The complete extension of the Biot–Tolstoy solution to the density contrast wedge with sample calculations

Abstract

The Biot–Tolstoy exact time-domain solution for the three-dimensional impulse response of an impenetrable wedge is extended to accommodate the isovelocity or density-contrast wedge. Fourier transformation of the time and axial variables, along with a Kantorovich–Lebedev transform applied to the cylindrical radial coordinate, leads to a solution in terms of residue series. When the wedge angle is a rational fraction of π, the residue series can be reduced to a finite sum which is evaluated for some special cases. The total pressure field consists of geometrical acoustics contributions, as predicted by Snell’s laws, plus a modified version of the Biot–Tolstoy diffraction field.