Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function |
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| Institution: | 1. School of Civil Engineering and Architecture, Xiamen University of Technology, Xiamen 361000, China;2. State Key Laboratory for Disaster Reduction in Civil Engineering, TongJi University, Shanghai 200092, China;3. College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China;1. Department of Civil Engineering, Istanbul Technical University, Istanbul, Turkey;2. Department of Civil Engineering, University of Southern California, Los Angeles, CA, USA;1. Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, No. 1, Xikang Road, Nanjing 210098, China;2. Department of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder, CO 80309-0428, USA;3. Department of Civil Engineering, The University of Hong Kong, Hong Kong, China;1. State Key Laboratory of Coast and Offshore Engineering, Dalian University of Technology, Dalian 116024, China;2. Institute of Earthquake Engineering, Dalian University of Technology, Dalian 116024, China;1. Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin 300072, China;2. Department of Civil Engineering, University of Southern California, Los Angeles, CA 90089–2531, USA |
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| Abstract: | A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly. |
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| Keywords: | Scattering Diffraction Stress-free waves Elastic half-space Cylindrical waves Bessel Hankel functions Boundary-valued Free-field Amplification |
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