commute,This is the viscous-diffusive internal gravity wave equation.

Put $Z=ϵ\zeta$ and seek the slowly attenuating wave envelope $A(Z)$ inAt order ${ϵ}^0$, the equation gives the internal gravity wave dispersion relationAt order ${ϵ}^1$, retaining one derivative of the slowly varying amplitude givesUsing $\omega =N\sin \theta$ therefore yieldsHence the leading viscous attenuation of an internal-wave beam isThe wave energy density, being quadratic in the amplitude, decays twice as rapidly in the exponent.

For the given internal gravity wave,The wavevector part of the Hamiltonian ray-tracing equations isApplying the initial data givesIn particular, $k(t)m(t)=k_0{m}_0$ is conserved.

Let $\theta$ be the angle made by the group velocity ray with the horizontal. The internal gravity wave dispersion relation givesEach sawtooth face changes height by $2h_0$ over horizontal distance ${\lambda }_T/2$, so its slope magnitude isThe internal-wave slope criticality criterion is $s_T<\tan \theta$. Hence reflection is subcritical forIn the corresponding internal-wave ray tracing sketch, every incident ray meets one planar face and leaves it into the fluid at the same angle $\theta$ to the horizontal. The reflected ray is steeper than either face, so it clears the sawtooth rather than running into an adjacent corner.

The boundary condition gives complex amplitude $w_0=Uk{h}_0$. Substitution of the internal gravity wave forminto the internal-wave equation givesSince $|\omega |=Uk$,For $k<N/U$, choose the sign of $m$ that gives upward energy propagation; with $\omega =-Uk$, this is $m>0$.

Let the velocity perturbation be $(u,w)$, pressure perturbation be $p$, and density perturbation be ${\rho }^{\prime }$ about the hydrostatic background $\hat{\rho }(z)$. The linearized inviscid Boussinesq approximation givesStable stratification means ${\hat{\rho }}_z<0$, and the buoyancy frequency isDifferentiate the momentum equations to eliminate $p$, use incompressibility, and then use the density equation to eliminate ${\rho }^{\prime }$. This yieldsFor $w\propto e^{i(kx+mz-\omega t)}$, the internal gravity wave dispersion relation is ${\omega }^2=N^2{k}^2/(k^2+m^2)$.

A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

A radiation condition selects the wave solution whose energy propagates away from its source. For an internal gravity wave, the vertical phase and group velocities have opposite signs, so upward radiation fixes the sign of the vertical wavenumber.

For a stationary internal gravity wave, the Taylor–Goldstein equation gives $m^2=N^2/U^2-U^{\prime \prime }/U-k^2$. In a propagating interval with slowly varying $m$, the WKB approximation has amplitude proportional to $m^{-1/2}$ and phase derivative $m$. The conditions $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$ fail at a regular turning level. With $U>0$, the upward branch has $m>0$ and negative intrinsic frequency; laboratory group velocity is $U(k^2,km)/(k^2+m^2)$ when the curvature term is negligible.

A fixed terrain obstacle forces a zero laboratory-frequency internal gravity wave through the kinematic boundary condition $w=U{h}_x$. Its intrinsic frequency is $-kU$. Propagation, radiation and the transient arrival of its permanent component are distinct questions.

A stratified layer between unstratified regions can trap internal gravity waves, because the exterior disturbance is an evanescent wave. Its discrete normal modes obey Robin boundary conditions representing the exterior layers. Periodic boundary forcing at a trapped normal mode frequency produces resonance in the ideal undamped model.

For upward internal gravity waves, let $B=c_{gz}{N}^2{A}_{\zeta }^2/\hat{\omega }$ be constant. Combining $c_{gz}=(N/k)\sin \theta {\cos }^2\theta$ with the monochromatic internal-wave overturning criterion gives ${\cot }^4\theta <B{k}^3\hat{\omega }/(N^3{\sin }^3\theta )$. Near a critical level the onset balance is $\cot \theta \sim (B{k}^3\hat{\omega }/N^3{)}^{1/4}$. This is compatible with $\hat{\omega }=N\cos \theta$; it fixes the overturning point rather than redefining the dispersion relation.