= Solution
A material interface obeys $D(y-\eta(x,t))/Dt=0$. In each <fluid> this gives $\phi_{jy}=\eta_t+(U_j+\phi_{jx})\eta_x$ at $y=\eta$, where $U_1=U$ and $U_2=V$. Discarding products of <perturbations> and evaluating at the undisturbed interface gives
$$
\boxed{\phi_{1y}=\eta_t+U\eta_x,\qquad\phi_{2y}=\eta_t+V\eta_x\quad(y=0).}
$$
For an inviscid <irrotational> <fluid>, unsteady Bernoulli is $\Phi_t+\tfrac12|\nabla\Phi|^2+p/\rho+gy=C(t)$, with total potential $\Phi=U_jx+\phi_j$. Subtract the undisturbed relation and linearize at the moving interface. The <pressure> <perturbation> there is $-\rho_j(\phi_{jt}+U_j\phi_{jx}+g\eta)$. With no <surface tension> the two pressures agree, giving
$$
\boxed{\rho_1(\phi_{1t}+U\phi_{1x}+g\eta)=\rho_2(\phi_{2t}+V\phi_{2x}+g\eta)\quad(y=0).}
$$
Use <complex amplitudes> and take real parts at the end. For $k>0$, solutions of <Laplace's equation> decaying away from the interface have the form
$$
\eta=Ae^{i(kx-\omega t)},\quad
\phi_1=B_1e^{ky}e^{i(kx-\omega t)},\quad
\phi_2=B_2e^{-ky}e^{i(kx-\omega t)}.
$$
Put $\Omega_1=\omega-Uk$ and $\Omega_2=\omega-Vk$. The kinematic conditions imply $B_1=-i\Omega_1A/k$ and $B_2=i\Omega_2A/k$. Substituting into the <pressure> condition gives
$$
\rho_1\left(g-\frac{\Omega_1^2}k\right)A=\rho_2\left(g+\frac{\Omega_2^2}k\right)A.
$$
A nonzero <amplitude> therefore requires
$$
\boxed{\rho_1(\omega-Uk)^2+\rho_2(\omega-Vk)^2=(\rho_1-\rho_2)gk.}
$$
To determine <stability>, let $R=\rho_1+\rho_2$ and $\overline U=(\rho_1U+\rho_2V)/R$. Completing the square yields
$$
(\omega-\overline Uk)^2=\frac{(\rho_1-\rho_2)gk}{R}-\frac{\rho_1\rho_2(U-V)^2k^2}{R^2}.
$$
For $U\neq V$, the right-hand side is negative exactly when
$$
k>\frac{(\rho_1^2-\rho_2^2)g}{\rho_1\rho_2(U-V)^2}.
$$
Then the two frequencies have opposite nonzero imaginary parts. The <root> with positive imaginary part gives <exponential growth> through $e^{-i\omega t}$. In terms of <wavelength> $\lambda=2\pi/k$, the unstable range is
$$
\boxed{\lambda<\lambda_c=\frac{2\pi\rho_1\rho_2(U-V)^2}{(\rho_1^2-\rho_2^2)g}.}
$$
This is the short-wavelength <Kelvin-Helmholtz instability> for a density-stable interface without <surface tension>. Equality is the neutral threshold; if $U=V$, the gravitational term is positive and this shear <instability> is absent.
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