Acoustic modes propagating along the free surface of granular media

In unconsolidated granular materials under gravity there exist acoustical waves propagating along the surface with anomalously low sound velocity. The presented theory describes these guided surface acoustic modes (GSAM) confined between the surface of the granular materials and in-depth layers with increasing rigidity. The analysis is based on the obtained original analytical solution of the Helmholtz equation that has never been used both in classical and quantum mechanics. This solution is valid for a particular rigidity profile, whereas the general case of grains with or without adhesion has been analyzed numerically. In contrast to the Rayleigh wave polarized in the sagittal (vertical) plane, which is the unique localized mode in a homogeneous solid, an infinite number of modes with sagittal polarization as well as an infinite number of shear horizontal modes have been found. The difference in physical mechanisms of localization is discussed, and the transformation of the GSAMs into the Rayleigh wave at the increasing adhesion is demonstrated: The first sagittal mode transforms into the Rayleigh one, while the others delocalize. The theory explains the experimentally observed magnitude of velocity for the acoustic waves in sand elliptically polarized in the sagittal plane.