AbstractIn this paper the problem of the interaction of an electron beam, with a moving beam of ions is considered. First, the steady state problem is solved when the beams are bounded by two parallel planes perpendicular to their directions of motion. Then this steady state is slightly disturbed and the subsequent propagation of this small disturbance is studied. It is found that the equations defining the transverse propagation fall in a separate group. In the present paper we have concentrated on the discussion of transverse propagation only, and the results have been obtained correct up to the third power of the ratio of the initial electron beam velocity to the velocity of light. It is found that the dispersion relation breaks up into four separate equations, and the propagation represented by each equation is considered in detail. Some of the important results worth mentioning are as follows:(I)Under the physical conditions present in a strong hydrogen discharge the wave velocity for the first few harmonics attains the extreme value when the wave length is approximately equal to three times the Debye distance.(II)When we take the wave number equal to zero the plasma oscillates&B a whole with amplitudes increasing or decreasing with time.