= Static Jeans growth-rate spectrum
{title2=$s(k)=\sqrt{4\pi G\rho_0-c_s^2k^2}$}
Under the <Jeans swindle>, longitudinal density disturbances in a static uniform gas have $\omega^2=c_s^2k^2-4\pi G\rho_0$. For $0<k<k_J=\sqrt{4\pi G\rho_0}/c_s$, one solution grows as $e^{s(k)t}$ with the displayed rate. It decreases monotonically with <wavenumber>, so the longest available wavelengths grow fastest and the growth-rate supremum is $\sqrt{4\pi G\rho_0}$ as $k\to0^+$. There is no preferred finite fastest wavelength in this model. The exact uniform $k=0$ density change is not an ordinary finite-potential Fourier perturbation of the fixed background, and finite-cloud boundaries restrict the available long modes.
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