Development of unstable cyclotron oscillations in inhomogeneous magnetic field

The development of short wavelength quasiclassical oscillations in inhomogeneous media is investigated using the Laplace transformation. The perturbations, which are arbitrary functions of space, are taken as a superposition of elementary perturbations which are initially delta -functions. An equation is found which relates the frequency of the elementary perturbation as a function of space and time. It is shown that the concept of a wave packet has only a limited value in an inhomogeneous medium, since, for a sufficiently sharp spatial variation of the oscillation growth rate, the interference of the elementary perturbations which form the packet can lead to its destruction. The investigation is carried out for unstable cyclotron oscillations in inhomogeneous magnetic fields. The development of perturbations in a monotonically varying magnetic field is investigated in detail. The amplification coefficient is shown to be independent of the type of perturbation. An approximate expression is found for the amplification coefficient for a non-monotonically varying magnetic field.