This paper investigates the details of reflection, transmission, and conversion of plane waves incident upon a fracture at arbitrary angles. The elastic compliance of fractures that is produced by the presence of a planar collection of void spaces and asperities of contact is modeled as a displacement‐discontinuity boundary condition between two elastic half‐spaces. Closed‐form expressions for the transmission and reflection coefficients on a fracture are derived by replacing the boundary conditions for a welded interface by those for a fracture into the standard procedure for plane wave analysis. The closed‐form expressions reveal that a single fracture can produce a variety of potentially diagnostic waves such as transmitted waves, reflected waves, converted waves, head waves, and P interface waves and introduce a finite group time delay to all these waves with respect to the incident wave. The amplitude and group time delay of the fracture‐induced waves are controlled by the fracture stiffness, wave frequency, and the Poisson's ratio of the medium. The head wave and inhomogeneous P interface waves are generated when an SV wave is incident upon a fracture, at and beyond a critical angle, respectively, which is determined by Snell's law. For some combinations of the fracture stiffness and the Poisson's ratio of the half‐spaces, no reflection or transmission of a P wave or an SV wave occurs.

Interface waves on a single fracture in an elastic solid are investigated theoretically and numerically using plane wave analysis and a boundary element method. The finite mechanical stiffness of a fracture is modeled as a displacement discontinuity. Analysis for inhomogeneous plane wave propagation along a fracture yields two dispersive equations for symmetric and antisymmetric interface waves. The basic form of these equations are similar to the classic Rayleigh equation for a surface wave on a half‐space, except that the displacements and velocities of the symmetric and antisymmetric fracture interface waves are each controlled by a normalized fracture stiffness. For low values of the normalized fracture stiffness, the symmetric and antisymmetric interface waves degenerate to the classic Rayleigh wave on a traction‐free surface. For large values of the normalized fracture stiffness, the antisymmetric and symmetric interface waves become a body S wave and a body P wave, respectively, which propagate parallel to the fracture. For intermediate values of the normalized fracture stiffness, both interface waves are dispersive. Numerical modeling performed using a boundary element method demonstrates that a line source generates a P‐type interface wave, in addition to the two Rayleigh‐type interface waves. The magnitude of the normalized fracture stiffness is observed to control the velocities of the interface waves and the partitioning of seismic energy among the various waves near the fracture.

Two-dimensional boundary element simulations are conducted to investigate the properties of Rayleigh-type fracture interface waves generated by a line source located near a single fracture. The fracture is modeled as a displacement–discontinuity boundary condition between two elastic half-spaces with identical properties. Numerical simulations are performed for different fracture stiffnesses, source polarizations, and source depths. Symmetric and antisymmetric fracture interface waves are observed with amplitudes and velocities that are controlled by the ratio of the fracture impedance to the half-space shear wave impedance, as predicted by plane-wave theory [Gu et al., J. Geophys. Res. 101, 827–835 (1996); Pyrak-Nolte and Cook, Geophys. Res. Lett. 14, 1107–1110 (1987)]. When the source is located off the fracture, these waves develop at incidence angles that decrease with source depth.

This paper describes a boundary integral equation method for simulating two‐dimensional elastic wave propagation in a rock mass with nonwelded discontinuities, such as fractures, joints, and faults. The numerical formulation is based on the three‐dimensional boundary integral equations that are reduced to two dimensions by numerical integration along the axis orthogonal to the plane of interest. The numerical technique requires the assembly and solution of the coefficient matrix only for the first time step, resulting in a significant reduction in computational time. Nonwelded discontinuities are each treated as an elastic contact between blocks of a fractured rock mass. Across such an elastic contact, seismic stresses are continuous and particle displacements are discontinuous by an amount which is proportional to the stress on the discontinuity and inversely to the specific stiffness of the discontinuity. Simulations demonstrate that such formulated boundary element method successfully models elastic wave propagation along and across a single fracture generated by a line source.