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Theoretical Analysis of Longitudinal Wave Attenuation in a Stressed Rock With Variable Cross-Section

In this work, the longitudinal wave propagation in stressed rock with variable cross-section is investigated analytically. Considered the stress-sensibility of dynamic elastic modulus and the viscosity of rock, a modified viscoelastic stress-strain relationship is established. Based on the continuity equation, motion equation and stress-strain relation equations, the wave propagation equation for a stressed rock with variable cross-section is obtained. The harmonic wave propagation is discussed in detail by calculating the attenuation coefficient in amplitude. The combined effects of static stress and geometry on the wave attenuation are analyzed. The results show that due to the variable static stress along the propagation path, the wave attenuation is space-dependent, and the distribution of attenuation coefficients may be remarkably different under different levels of static stress. The wave attenuation in a stressed rock with variable cross-section is also frequency-dependent, and the influence of static stress on the lower-frequency wave components is more obvious compared with that on the higher-frequency wave components. Comparing the wave attenuation among rocks with three different geometries, we conclude that the wave attenuation depends on actual normal static stress, the cross-sectional areas and the changing rates of cross-sectional area.