Numerical computations of time-domain diffractions from wedges and reflections from facets

Abstract

Numerical computations of time-domain impulsive functions require a low-pass-filter operation to satisfy the Nyquist sampling rule. For example, the exact impulsive solutions for diffraction and reflection from a rigid wedge [M. A. Biot and I. Tolstoy, J. Acoust. Soc. Am. 29, 381–391 (1957)] both have singularities at their initial arrival times. These particular solutions can be approximated and low-pass filtered to satisfy the sampling rule in numerical computations. In evaluating specular reflection from a finite plane facet, a reinterpretation of the physical meanings of the terms resulting from the impulsive solution of the Fresnel–Kirchhoff integral is introduced [M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Sec. A.9, and A. W. Trorey, Geophysics 35, 762–784 (1970)]. It is believed that the solution contains terms associated only with reflection, and that the interpretation of the Rubinowicz representation of the boundary terms as boundary diffraction waves is physically incorrect. As a consequence, for amplitude calculations of a specular facet reflection, a working hypothesis is proposed, namely that the reflected amplitude is proportional to the incident signal where the constant of proportionality is a function of the geometry, facet width, and the signal waveform. By numerical studies, the constant of proportionality g(v) can be expressed as a polynomial in v, where v=w(λAr)1/2, w is the facet width, λA is the wavelength corresponding to the peak frequency of the signal, and r is the distance from a colocated source and receiver to the facet. The function g(v) depends on the waveform of the signal (i.e., a boxcar, single cycle of a sinewave, etc.). As v tends to zero, the facet reflection tends to zero. For large v (>2 or 3), g(v) tends to one and the facet reflection is effectively that from a full plane.