Static Jeans growth-rate spectrum ($s(k)=\sqrt{4\pi G{\rho }_0-c_s^2{k}^2}$)

Under the Jeans swindle, longitudinal density disturbances in a static uniform gas have ${\omega }^2=c_s^2{k}^2-4\pi G{\rho }_0$. 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 $k\to 0^{+}$. 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.