Experiments with acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. Comparison of the determined dispersion relations for guided surface acoustic modes with a theoretical model reveals the dependencies of the elastic moduli of the granular medium on pressure. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through non-affine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure. Unexpectedly, and in disagreement with the most of the available theories, the bulk modulus depends on pressure in a very similar way to the shear modulus.

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.

A theoretical description of a natural acoustical waveguide existing in unconsolidated granular materials due to a gravity-induced stiffness gradient is proposed. The analytical theory for the acoustic modes propagating in the medium with a power-law type inhomogeneity uses some original solutions of the Helmholtz equation that have not been derived before either in classical or in quantum mechanics. The dispersion relations and a physical mechanism of localization for these modes indicate their essential difference both from the Rayleigh surface acoustic waves and the waveguide modes in homogeneous plates.