In this paper, brief review of new and early publications devoted to the interaction of electron streams and the enhancement due to the interference of partial waves in streams, and due to instability is adduced. The lay emphasis on the consideration the mechanism of periodic interaction of electron streams as applied to option of the variant double-stream traveling-wave tube, on the example of system consisting of two electron beams divided by the metallic perforated screen, which interact with the electromagnetic field. The addressing to this topic is due primarily to the fact that the two-stream instability and regime of the interference amplification, which take place with the continuous and periodic interaction of two different-speed electron streams, will allow creating new terahertz range devices. The periodic nature of the interaction is expressed in the fact that the wave processes in the streams are represented by an infinite set of spatial harmonics, and the interaction of the streams is reduced to their continuous interaction between themselves and with field of electrodynamics system. A one-dimensional model of continuous and discrete two-beam interaction is considered. The dispersion equation is obtained and the functional connection of its roots and the amplification factor from the parameter of desynchronism are constructed. Variable data are described by superposition of five partial waves for two electron beams interacting with electromagnetic field without backward radiation. Gain is determined either by electron-wave interaction or by interaction with circuit according to value of space-charge parameter. The variant of the device in which the discrete interaction is realized shows a significant increase in the gain and an increase in the band of the desynchronism parameter between the average flow velocity and the phase velocity of the unperturbed wave, in which there is an amplification compared to a one-beam traveling-wave tube and traveling-wave tube with two continuously interacting streams.