23 Electromagnetic Waves Generated by Line Current Pulses Andrei B. Utkin INOV - INESC Inovação Portugal 1. Introduction Solving electromagnetic problems in which both the source current and the emanated wave have complicated, essentially nonsinusoidal structure is of paramount interest for many real- word applications including weaponry, communications, energy transportation, radar, and medicine (Harmuth, 1986; Fowler at al., 1990; Harmuth et al., 1999; Hernández-Figueroa et al., 2008). In this chapter we will focus on electromagnetic fields produced by source-current pulses moving along a straight line. The explicit space-time representation of these fields is important for investigation of man-made (Chen, 1988; Zhan & Qin, 1989) and natural (Master & Uman, 1984) travelling-wave radiators, such as line antennas and lightning strokes. Traditional methods of solving the electromagnetic problems imply passing to the frequency domain via the temporal Fourier (Laplace) transform or introducing retarded potentials. However, the resulted spectra do not provide adequate description of the essentially finite-energy, space-time limited source-current pulses and radiated transient waves. Distributing jumps and singularities over the entire frequency domain, the spectral representations cannot depict explicitly the propagation of leading/trailing edges of the pulses and designate the electromagnetic-pulse support (the spatiotemporal region in which the wavefunction is nonzero). Using the retarded potentials is not an easy and straightforward technique even for the extremely simple cases, such as the wave generation by the rectangular current pulse — see, e.g., the analysis by Master & Uman (1983), re- examined by Rubinstein & Uman (1991). In the general case of the sources of non-trivial space-time structure, the integrand characterizing the entire field via retarded inputs can be derived relatively easily. In contrast, the definition of the limits of integration is intricate for any moving source: one must obtain these limits as solutions of a set of simultaneous inequalities, in which the observation time is bounded with the space coordinates and the radiator's parameters. The explicit solutions are thus difficult to obtain. In the present analysis, another approach, named incomplete separation of variables in the wave equation, is introduced. It can be generally characterized by the following stages: • The system of Maxwell's equations is reduced to a second-order partial differential equation (PDE) for the electric/magnetic field components, or potentials, or their derivatives. • Then one or two spatial variables are separated using the expansions in terms of eigenfunctions or integral transforms, while one spatial variable and the temporal variable remain bounded, resulting in a second-order PDE of the hyperbolic type, which, in its turn, is solved using the Riemann method. www.intechopen.com