Subluminal wave bullets: Exact localized subluminal solutions to the wave equations

Abstract

In this work it is shown how to obtain, in a simple way, localized (nondiffractive) subluminal pulses as exact analytic solutions to the wave equations. These ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the noncausal (backward moving) components that so frequently plague the previously known localized waves. Such solutions are suitable superpositions of---zeroth order, in general---Bessel beams, which can be performed either by integrating with respect to (w.r.t.) the angular frequency $\ensuremath{\omega}$, or by integrating w.r.t. the longitudinal wave number ${k}_{z}$: Both methods are expounded in this paper. The first one appears to be powerful enough; we study the second method as well, however, since it allows us to deal even with the limiting case of zero-speed solutions (and furnishes a way, in terms of continuous spectra, for obtaining the so-called ``frozen waves,'' so promising also from the point of view of applications). We briefly treat the case, moreover, of nonaxially symmetric solutions, in terms of higher-order Bessel beams. Finally, some attention is paid to the known role of special relativity, and to the fact that the localized waves are expected to be transformed one into the other by suitable Lorentz transformations. In this work we fix our attention especially on acoustics and optics: However, results of the present kind are valid whenever an essential role is played by a wave equation (such as electromagnetism, seismology, geophysics, gravitation, elementary particle physics, etc.).