JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. B5, PAGES 3597-3602, APRIL 10, 1987 Nonlinear Generation of Elastic Waves in Crystalline Rock PAUL A. JOHNSON AND THOMAS J. SHANKLAND Earth and Space SciencesDivision, Los Alamos National Laboratory, Los Alamos, New Mexico RICHARD J. O'CONNELL Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts JAMES N. ALBRIGHT Earth and SpaceSciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico The nonlinear interaction of two elastic waves at frequencies fl and f2 in an elastically nonlinear material can give riseto a collimatedwave at the difference frequency fl --f2. Because the amplitudeof 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 Af=fx -f:, (2) the amplitude of the nonlinear signalmust 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. INTRODUCTION Nonlinear elastic properties and their effects have received considerable study in the field of acoustics. Westervelt [1963] showed that two near-source, collinear (i.e., parallel), high- frequency "carrier" waves could interact to produce sound with frequencies equal to the sum and difference of the high- frequency carriers while retaining a radiation pattern charac- teristic of the carriers. The radiation pattern, formed when two monochromatic collinear carriers were injected into an elasti- cally nonlinear medium, was shown to be similar to the pat- tern caused by a long linear array of signal sources in the medium itself. The collinear configuration has been called an end fire or parametric array by Bellin and Beyer [1962], who produced evidence showing parametric array formation in water-filled tanks and in air. Muir and Willette [ 1972] demon- strated that the far-field radiation pattern of the beams at the sum and differencefrequencies created by a parametric array was narrow and had no side lobes. Unterberger et al. [1981] conductedexperiments of parametric array formation in a salt dome. The signal at the difference frequency Aris of particular interest since its amplitude decaysmore slowly with distance because of its longer wavelength.Thus despite the fact that the energy conversionfrom the primary input beams to the differ- ence frequency beam is inefficient [Taylor and Rollins, 1964], the combined effectsof collimation and lower spatial attenu- ation produce a useful low-frequency acoustic source; ex- ploitation of nonlinear beam generation has led to devel- Copyright 1987 by the American Geophysical Union. Paper number 6B6260. 0148-0227/87/006 B-6260505.00 opment of new technologiesin echo sounding [Nichols, 1971], underwater communications, and probing of shallow sedi- ments beneath water [Muir, 1976]. In experiments by other workers, the acoustic medium has been a uniform, uncracked material such as water, salt, metal, glass, or single crystals in which the nonlinearity arises pri- marily from the nonlinear elasticity of the material itself I-Hiki and Mukai, 1973]. We have used rocks, which are inherently much more nonlinear than the above mentioned materials, because rocks contain numerous microcracks that give rise to large changes of velocity with pressure [Birch, 1960]. In geo- physics, elastic nonlinearity has usually been studied in con- nection with equations of state in which nonlinear contri- butions appear as third-order or higher terms in the elastic free energy expansion. These higher-order contributions are much greater in microcracked material [Watt et al., 1976]. Since the amplitude of the nonlinear signal is proportional to the value of the higher-order terms [Hiki and Mukai, 1973], the larger nonlinear terms in microcracked material should be reflected in a larger amplitude of the difference frequency beam. Therefore, using rocks in nonlinear experiments should enhance the amplitude of the difference frequency beam as compared with the amplitude generated in uncracked materi- als. Eventually, we hope to study the nonlinearity of rocks through experimental observations of the difference frequency beam amplitude. Such studies could show how effective bulk and shear moduli vary with pressure. In presenting experimental observations of the nonlinear interaction of elastic waves in crystalline rocks, a necessary first step is to confirm that the observed nonlinearity occurs within the bulk of the rock rather than in associated electronic apparatus. Our work has taken two principal directions as a 3597