Rayleigh waves in liquid layer resting over an initially stressed orthotropic half-space under self-weight
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ORIGINAL PAPER
Rayleigh waves in liquid layer resting over an initially stressed orthotropic half-space under self-weight Sanjeev Anand Sahu 1 & Soniya Chaudhary 1 & Pradeep Kumar Saroj 2 & Amares Chattopadhyay 1
Received: 27 March 2015 / Accepted: 16 February 2017 # Saudi Society for Geosciences 2017
Abstract The study of surface waves (Rayleigh wave) finds their virtuous applications in a numerous geological and geophysical fields including water, oil, gas, and other subsurface geological probing and exploration. The present paper efforts to investigate the influence of initial stress, Earth magnetism, and gravity on propagation of Rayleigh waves. Considered model is consist of a liquid layer lying over a magnetoelastic orthotropic half-space under self-weight and initial stress. Method of separation of variable is used to solve the equation of motion. Solutions of governing equations are obtained in terms of displacement. Frequency relation for Rayleigh wave has been obtained and matched with classical Rayleigh wave equation. In addition to classical case, some existing results have been deduced as particular case of the present study. Obtained results have been shown through numerical illustrations. It is found that the considered parameters (initial stress, Earth magnetism, and gravity) have prominent effect on phase velocity of Rayleigh wave. Graphical representations have been made to exhibit the velocity profile of Rayleigh waves for different cases with the help of MATHEMATICA. The present study may be useful for seismologists and engineers who are concern with applications of wave propagation in magnetoelastic orthotropic medium. Keywords Gravity field . Initial stress . Orthotropic . Magnetoelastic . Rayleigh wave * Sanjeev Anand Sahu [email protected]
1
Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
2
Department of Mathematics, National Institute of Technology, Calicut, Kerala 826004, India
Nomenclature Cij are the stiffness tensor components in the contraction notation ! E is the electric intensity g is the Earth gravity h is the perturbed magnetic field over constant primary magnetic field vector ! H0 is the primary magnetic field vector ! j is the electric current density P is the initial stress s is the specific heat per unit mass T0 is the initial temperature T is the absolute temperature ! u is the component of displacement vector λ and μ are Lame’s constant μe is the magnetic permeability τ is the function of depth τij is the stress component
Introduction The characterization of surface waves in initially stressed magnetoelastic orthotropic medium has great significance in geophysics and Earth system sciences. Surface waves carry the greatest amount of energy from shallow shocks and are of primary cause of destruction due to earthquakes. Problems of surface waves in an orthotropic elastic medium are very important due to their extensive applications in various branches of science and technology. The assumption that t
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