Nonlinear and dispersive acoustic wave propagation

Because of cracks and poor consolidation, rocks may have large third- and fourth-order nonlinear elastic moduli. Even though strains are small, nonlinear effects may be important in acoustic wave transmission experiments. A nonlinear and dispersive extension of Hooke's law is proposed. Combined with Newton's law, this gives a nonlinear and dispersive acoustic wave equation. For some combinations of nonlinear and dispersive parameters, the wave equation can be reduced to the Korteweg-deVries equation, such that analytical solutions can be obtained. Finite-difference simulations with an initial Ricker wavelet show that the nonlinear terms in the wave equation steepen the wavefront and higher harmonics in the frequency spectrum. When dispersion is included, a nonlinear stress-strain relation with hysteresis is observed.