Abstract

Plane-wave directivity patterns for both P- and S-waves approaching a free surface are well known (Knopoff et al., 1957, Figure 3–5). These have been shown to apply in a reciprocal manner to time-harmonic S-waves emanating from vertical and horizontal sources (Miller and Pursey, 1954; Cherry, 1962) in both two-dimensional (2-D) and three-dimensional (3-D) cases. Knopoff and Gilbert (1959) showed that the plane-wave directivity patterns also apply to the first motions seen in the impulsive-source case (3-D) and Pilant (1979, sec. 9-6) showed that they held in the equivalent 2-D problem. Theoretical expressions for these patterns are given by Pilant (ibid) as  
Dzs(θ)=2sinθscosθs(a2sin2θs)½(12sin2θs)2+4sin2θscosθs(a2sin2θs)½,
(1)
and  
Dxs(θ)=cosθs(12sin2θs)(12sin2θs)2+4sin2θscosθs(a2sin2θs)½,
(2)
where θs is measured from the vertical and the positive z-axis is into the medium. The x-axis lies along the free surface and the quantity a = vs/vp. For angles greater than critical (θc = sin−1a), the proper expression for the square root is given by  
(a2sin2θs)½i(sin2θsa2)½.
(3)
Thus for angles of incidence (or take-off) greater than θc, both Dzs(θ) and Dxs(θ) become complex numbers and lead to phase-shift induced waveform changes as the S-waves interact with the free surface. The functions Re [Dzs(θ)] and Re [Dxs(θ)] are shown in Figure 1 for the angular range 34–37 degrees which includes the angle θc = 35.26 degrees. For this example, a = 3−1/2, corresponding to a Poisson's ratio equal to one-quarter. The null in Dzs(θ) and the maximum in Dxs(θ) are clearly seen.
This content is PDF only. Please click on the PDF icon to access.

First Page Preview

First page PDF preview
You do not currently have access to this article.