The main objective of this paper is to show the comparison of two models of infinite absorbing layer with increasing damping in numerical investigations of elastic wave propagation in unbounded structures. This has been achieved by the Authors by a careful investigation of two different engineering structures characterised by gradually increasing geometrical and mathematical description complexities. The analysis included propagation of longitudinal elastic waves in a 1-D half-infinite isotropic rod, modelled according to the classical 1-mode theory of rods as well as propagation of coupled shear and flexural elastic waves in a 1-D half-infinite isotropic beam modelled according to the Timoshenko beam theory. The comparison of both models has been not only presented by the Authors, but also advantages and disadvantages of both of them have been discussed.