Nonlinear and dispersive acoustic wave propagation

Abstract

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.