On the time dependence of a narrow-band signal in a dispersive medium far from the emission point
The propagation and distortion of a narrow-band signal in a homogeneous dispersive medium are considered. It is shown that in the transition from the near zone to the asymptotic zone the time dependence of the signal intensity changes but then stabilizes in the asymptotic zone. It no longer changes up to the path length, at which the distortion of the amplitude-frequency spectrum of the signal begins, for example, its carrier frequency begins to change or the signal loses its narrow-band status, and it becomes necessary to take into account the absorption dispersion. In this case, the time dependence of the signal intensity in the asymptotic zone does not always coincide with the intensity of its Fourier spectrum at the starting point on a certain scale. The conditions under which this coincidence takes place or, on the contrary, does not take place, are formulated. It turns out that this coincidence does not take place for signals that are narrowband "in general", the spectrum width of which is small compared to the carrier frequency, but only for signals that are "narrowband for a given medium", the spectral width of which is small compared to the distance from the carrier frequency to nearest singularity of the wave number on the complex plane. In this case, this coincidence takes place not only in the region of applicability of the second approximation of the classical dispersion theory, but also in the entire asymptotic zone up to the path lengths at which the amplitude-frequency spectrum of the signal begins to change.