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Abstract

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 θ is the coordinate of the th 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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