NONLINEAR TUBE WAVES

The nonlinear characteristics of an acoustic tube wave propagating alo ng the axis of a fluid-filled circular borehole in an elastic solid th at is locally isotropic but whose properties may vary radially is cons idered. The analysis is carried out in the quasistatic limit. All term s through quadratic in the amplitude of the wave are considered and th e amplitude of second-harmonic generation and the pressure dependence of the tube wave speed, dV(T)/dp, are expressed in terms of the fluid and formation nonlinear parameters. The results show that if there is no radial variation of the shear modulus of the solid then both the am plitude of second-harmonic generation and dV(T)/dp are independent of the third-order elastic constants of the solid and nearly equal to tho se of the fluid alone. If there is a radial variation of the shear mod ulus then the numerical calculations indicate that both the amplitude of second-harmonic generation and dV(T)/dp can be completely dominated by the nonlinear parameters of the solid. A perturbation theory valid for the case in which the shear modulus is nearly constant is derived that demonstrates that the nonlinear response is scaled by the value of the third-order parameters of the solid, leveraged by the degree an d depth of alteration of the shear modulus.