In this paper we present a new consolidated approach to the problem of determining the electromagnetic field resulting from a finite electric current source outside two electrically different media. This approach rests on properties of generalized functions and the solution of a given problem depends essentially only on the gradients of the characteristic functions of the various regions. In particular we study the scattered fields from a flat earth, from a two-layered earth, from an earth consisting of two different adjacent electrical media (the "mixed-path" problem), and from a rough surface. We show that each of these problems can be modeled and solved from one set of equations; in each case we carry the analysis to a stage where it agrees with that presented by the authors in specific detailed papers that treat each problem separately. As such the solutions can be shown to agree with classical approaches. The technique used here contains a minimum of sophisticated analysis and has the following properties: The source field is arbitrary, the resulting equations directly relate the unknown electric and known source field intensities without the use of intermediate Hertz potentials, and the boundary conditions evolve directly from the formulation as auxiliary equations.

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We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

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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.