The Burton, Cabrera and Frank (BCF) theory plays a key conceptual role in understanding and modeling the crystal growth of vicinal surfaces. In BCF theory the adatom concentration on a vicinal surface obeys to a diffusion equation, generally solved within quasi-static approximation where the adatom concentration at a given distance x from a step has a steady state value n(x). Recently, we show that going beyond this approximation (Ranguelov and Stoyanov, 2007) [6], for fast surface diffusion and slow attachment/detachment kinetics of adatoms at the steps, a train of fast-moving steps is unstable against the formation of steps density waves. More precisely, the step density waves are generated if the step velocity exceeds a critical value related to the strength of the step step repulsion. This theoretical treatment corresponds to the case when the time to reach a steady state concentration of adatoms on a given terrace is comparable to the time for a non-negligible change of the step configuration leading to a terrace adatom concentration n(x, t) that depends not only on the terrace width, but also on its "past width". This formation of step density waves originates from the high velocity of step motion and has nothing to do with usual kinetic instabilities of step bunching induced by Ehrlich-Schwoebel effect, surface electromigration and/or the impact of impurities on the step rate. The so-predicted formation of step density waves is illustrated by numerical integration of the equations for step motion. In order to complete our previous theoretical treatment of the non stationary BCF problem, we perform an in-situ reflection electron microscopy experiment at specific temperature interval and direction of the heating current, in which, for the first time, the step density waves instability is evidenced on Si(111) surface during highest possible Si adatoms deposition rates. (C) 2016 Elsevier B.V. All rights reserved.