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  
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  
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.
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