We formulate the limits of applicability of the hydrodynamic equations and prove the necessity of introducing a correction to the potential energy transfer in the heat conductivity equation, which allows developing the hydrodynamic theory of the propagation of sound waves with small amplitudes. We show that this correction affects almost all predictions of the standard hydrodynamic theory. In particular, this correction allows extending the applicability domain of the hydrodynamic theory to the case of an arbitrarily viscous liquid. Moreover, in total accordance with the experimental data, the theory predicts that the sound speed and the damping rate remain finite at all frequencies up to frequencies of the order of $10^{-12}$sec${}^{-1}$, while the hydrodynamic equations make no sense at higher frequencies and sound wave propagation in the medium consequently becomes impossible. We show that the dimensionless dispersion equation contains only one material parameter. We predict the existence of the highly damped second sound.