= Dispersion relation for two density interfaces in uniform shear
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> have
$$
\widehat w=b_+e^{-k|z-h/2|}+b_-e^{-k|z+h/2|}.
$$
The <jump conditions for stratified inviscid shear flow> give the determinant equation
$$
[(\Delta U/2-c)^2-d][(-\Delta U/2-c)^2-d]-d^2e^{-2kh}=0,\qquad d=\frac{g\Delta\rho}{4\rho_0k}.
$$
Writing $\alpha=kh/2$, $J=g\Delta\rho h/(\rho_0\Delta U^2)$ and $\widetilde c=2c/\Delta U$, this is
$$
\widetilde c^4-(2+J/\alpha)\widetilde c^2+(1-J/(2\alpha))^2-\left[J/(2\alpha)\right]^2e^{-4\alpha}=0.
$$
The two roots for $\widetilde c^2$ 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>.
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