Two fundamental fixed-speed representations of localized pulse solutions to the scalar wave equation

Abstract

Summary form only given. A study is undertaken of two fundamental representations suitable for the derivation of localized wave (LW) pulse solutions to the scalar wave equation. The first one uses superpositions over products of plane waves moving in opposite directions along the characteristic variables z-ct and z+ct. This bidirectional representation, introduced in Besieris et al. (1989), has proved instrumental in advancing our understanding of focus wave mode-like pulses. The second representation, based on the Lorentz invariance of the scalar wave equation, uses products of plane waves propagating along either subluminal boost variables or superluminal boost variables. The superluminal boost representation is particularly suitable for the derivation of X wave-like solutions. Subluminal and superluminal Lorentz transformations are used to derive closed-form LW solutions to the scalar wave equation by boosting known solutions of other equations. Several solutions are deduced in this manner and their properties are discussed. Of particular interest is the derivation of a novel finite energy LW solution, named the modified focus X wave pulse. It is characterized by low sidelobe levels, a desirable property for applications. The two fundamental representations described above involve multiplicative plane waves moving with fixed speeds along the preferred direction z. They are contrasted to the conventional Fourier synthesis which consists of an additive superposition of plane waves, each one characterized by a speed c perpendicular to its wavefront.