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For elastic wave scattering problems in unbounded anisotropic media, theexistence of backward waves makes classic truncation techniques failcompletely. This paper is concerned with an exact truncation technique forterminating backward elastic waves. We derive a closed form of elastrodynamicGreen's tensor based on the method of Fourier transform and design twofundamental principles to ensure its physical correctness. We present arigorous theory to completely classify the propagation behavior of Green'stensor, thus proving a conjecture posed by B\'ecache, Fauqueux and Joly (J.Comp. Phys., 188, 2003) regarding a necessary and suffcient condition of thenon-existence of backward waves. Using Green's tensor, we propose a newradiation condition to characterize anistropic scattered waves at infinity.This leads to an exact transparent boundary condition (TBC) to truncate theunbounded domain, regardless the existence of backward waves or not. We developa fast algorithm to evaluate Green's tensor and a high-accuracy scheme todiscretize the TBC. A number of experiments are carried out to validate thecorrectness and efficiency of the new TBC.

On exact truncation of backward waves in elastrodynamics