American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-5, pp-168-183 www.ajer.org Research Paper Open Access www.ajer.org Page 168 Plane Waves in Transversely Isotropic Viscothermoelastic Medium with Two Temperatures and Rotation Rajneesh Kumar 1 , Lajvinder Singh Reen 2 , S.K. Garg 3 1 Department of Mathematics, Kurukshetra University, Kurukshetra, 136119, Haryana, India 2 Department of Mathematics, Seth Jai Prakash Mukand Lal Institute of Engineering & Technology Radaur, 135133, Haryana, India 3 Department of Mathematics, Deen Bandhu Chhotu Ram University of Science and Technology Murthal, 131039, Haryana, India ABSTRACT: The present investigation is to study the plane wave propagation and reflection ofplane waves in a homogeneous transversely isotropic viscothermoelastic medium with two temperature and rotation in the context of GN type-II and type-III (1993) theory of thermoelasticity. It is found that, for two dimensional assumed model, there exist three types of coupled longitudinal waves, namely quasi-longitudinal wave (QL), quasi-transverse wave (QTS) and quasi -thermal waves (QT). The different characteristics of waves like phase velocity, attenuation coefficients, specific loss and penetration depth are computed numerically and depicted graphically. The phenomenon of reflection coefficients due to quasi-waves at a plane stress free with thermally insulated boundary is investigated. Theratios of the amplitudes of the reflected waves to that of incident waves are calculated as a non-singular system of linear algebraic equations. These amplitude ratios are used further to calculate the shares of different scattered waves in the energy of incident wave. The conservation of energy at the free surface is verified. The effect of viscosity on the energy ratios are depicted graphically and discussed. Some special cases of interest are also discussed. Keywords: Phase velocity, Attenuation coefficients, Energy ratios, Penetration depth, viscothermoelasticity. I. INTRODUCTION The problem of elastic wave propagation in different media is an important phenomenon in the field of seismology, earthquake engineering and geophysics. The elastic wave propagating through the earth (seismic waves) have to travel through different layers and interfaces. These waves have different velocities and are influenced by the properties of the layer through which they travel. The signals of these waves are not only helpful in providing information about the internal structures of the earth but also helpful in exploration of valuable materials such as minerals, crystals and metals etc. This technique is one of the most suitable in terms of time saving and economy. As the importance of anisotropic devices has increased in many fields of optics and microwaves, wave propagation in anisotropic media has been widely studied over in the last decades. The anisotropic nature basically stems from the polarization or magnetization that can occur in materials when external fields pass by. Mathematical modeling of plane wave propagation along with the free boundary of an elastic half-space has been subject of continued interest for many years. Keith and Crampin (1977) derived a formulation for calculating the energy division among waves generated by plane waves incident on a boundary of anisotropic media. Wave propagation in a microstretch thermoelastic diffusion solid has been investigated by Kumar (2015). Reflection of plane waves at the free surface of a transversely isotropic thermoelastic diffusive solid half-space has been discussed by Kumar and Kansal (2011).Wave propagation has remained thestudy of concern of many researchers (Marin Marin (2013),Kumar and Mukhopadhyay(2010), Lee and Lee (2010),Kumar and Gupta (2013), Othman (2010),Kaushal, Kumar and Miglani(2011), Kumar, Sharma and Ram (2008),Kaushal, Sharma and Kumar(2010)). The theoretical study and applications in viscoelastic materialshavebecomeanimportanttaskforsolidmechanics with the rapid development of polymer science and plastic industry as well as with the wide use of materials under high temperature in modern technology and application of biology and geology in engineering.