Reflection and refraction of three dimensional plane quasi-P waves at a corrugated surface between distinct triclinic el
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Reflection and refraction of three dimensional plane quasi-P waves at a corrugated surface between distinct triclinic elastic half spaces A. Chattopadhyay · Pato Kumari · V. K. Sharma
Received: 15 September 2010 / Accepted: 20 July 2011 / Published online: 8 August 2011 © Springer-Verlag 2011
Abstract The present work deals with the reflection and transmission of threedimensional plane quasi-P (qP) waves at a corrugated surface between distinct triclinic elastic half spaces. A method has been developed to find the analytical expressions of all the three phase velocities of qP, quasi-SV (qSV) and quasi-SH (qSH) waves in three dimensions. Closed form expressions in three-dimensions for the amplitude ratios of reflection and transmission coefficients of qP, qSV and qSH waves are obtained. Graphs for reflection and transmission coefficients are drawn for different polar and azimuthal angles. These reflection and transmission coefficients are computed numerically and shown graphically for fixed values of relevant parameters. Graphical user interface software in MATLAB has been designed to generalize the result of the study, by allowing one to choose arbitrary suitable values for relevant parameters. It is found that reflection and transmission coefficients are significantly different in three dimensional case compared to the two dimensional case. Keywords Three-dimensional waves · Corrugated surface · Polar angle · Azimuthal angle · Reflection · Refraction · Triclinic medium · Quasi-P · Quasi-SV · Quasi-SH waves · GUI Mathematics Subject Classification (2000)
86A15
A. Chattopadhyay Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India e-mail: [email protected] P. Kumari Department of Mathematics, Jaypee Institute of Information Technology, Noida 201307, India e-mail: [email protected] V. K. Sharma (B) ISSA, Defence Research and Development Organization, Delhi 110054, India e-mail: [email protected]
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List of symbols u τi j Ci j ei j ρ u¨ i p = ( p1 , p2 , p3 ) c k d = (d1 , d2 , d3 ) A η d φ θ k0 ki , (i = 1, 2, 3) ki , (i = 4, 5, 6) L j ( j = 1, 2, 3) A j ( j = 1, 2, 3) B j ( j = 1, 2, 3) E j ( j = 1, 2, 3) C j ( j = 1, 2, 3) D j ( j = 1, 2, 3) G j ( j = 1, 2, 3) Rq P Rq SV Rq S H Tq P Tq SV Tq S H Rqn P , Rqn SV , Rqn S H TqnP , TqnSV , TqnS H RqnP , RqnSV , RqnS H TqnP , TqnSV , TqnS H
Int J Geomath (2011) 2:219–253
Displacement vector Component of stress tensor Stiffness coefficients Components of strain tensor Density of medium Components of acceleration Unit propagation vector Phase velocity Wave number Unit polarization vector Wave amplitude Phase function Amplitude of the corrugation Polar angle Azimuthal angle Angular wave number of incident qP wave Angular wave number for reflected qP, qSV, qSH waves Angular wave number for transmitted qP, qSV, qSH waves Components of polarization vector for incident qP wave Components of polarization vector for reflected qP wave Components of polarization vector for reflected qSV wave Components of polarization vector for
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