Sub-luminal wave bullets: Exact Localized subluminal Solutions to the Wave Equations

TL;DR

This paper presents a simple method to derive exact, localized subluminal wave pulses that propagate without distortion in homogeneous media, avoiding non-causal components and applicable across various wave-based fields.

Contribution

It introduces new analytic solutions for subluminal localized waves, employing superpositions of Bessel beams via two integration methods, including a novel approach for zero-speed and frozen wave solutions.

Findings

Exact subluminal localized wave solutions derived

Solutions applicable for arbitrary frequencies and bandwidths

Method for obtaining zero-speed and frozen waves explored

Abstract

In this work it is shown how to obtain, in a simple way, localized (non- diffractive) subluminal pulses as exact analytic solutions to the wave equations. These new 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 non-causal components so frequently plaguing the previously known localized waves. The new 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, or by integrating w.r.t. the longitudinal wavenumber: 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 dealing even with the limiting case of zero-speed solutions (and furnishes a new 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 non-axially symmetric solutions, in terms of higher order Bessel beams. At last, particular attention is paid to the 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. The analogous pulses with intrinsic finite energy, or merely truncated, will be constructed in another paper. In this work we fix our attention especially on electromagnetism and optics: but results of the present kind are valid whenever an essential role is played by a wave-equation (like in acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.)