For a harmonic wave of single frequency impinging on a fracture, the Pyrak-Nolte displacement discontinuity model (DDM) can be used to determine the reflection and transmission coefficients of the resultant harmonic waves. However, it is often the case in the field that a wave impinging on a fracture will be a pulse of finite duration which contains a spectrum of frequencies. Here, for a P-wave impinging a fracture at normal incidence, the DDM is extended to consider arbitrarily shaped pulse waves using the discrete Fourier Transform. The method presented calculates two resultant spectra for the transmitted and reflected wave as well as their corresponding energy coefficients. The impinging pulse wave is made dimensionless with respect to a defined "characteristic" frequency; this allows for a dimensionless comparison to the energy response of a single frequency harmonic wave using the Pyrak-Nolte curves. The results show that the total reflected energy of a finite-pulse incoming wave will be lower than that of a harmonic wave having the same characteristic frequency.
The study of waves interacting with fractures is of upmost importance for the detection of joints, bedding planes, faults, and fractures (Myer et al 1985; Pyrak-Nolte et al 1987), determining the mechanical properties of the fractures (Pyrak-Nolte and Nolte 2016), and modeling the reflection of blast waves during excavation operations (Hao et al 2001).
When a P-wave, SH-wave, SV-wave, or any relative combination of these waves, impinges on a fracture, at an arbitrary angle of incidence, it is transformed into a resultant transmitted and reflected wave, each respectively containing a proportion of the impinging wave energy. If the fracture is sufficiently thin compared to the wavelength of the impinging wave, such that the inertia of the fracture is negligible, the stress can be considered continuous across the fracture, whilst the displacement is discontinuous. This interface boundary condition is referred to as the displacement discontinuity model (DDM; Pyrak-Nolte et al 1990a), also known as the slip-interface model (Schoenberg 1980). It is used for both analytical and numerical calculations (Coates and Schoenberg 1995).