This paper extends the time domain wave splitting and invariant imbedding method to an inhomogeneous wave equation with a source term: uxx−utt+A(x)ux=2D(x)i′(t). The direct scattering and inverse source problems of this equation are studied. Operators J̃± that map the source function into the scattered waves at the edges of the slab are defined. A system of coupled nonlinear integrodifferential equations for these scattering operator kernels is obtained. The direct scattering problem is to obtain the scattering operator kernels J± and R+ when parameters A and D are given. The inverse problem is to simultaneously reconstruct A(x) and D(x) from the scattering operator kernels R+(0,t), 0≤t≤2 and J−(0,t), 0≤t≤1. Both numerical inversion algorithms and the small time approximate reconstruction method are presented. A Green’s function technique is used to derive Green’s operator kernel equations for the calculation of the internal field. It provides an alternative effective and fast way to compute the scattering kernels J±. For constant A and D the Green’s operator kernels and source scattering kernels are expressed in closed form. Several numerical examples are given.
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