= Vortex sheet in a symmetric rigid channel
{title2=$\omega=k(U_1+U_2)/2\pm i|k(U_1-U_2)|/2$}
Consider an equal-density <vortex sheet> halfway between rigid walls, with equal layer depths $h$, base <velocities> $U_1,U_2$ and no restoring interfacial force. The irrotational perturbations have vertical factors $\cosh(k(y-h))$ above and $\cosh(k(y+h))$ below. Their <kinematic boundary conditions> and <pressure continuity> give
$$
\coth(|k|h)\left[(\omega-kU_1)^2+(\omega-kU_2)^2\right]=0.
$$
The common depth factor cancels, yielding the displayed <dispersion relation> roots and <Kelvin-Helmholtz instability> for nonzero shear. The real <phase velocity> of the pattern is $(U_1+U_2)/2$ and the growing branch has <temporal growth rate> $|k(U_1-U_2)|/2$. This exact depth cancellation depends on equal depths and equal <mass densities>; a single nearby wall or unequal layers changes the inertia factors.
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