Transient waves from internal sources in non-stationary media — Numerical implementation

Abstract

In this paper, the focus is on numerical results from calculations of scattered direct waves, originating from internal sources in non-stationary, dispersive, stratified media. The mathematical starting point is a general, inhomogeneous, linear, first order, 2 × 2 system of equations. Particular solutions are obtained, as integrals of waves from point sources distributed inside the scattering medium. Resolvent kernels are used to construct time-dependent fundamental wave functions at the location of the point source. Wave propagators, closely related to the Green functions, at all times advance these waves into the surrounding medium. Two illustrative examples are given. First waves, propagating from internal sources in a Klein-Gordon slab, are calculated with the new method. These wave solutions are compared to alternative solutions, which can be obtained from analytical fundamental waves, solving the Klein-Gordon equation in an infinite medium. It is shown, how the Klein-Gordon wave splitting, which transforms the Klein-Gordon equation into a set of uncoupled first order equations, can be used to adapt the infinite Klein-Gordon solutions to the boundary conditions of the Klein-Gordon slab. The second example hints at the extensive possibilities offered by the new method. The current and voltage waves, evoked on the power line after an imagined strike of lightning, are studied. The non-stationary properties are modeled by the shunt conductance, which grows exponentially in time, together with dispersion in the shunt capacitance.