Bending waves in crumpled sheets

Crumpled paper has recently emerged as a model for disordered media. Here we use wave propagation to probe aluminum foils crumpled into balls made by hand or into cylinders obtained by confinement in a container. Surprisingly, the raw dispersion relations appear to differ from sample to sample. They correspond to bending waves that follow an effective path that is shorter than the distance between the input and output points. This can be interpreted in terms of two modes of propagation: slow bending waves and a fast mode whose possible origin is discussed. In addition, the effective paths behave differently in spheres and in cylinders. These results enable the characterization of the sample structure and point toward the geometric rigidity of the configurations.