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We show that equations of Newtonian hydrodynamics and gravity describingone-dimensional steady gas flow possess nonlinear periodic solutions. In thecase of a zero-pressure gas the solution exhibits hydrodynamic similarity andis universal: it is a lattice of integrable density singularities coincidingwith maxima of the gravitational potential. With finite pressure effectsincluded, there exists critical matter density that separates two regimes ofbehavior. If the average density is below the critical, the solution is adensity wave which is in phase with the wave of the gravitational potential. Ifthe average density is above the critical, the waves of the density andpotential are out of phase. Traveling plane gravitostatic waves are alsopredicted and their properties elucidated. Specifically, subsonic wave is madeout of two out of phase oscillations of matter density and gravitationalpotential. If the wave is supersonic, the density-potential oscillations are inphase.

Steady flows, nonlinear gravitostatic waves and Zeldovich pancakes in a Newtonian gas