The transient radiation of low-frequency elastic waves through isotropic and homogeneous soft media is investigated using the Green's function approach. A careful analysis of the coupling term is performed and yields the introduction of a very near field region in which its amplitude behaves as 1/r. To address the calculation of impulse responses, a simplified Green's function is proposed for semi-infinite media and compared to exact solutions. Impulse response calculations are successfully compared with experimental measurements obtained for circular radiators of different diameters using transient elastography. Results presented in this paper provide a better understanding of the role of the coupling term in elastography and should be used to compensate diffraction and coupling effects observed in transient elastography.