= Kelvin-Helmholtz dispersion relation with gravity
{c}
{title2=$\rho_1(U_1-c)^2+\rho_2(U_2-c)^2=g(\rho_2-\rho_1)/k$}
Two deep layers of <incompressible flow> with upper <mass density> $\rho_1$, lower <mass density> $\rho_2$, and tangential <velocities> $U_1,U_2$ have this <dispersion relation> when <surface tension> is absent. Decaying <velocity potentials> and the linearized <kinematic boundary condition> determine their amplitudes; the <dynamic boundary condition for an inviscid interface> gives the displayed relation. For $k>0$ the squared departure of the <phase velocity> $c$ from the density-weighted mean <velocity> is $g(\rho_2-\rho_1)/[k(\rho_1+\rho_2)]-\rho_1\rho_2(U_1-U_2)^2/(\rho_1+\rho_2)^2$. A negative value gives <Kelvin-Helmholtz instability> despite stable <density stratification>.
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