Wannier-Stark Ladders in Torsional Waves

We study the normal modes of torsional waves in an elastic rod consisting of a set of $n $circular cylinders of varying length determined by a parameter $\gamma $. We present experimental, theoretical, and numerical results. It is shown that some analogies to the Wannier-Stark ladders, originally introduced by Wannier (1960), are exhibited by this classical system. The ladders consist of a series of equidistant energy levels for the electrons in a crystal in the presence of a static external electric field, the nearest-neighbor spacing being proportional to the intensity of the external field. For the case of torsional waves in the rod, we have observed a similar behavior: the vibrations of the rod show resonances of equidistant frequencies, the nearest neighbor spacing being proportional to $\gamma $, associated with the geometry of the rod. One should point out, however, that the analogy is not perfect. \newline \newline References: \newline Wannier G. H. (1960) Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field, Phys. Rev. \textbf{117}, 432-439; Wannier G. H.