Edge Waves as a Solution of the Wave Equation
Published:January 01, 2008
2008. "Edge Waves as a Solution of the Wave Equation", Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting, Kamill Klem-Musatov, Arkady M. Aizenberg, Jan Pajchel, Hans B. Helle
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A line formed by points of discontinuity of the interface or its first tangential derivatives is called an edge. A point of the edge is considered to be regular if the corresponding line is continuous along with its first tangential derivative. The edge is considered smooth if its points are all regular.
In the near vicinity of any point of a smooth edge, we can neglect its curvature and approximate the interface by two half planes touching at the edge (Figure 1). Then reflected/transmitted wavefields must satisfy the boundary conditions at both half planes. By using cylindrical coordinates (r, θ, z) where axis z coincides with the edge, we can write the boundary conditions as where θℓ is the coordinate of the ℓth half plane. Then the reflection/transmission problem reduces to the integration of equations 139 of Chapter 1 under conditions 1 above.
Also rewrite equation 136 of Chapter 1 in cylindrical coordinates. Let points x1 = x2 = x3 = 0 and r = z = 0 coincide, axes x1 and z coincide, and planes x2 = 0 and θ = 0 coincide. Then and the wave-vector components 137 of Chapter 1 can be written as where krm is the projection of km on the plane z = constant and αm is the angular cylindrical coordinate of this projection (Figure 2). By substituting these equations into 136 of Chapter 1, we obtain
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Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting
In Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting (SEG Geophysical Monograph Series No. 14), the theoretical framework of the edge and tip wave theory of diffractions has been elaborated from fundamental wave mechanics. Seismic diffractions are inevitable parts of the recorded wavefield scattered from complex structural settings and thus carry back to the surface information that can be exploited to enhance the resolution of details in the underground. The edge and tip wave theory of diffractions provides a physically sound and mathematically consistent method of computing diffraction phenomena in realistic geologic models. In this book, theoretical derivations are followed by their numerical implementation and application to real exploration problems. The book was written initially as lecture notes for an internal course in diffraction modeling at Norsk Hydro Research Center, Bergen, Norway, and later was used for a graduate course at Novosibirsk State University in Russia. The material is drawn from several previous publications and from unpublished technical reports. Edge and Tip Diffractions will be of interest to geoscientists, engineers, and students at graduate and Ph.D. levels.