Statistical characteristics are calculated for stationary velocity fluctuations in a one-dimensional open channel flow with a given vertical velocity profile and with one-dimensional irregular bottom waves, characterized by a spectral density function. The calculations are based on an approximate calculation of the velocity fluctuations in the fluid generated by a harmonic corrugation of the bottom. As linearized dynamical equations are used, the velocity fluctuations caused by random bottom disturbances may be obtained by superposition. The dynamics of the motion is assumed to be governed by the Orr-Sommerfeld equation representing the internal wave motion in the fluid. This equation is solved in an approximate manner by reducing it in the upper layer of the fluid to the Rayleigh equation. Close to the bottom we simplify it to a shape still containing the essentials of the viscous behaviour of the flow. Numerical examples and a tentative qualitative comparison with experimental data are given.