Influence of the weak electric fields of the order of a few V/cm, inevitably existing in plasma discharges, on propagation of dust density waves is theoretically investigated for typical experimental conditions. The model predicts significant modifications of the dispersion of the dust acoustic modes due to equilibrium ion and dust flows and an existence of cut-off wavenumbers, corresponding to aperiodically growing dust density perturbations. The theory explains the preferential range of wavenumbers, where the dust density waves can be effective excited. Examples of dispersion curves are calculated for realistic complex plasma parameters and comparison with recent experiments is presented.