Weakly nonlinear oscillations of a compliant object buried in soil.

A nonlinear model equation in Rayleigh-Plesset form is developed for volume oscillations of a compliant object buried close to the surface in soil. The equation takes into account the stress-free boundary condition on the surface of the ground. The model is fully nonlinear given exact relations for the elastic potential energy stored in deformation of the object and the soil. Expansions of the potential energies for weak nonlinearity are provided in terms of elastic constants that can be determined experimentally. When the shear modulus is allowed to increase with depth below the surface, the natural frequency predicted by the model first decreases and thereafter increases with burial depth, in agreement with reported observations. Perturbation solutions are derived for the displacements on the surface of the ground at the second harmonic and difference frequency due to the nonlinear response of the object to acoustic excitation.