Stop-pass behavior of acoustic waves in a 1D fractured system

This study examines the dispersion and the stop-pass band behavior of acoustic waves propagating across periodically spaced and non periodically spaced parallel fractures. Laboratory ultrasonic wave measurements performed on a stack of synthetic fractures (identical steel plates with roughened interfaces) and numerical propagator matrix simulations show spectra with distinct stop-pass band structures that develop with decreasing fracture stiffness. To understand the physics behind these observations, an exact dispersion equation for wave propagation through an infinite series of equally spaced fractures is derived using displacement-discontinuity boundary conditions to model the constitutive behavior of the fractures and Floquet's (Bloch's) theory for the periodic boundary conditions. Both the measured and numerically simulated stop-pass band structures show good agreement with the theoretical predictions. Furthermore, the theory reveals that the left boundary of the stop-bands contains information about the fracture stiffness, suggesting the possibility of determining the stiffness of the parallel fractures from seismic waves. This paper also discusses the effects of fractured systems with random distributions of fracture spacings and stiffnesses on the stop-pass band structures of seismic waves in fractured rock.