Propagation in a dispersionless linear medium whose dielectric constant varies slowly in space and time is analyzed. The medium is represented as a series of small and moving dielectric constant discontinuities, with a uniform medium between adjacent moving steps. In the simplest approximation the reflections generated at the steps are ignored but the change in transmitted amplitude is taken into account. This extends earlier work by Tsai and Auld and shows that the electric field is proportional to (ωk/ε)1/2 as ω, k, and ε vary along the path taken by a given phase of the propagating wave. This amplitude variation is shown to be equivalent to the conservation of photon numbers. The neglected higher-order multiple reflections are discussed in an evaluation of the validity of the approximation. We also review the solution for the truly nonlinear problem. In this case, the same nonlinearities which couple a large pump wave and a small signal cause different portions of the pump wave to travel with different velocities. A given portion of the small signal stays with a particular value of the dielectric constant and therefore no reflections are generated. The E∼(ωk/ε) 1/2 result then becomes exact. Finally, a novel amplifying device called "radiation klystron" is discussed. In this device the small signal to be amplified controls the propagation of a high-level high-frequency signal. © 1973 American Institute of Physics.