Reflection and refraction of elastic waves at the interface of an elastic solid and partially saturated soils
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O R I G I NA L PA P E R
M. Kumar · A. Singh · M. Kumari · M. S. Barak
Reflection and refraction of elastic waves at the interface of an elastic solid and partially saturated soils
Received: 27 April 2020 / Revised: 15 July 2020 / Accepted: 5 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The present problem deals with the reflection and refraction of plane waves at the interface of an elastic solid and partially saturated soils. A mathematical model of partially saturated soils developed by Ghasemzadeh and Abounouri (Soil Dyn Earthq Eng 51:1–8, 2013) is employed. The incidence of two main waves (P and SV ) is considered. As a result of the incident wave, two reflected waves (in an elastic solid) and four refracted waves (in partially saturated soils) are generated. The elastic solid behaves non-dissipative, while partially saturated soils behave dissipative due to the presence of viscous pore fluids, as a result of which the incident and reflected waves are homogeneous and all the refracted waves are inhomogeneous. The coefficients of reflection and refraction for a given incident wave result in a non-singular system of linear equations. Further, these reflection and refraction coefficients are used to obtain the energy shares of various reflected and refracted waves. This study has been numerically tested to determine the effects of various properties of partially saturated soils on the various energy shares. The numerical example confirms that during the reflection or refraction process conservation of incident energy is obtained at each angle of incidence.
1 Introduction The problems based on the reflection and refraction techniques at the interface of two media are of great interest from both practical and theoretical point of views. These techniques are widely used in the exploration geophysics, petroleum engineering, building industry, and various other fields. The propagation characteristics of seismic waves are influenced by material properties through which they travel. The phenomenon of reflection and refraction (i.e., some of the incident energy reflected back from the interface and some refracted through the interface) is occurring due to the discontinuities that encounter between two different materials during the seismic wave propagation. The recorded signals of reflected and refracted waves from the interface provide a lot of information about various key features of the medium through which they propagate. Therefore, this technique (reflection/refraction) is mainly used in the exploration of minerals and hydrocarbons present inside M. Kumar Department of Mathematics, Dr. B. R. Ambedkar Govt. College, Dabwali, Sirsa, Haryana 125104, India E-mail: [email protected] A. Singh · M. S. Barak Department of Mathematics, Indira Gandhi University Meerpur, Rewari, Haryana 122503, India E-mail: [email protected] M. S. Barak E-mail: [email protected] M. Kumari (B) Department of Mathematics, M. P. College for Women, Mandi Dabwali, Sirsa, Haryana 125104, India E-
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