Nonlinear differential equations are derived for a plane, longitudinal steady-state wave propagating perpendicularly to a magnetic field in a thermal, collisional, weakly ionized plasma in the two-fluid approximation. They describe the variation along the propagation direction of the longitudinal electric field and the two components of velocity of each species. In the limit of small amplitude of oscillation, the equations lead to the dispersion relation obtained by Kato and Hirata (1967). Numerical solutions for various possible values of parameters in the ionospheric E layer are obtained with a computer. When the electron jet velocity is larger than the ion thermal velocity, primarily ion waves of a few tens of khz frequency and a few cm wavelength are obtained. The amplitude of the waves has a maximum value and it decays in the laboratory frame with an e-folding distance of a fraction of a meter. Radar backscatter observations (with metric wavelengths) seem to be from the envelope of the wave. The electric field of the maximum amplitude is 1–2 v/m, which is ∼105 times the microfield energy density and a few per cent of the kinetic energy density of the ion motion relative to the wave. It increases with wave velocity but not with electron jet velocity.