= Isothermal displacement energy of a self-gravitating slab
{title2=$\omega^2I=c_s^2\int|\delta\rho|^2/\rho\,dz-(4\pi G)^{-1}\int(|\phi_z|^2+k^2|\phi|^2)\,dz$}
For an isothermal gas slab, a horizontal <Fourier mode> has $\delta\rho=-ik\rho\xi_x-(\rho\xi_z)_z$, $W=c_s^2\delta\rho/\rho+\phi$, $\omega^2\xi_x=ikW$, $\omega^2\xi_z=W_z$ and $(\partial_z^2-k^2)\phi=4\pi G\delta\rho$. Multiply momentum by $\rho\boldsymbol\xi^*$ and integrate by parts, then use the <Poisson equation for Newtonian gravity>. Vanishing <pressure>-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int\rho|\boldsymbol\xi|^2dz>0$. Thus every admissible squared <eigenfrequency> is real; a negative value gives gravitational growth rather than an oscillatory instability.
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