## ABSTRACT

Let us assume that the electron density varies only in time. One can then construct the basic solutions by assuming that the field variables vary harmonically in space. This chapter shows that unlike the case of spatial discontinuity, for a temporal discontinuity, the real power density of the source wave is not equal to the sum of real power densities of three modes. The computation of the energy density of each of the first two modes involves the stored energy in the electric and magnetic fields in free space and the kinetic energy of the electrons. The energy density of the third mode involves the energy of the wiggler magnetic field and the kinetic energy of the electrons. A step profile is a useful approximation for a fast profile with a small rise time. The chapter considers a perturbation technique to compute the correction terms for a fast profile.