Nonlinear generation of elastic waves in crystalline rock

Abstract

The nonlinear interaction of two elastic waves at frequencies ƒ1 and ƒ2 in an elastically nonlinear material can give rise to a collimated wave at the difference frequency ƒ1 ‐ ƒ2. Because the amplitude of a difference frequency beam is proportional to the degree of elastic nonlinearity of the material through which it passes, amplitude should be higher in a material containing microcracks such as rock than it is in uncracked materials such as metals, single crystals, or water in which nonlinear elastic interactions have previously been observed. The “nonlinear signal” is important for investigating the nonlinear properties of rocks. Such a beam has already proved useful as a low‐frequency acoustic source in water and may ultimately be useful in geophysical exploration. In this paper, our observations of nonlinear signal generation in experiments with crystalline rocks are presented. Three criteria must be fulfilled in such experiments to establish that nonlinear interactions take place in the rock and not in the associated experimental apparatus: (1) The frequency of the observed nonlinear signal must precisely equal the difference frequency Δƒ = ƒ1 ‐ ƒ2, (2) the amplitude of the nonlinear signal must be proportional to the product of the amplitudes of the primary beams, and (3) the trajectory of the nonlinear signal, which is a function of the input trajectories, wave types, frequencies, and rock velocities, must match that predicted by theory. We observed signals that satisfy the above three criteria in the frequency range from 0.1 to 1.0 MHz.