The application of integration by the method of stationary phase to resonant oscillations excited by a small pulsed dipole is outlined. Both the growth and the decay of the oscillations near the plasma frequency are determined by this method at a fixed distance from the dipole, first in the absence and then in the presence of an external magnetic field. It is shown that Landau damping must be taken into account in the calculation of the growth but not of the decay. The oscillations are shown to spread out with a speed that is about half the mean thermal speed of electrons.Only the decay, not the growth, of the oscifiations near harmonics of the cyclotron frequency can be calculated by the same method. It is shown, moreover, that the amplitude, calculated for an observation point that moves away with sateffite velocity in an ionospheric environment, is only valid for time delays longer than about a minute. Such a result is therefore of no practical interest because the resonances observed from sateffites only last a few milliseconds. The erroneous nature of using such a result and the need for a different approach, such as that used in earlier work by the first author, are thus demonstrated.