UBC Theses and Dissertations
Reflected wave propagation in a wedge Ishill, Hiroshi
The behavior of elastic body waves in a dipping layer overlying an elastic medium has been theoretically investigated by a multiple reflection formulation. Although the diffracted wave is not included in this formulation, its importance is studied by investigation of the amplitude discontinuities within the wedge. For a plane SH wave incident at the base of the dipping layer perpendicular to strike, a series solution has been obtained. Numerical values of the amplitude, phase and phase velocity are calculated on the surface. For waves propagating in the up-dip direction the amplitude versus frequency curves for a constant depth to the interface change slowly with increasing dip for dip angles less than 20°. However for waves propagating in the down-dip direction the character of the amplitude curves change rapidly. In these cases, it is found that the diffracted wave plays an important role. In addition to satisfying the boundary conditions at the surface and the lower boundary of the wedge, the diffracted wave must also satisfy additional conditions along a dipping interface between the wedge boundaries due to the geometrical nature of the reflected wave solution. It is found that the phase velocities vary rapidly with both period of the wave and depth to the interface. For incident plane P and SV waves, the complexity of the problem due to the converted waves does not allow the solution to be expressed in series form. However, a computational scheme has been developed which allows the calculation of the disturbance due to the multiply reflected waves. For both incident P and SV waves, numerical values of displacements and displacement ratios are calculated on the surface. It is found that the displacement ratios for incident SV waves are much more sensitive to dip than are there for incident P waves. For incident P and SV waves propagating in the down-dip direction with a propagation direction [symbol omitted],β = 120°, the amplitude ratio versus frequency curves for constant depth to interface do not have significant peaks for dip angles greater than 15°. The maximum discontinuities caused by the outgoing wave are also calculated to determine the role of the diffracted wave. As subsidiary problems the energy relations between waves at an interface between elastic media are determined in terms of propagation direction in a cylindrical system and the complex propagation direction is interpreted using the Rayleigh wave. The final study is to determine by a reflected wave formulation the displacements due to periodic and impulsive line sources of SH waves in the wedge overlying an elastic medium. A formal solution is found by which the contributions due to head and reflected waves are determined by evaluation of the integrals by the method of steepest descent. Using ray paths, the contributions of the integrals have been interpreted. The range of existence of head waves has been examined and the discontinuities associated with diffracted waves studied. In the case of a free or rigid lower boundary of the wedge, the dispersion relation has been determined.
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