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