Two Fundamental Representations of Localized Pulse Solutions To the Scalar Wave Equation - Abstract

Abstract

A study is undertaken of two fundamental representations suitable for the derivation of localized pulse (LW) 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 an earlier publication, has proved instrumental in advancing our understanding of Focus Wave Mode (FWM)-like pulses. The second representation, based on the Lorentz invariance of the scalar wave equation, uses products of plane waves propagating along the subluminal and superluminal boost variables. This representation is suitable for the derivation of X-wave-type 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, e.g., the 2-D scalar wave equation, the Helmholtz equation and Laplace's equation. Several of these LW 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, e.g., in pulse echo techniques used in medical imaging.