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Two equal stable density jumps of size $\Delta \rho /2$ at $z=\pm h/2$ lie in the global linear shear flow $U=\Delta Uz/h$. Decaying normal modes haveThe jump conditions for stratified inviscid shear flow give the determinant equationWriting $\alpha =kh/2$, $J=g\Delta \rho h/({\rho }_0\Delta U^2)$ and $\tilde{c}=2c/\Delta U$, this isThe two roots for ${\tilde{c}}^2$ are real; one is negative exactly when $2\alpha /(1+e^{-2\alpha })<J<2\alpha /(1-e^{-2\alpha })$. At large $\alpha$, this narrow band centres on $J=2\alpha$, where the isolated counterpropagating interfacial gravity waves have the same zero laboratory speed. This realizes counterpropagating wave instability.

Use the standard ideal-fluid assumptions underlying the stated relation: both fluids are incompressible, inviscid and unbounded away from the interface, with no surface tension. An irrotational flow has $u_j=\nabla {\varphi }_j$, and incompressibility gives $\Delta {\varphi }_j=0$ in each fluid. The unsteady Bernoulli equation iswhere the arbitrary time functions can be absorbed into the potentials. On $z=\zeta (x,t)$ the material-interface and normal-stress conditions areDisturbance velocities decay as $z\to \pm \infty$. The equilibrium pressures are $p_{\ast }-{\rho }_{j}gz$.

Linearize at the resting flat interface. The equations become $\Delta {\varphi }_j=0$, ${\zeta }_t={\varphi }_{j,z}$ on $z=0$, andFor $k>0$, write $\zeta =\eta e^{i(kx-\omega t)}$. Decay selects ${\varphi }_1=A_1{e}^{-kz}{e}^{i(kx-\omega t)}$ above and ${\varphi }_2=A_2{e}^{kz}{e}^{i(kx-\omega t)}$ below. The kinematic conditions give $A_1=i\omega \eta /k$ and $A_2=-i\omega \eta /k$. Substituting into the dynamic condition givesFor a signed wave number replace $k$ by $|k|$. A heavier lower fluid gives stable interfacial gravity waves; a heavier upper fluid gives ${\omega }^2<0$ and Rayleigh-Taylor instability. The density ordering is not silently assumed positive in the derivation.

For large $\alpha$, the evanescent interaction between the two interfaces is proportional to $E=e^{-2\alpha }$. Ignoring it first gives isolated interfacial gravity waves with phase speedsThe lower wave propagating positively relative to its local current and the upper wave propagating negatively have equal laboratory speed whenTheir common speed is zero. This is counterpropagating wave instability: the interfaces communicate through their decaying velocity fields, and near resonance their wave phases can lock while extracting energy from the mean shear.

The instability interval has asymptotic endpoints $J_{\pm }=2\alpha [1\pm E+O(E^2)]$ and widthNear exact resonance the interaction is weak but destabilizing: at $q=1$, $C_{-}^2=2-\sqrt{4+E^2}=-E^2/4+O(E^4)$, so $\mathrm{Im}C\sim E/2$. Thus the resonant band narrows and growth weakens exponentially with separation measured in decay lengths. The resonance here is between two density interfaces in globally uniform shear, not a resonance modified by additional vorticity jumps.

For large $\alpha$, the coupling $e^{-kh}=e^{-2\alpha }$ is exponentially weak. If each interface were isolated, its interfacial gravity waves would have dimensional phase velocities $U(z_{\pm })\pm \sqrt{d}$, orThe upper-interface wave travelling against its positive local current has speed $1-\sqrt{J/(2\alpha )}$. The lower-interface wave travelling against its negative current has speed $-1+\sqrt{J/(2\alpha )}$. Their laboratory speeds match at zero when $J=2\alpha$. Weak interaction then produces counterpropagating wave instability rather than two independent travelling waves.

The band has the large-wavenumber form $J=2\alpha \pm 2\alpha e^{-2\alpha }+O(\alpha e^{-4\alpha })$. Exactly at resonance, $\mathrm{Im}\tilde{c}\sim e^{-2\alpha }/2$, soThus the instability window and growth rate both become exponentially small as the interfaces become weakly coupled.