As a preliminary study to more complex situations of interest in small-scale technology, this paper envisages the elementary propagation properties of elastic waves in one-spatial dimension when some of the properties (mass density, elasticity) may vary suddenly in space or in time, the second case being of course more original. Combination of the two may be of even greater interest. Toward this goal, a critical examination of what happens to solutions at the crossing of pure space-like and time-like material discontinuities is given together with simple solutions for smooth transitions and numerical simulations in the discontinuous case. The effects on amplitude, speed of propagation, frequency changes and the appearance of a Doppler-like effect are demonstrated although the whole physical system remains linear.