Focused waves and the scalar wave equation

Abstract

New solutions of the free space wave equation are studied and particular forms of these new solutions lead naturally to the localized wave solutions that have recently been reported in the literature. Localized waves exhibit a high degree of spatial and temporal localization. Choosing the characteristic variables, (z−ct, z+ct), the Green’s function is constructed for the scalar wave equation. With this form for the Green’s function, it is shown how a relatively simple line source of infinite extent, aligned with the propagation axis, gives rise to a propagated field which exhibits a degree of transverse focusing.