The optical analogues of Bloch oscillations and their associated Wannier-Stark ladders have been recently analyzed. In this paper we propose an elastic realization of these ladders, employing for this purpose the torsional vibrations of specially designed one-dimensional elastic systems. We have measured, for the first time, the ladder wave amplitudes, which are not directly accessible either in the quantum mechanical or optical cases. The wave amplitudes are spatially localized and coincide rather well with theoretically predicted amplitudes. The rods we analyze can be used to localize different frequencies in different parts of the elastic systems and viceversa. PACS numbers: 43.35.+d,63.20.Pw,43.40.Cw Recently, undulatory systems showing analogues of Bloch oscillations and Wannier-Stark ladders (WSL) attracted increasing attention in several fields of physics [1,2,3,4,5]. As shown by Bloch, electrons in a periodic potential have extended solutions. The same is true for the behavior of an electron under the action of a static electric field. In contrast, and opposite to intuition, when both the periodic potential and the electric field are present, the solutions are localized; this is only true when band to band Zener tunneling is negligible or the system is short enough. The spectrum then shows equally spaced resonances known as Wannier-Stark ladders, the nearest-neighbor level spacing being proportional to the intensity of the external field [6]. In the time domain, the Wannier-Stark ladders yield the so called Bloch oscillations which consist in a counterintuitive effect where the electrons show an * Permanent address:

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The flexural vibrations of a locally periodic rod, which consists of N unit cells, are discussed both from the experimental and theoretical points of view. Timoshenko's beam theory and the transfer matrix method are used to calculate the normal-mode frequencies and amplitudes. The theoretical values are then compared with the experimental ones, which are obtained using an electromagnetic acoustic transducer (EMAT). Good agreement between the numerical results and the experimental measurements is obtained. It is shown that as N grows, a band spectrum emerges.

PACS 43.20.Fn -Scattering of acoustic waves PACS 24.30.Cz -Giant resonances PACS 43.40.At -Experimental and theoretical studies of vibrating systems Abstract -The doorway state phenomenon has been recently analysed in many different systems, both quantum and classical. The systems range from nuclei to sedimentary valleys, therefore covering a range in size of 19 orders of magnitude. It also applies to systems with chaotic spectra as well as to integrable systems. In all these works, the doorway state has been discussed only in the energy or frequency domains. In this letter we present numerical and experimental results for a quasi-one-dimensional elastic system which presents a doorway state and, for the first time, the temporal evolution of the phenomenon is measured directly.

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