Transient waves in nonstationary media : with applications to lightning and nonlinear media

Abstract: Propagation of transient waves in nonstationary, inhomogeneous, dispersive, stratified media is considered. Waves originating from sources exterior to the scatterer as well as from internal sources are treated. Algorithms are developed and illustrated by computations of wave phenomena in stationary, nonstationary and weakly nonlinear media. In the theoretical part, the underlying hyperbolic equation is a general, homogeneous, linear, first order 2x2 system of equations. The coefficients depend on one spatial coordinate and time. Memory effects are modeled by integral kernels, which are functions of two different time coordinates. The analysis builds on generalization of the wave splitting concept, originally developed for time-invariant media. Imbedding and Green functions (propagator kernels) equations are derived for the external source problem. The wave propagators of the internal source problem are based on generalized Green functions equations. Special attention is paid to characteristic curves and discontinuities. Particular solutions are obtained as integrals of fundamental waves from distributed point sources. Resolvent kernels and wave propagators are essential. Direct and inverse computational algorithms are developed for the nonstationary, homogeneous semi-infinite medium. Generalized susceptibility kernels with one spatial and two time coordinates are used. A function depending on two time coordinates is reconstructed. Furthermore, direct scattering algorithms for internal sources are implemented. Waves in a Klein-Gordon slab are calculated and compared to alternative solutions obtained from analytical fundamental waves of an infinite Klein-Gordon medium. In a second example, the current and voltage waves, evoked on the power line after an imagined strike of lightning, are studied. The nonstationary properties are modeled by the shunt conductance, together with dispersion in the shunt capacitance. The nonstationary theory is used to study direct wave propagation phenomena in weakly nonlinear media by linearization. Two different iterative procedures to find the nonlinear solutions are discussed. One leads into a truly nonstationary, mixed initial boundary value problem with a linear equation characterized by time-dependent coefficients and source terms. This procedure is applied to a pulse generator for high-frequency switching. The alternative time-invariant procedure, which is a variation of the nonlinear Born approximation, is used to calculate wave propagation in Kerr media.