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Use ${\varphi }_j=U_{j}x-\frac{1}{2}{U}_j^{2}t+{\varphi }_j^{\prime }$ and retain terms linear in the displacement and perturbation potentials. Evaluation at $z=\eta$ may then be replaced by evaluation at $z=0$: the correction to a perturbation is second order. The kinematic boundary conditions becomeThe linearized pressure continuity isThe decaying normal modes give $-k{A}_1=ik(U_1-c)B$ and $k{A}_2=ik(U_2-c)B$. Thus $A_1=-i(U_1-c)B$, $A_2=i(U_2-c)B$. Substitution into the dynamic condition yields the Kelvin-Helmholtz dispersion relation with gravity,Put $\rho ={\rho }_1+{\rho }_2$ and $\overline{U}=({\rho }_1{U}_1+{\rho }_2{U}_2)/\rho$. Completing the square givesFor $k>0$, the temporal growth rate is $k{c}_i$. A growing complex conjugate branch therefore exists precisely when the radicand of the square root is negative:At equality the two phase velocities coincide and there is no exponential growth. Below it the roots describe real internal gravity waves. The denser lower fluid gives stable density stratification, but the tangential shear can overcome it. The assumption $U_1\ne U_2$ is needed for the displayed threshold; equal streams have no such shear instability.

For a single equal-density vortex sheet with stream speeds $0$ and $V$, the gravitational contribution vanishes. The Kelvin-Helmholtz dispersion relation with gravity reduces to $c^2+(V-c{)}^2=0$, with roots $c=V/2\pm i|V|/2$. Its positive temporal growth rate is ${\gamma }_{\mathrm{sheet}}=k|V|/2$.

Both jet parity branches have the common growth rate derived above, henceEquivalently, $2\sqrt{s}\le 1+s$ follows from $(\sqrt{s}-1{)}^2\ge 0$. Equality is approached when the two interfaces are separated by many disturbance decay lengths, $kL\to \infty$. The zero-shear case has both growth rates zero. For small $kL$, the growth rate ratio is approximately $2\sqrt{kL}$: interaction between the sheets suppresses amplification relative to an isolated sheet.