Modelling shear-wave singularities in an orthorhombic medium
R. James Brown, Stuart Crampin, Eric V. Gallant
In this paper we report on and compare the results of numerical and physical modelling experiments that focus on the behaviour of shear waves near a point singularity in an orthorhombic material. In the neighbourhood of such singularities, at least one of which is known to exist in a symmetry plane of an orthorhombic material, shear waves may exhibit rapid polarization variations approaching 180 degrees. Our objectives are: to predict theoretically the location of a point singularity in the industrial laminate Phenolic CE using its stiffnesses as determined from experimental measurements; and to locate such a singularity by direct observation of wave propagation in its vicinity in a controlled physical experiment.
Piezoelectric transducer sources and receivers are placed in antipodal positions on a sphere of orthorhombic phenolic laminate and used to acquire traces shot along different directions through the phenolic. Data required for determination of stiffnesses are acquired, as well as traces along two profiles involving only shear sources and receivers. These profiles clearly show rapid variations of polarization direction near the singularity. We believe that this is the first time that a point singularity has been directly identified in either laboratory or field measurements, although the effects of propagation near point singularities have been recognized in VSP data by Bush and Crampin in the Paris Basin and by Yardley and Crampin above the Austin Chalk.
The location of the point singularity on the group-velocity or wave surfaces agrees extremely well with the location computed for the phase-velocity or slowness surfaces from the experimentally determined stiffnesses of the sphere. The slight observed shift of the singularity from phase-velocity to group-velocity surfaces agrees both in magnitude and sense with theory and with previous physical modelling results. We also find that the locations of point singularities are quite sensitive to variations in stiffness values. Small changes in stiffnesses from a cube to a sphere of phenolic cause a shift of some 15 degrees in the position of the point singularity. This could imply that determinations of the directions of point singularities in field cases will act as a very good constraint on the stiffness values of a particular rockmass, which are directly related to its lithologic and fabric-related properties.