Research Article The Analytical Form of the Dispersion Equation of Elastic Waves in Periodically Inhomogeneous Medium of Different Classes of Crystals Nurlybek A. Ispulov, 1 Abdul Qadir, 2 Marat K. Zhukenov, 1 Talgat S. Dossanov, 1 and Tanat G. Kissikov 3 1 S. Toraighyrov Pavlodar State University, Pavlodar 140008, Kazakhstan 2 Sukkur Institute of Business Administration, Sindh, Pakistan 3 University of California, Davis, CA 95616, USA Correspondence should be addressed to Abdul Qadir; aqadir@iba-suk.edu.pk Received 28 March 2016; Revised 28 June 2016; Accepted 16 November 2016; Published 29 January 2017 Academic Editor: Andr´ e Nicolet Copyright © 2017 Nurlybek A. Ispulov et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te investigation of thermoelastic wave propagation in elastic media is bound to have much infuence in the felds of material science, geophysics, seismology, and so on. Te heat conduction equations and bound equations of motions difer by the difculty level and presence of many physical and mechanical parameters in them. Terefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of frst-order diferential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained. 1. Introduction Te theory of thermoelasticity deals with the study of mutual interactions of thermal and mechanical felds in elastic bodies [1, 2]. It has vast applications in the various branches of Physics as well as in engineering, like materials engineering, mechanical engineering, nuclear engineering, and so forth. Teory of thermoelasticity is based on assumption that temperature distribution in an elastic object is governed by hyperbolic type parabolic-type partial diferential equa- tion as described by Fourier law of heat conduction [3– 5]. According to Fourier law any thermal impulse is felt everywhere instantly in an object. Obviously it raised some serious concerns due to its unrealistic point of view. In order to circumvent this problem and to make it realistic a generalized theory of thermoelasticity was proposed which takes into account a fnite thermal relaxation time. In this the- ory the temperature distribution is governed by hyperbolic type equations, which results in heat propagation in solids being considered as wave phenomenon instead of difusion phenomenon. In order to investigate the wave propagation in aniso- tropic inhomogeneous medium a new method of matrizant was developed. Tis method allows investigation of wave propagation in anisotropic medium with various physical and mechanical properties [6–8]. In 1950 Tompson [9] proposed a matrix method in order to investigate the elastic wave propagation in isotropic stratifed media. Haskell also enhanced the method in 1953 [10]. Afer that major work was carried out by Stroh and others [11, 12]. He analytically investigated the dislocations in anisotropic medium by expressing frst-order motion equations using (6×6) matrix. In order to investigate the insu- lators made up of piezoelectric materials, six-dimensional framework was enhanced to eight-dimensional formalism. Matrix method also paved the way for carrying out numerical simulation in anisotropic media [13, 14]. Various researchers have investigated the ordered structures and layered medium Hindawi Advances in Mathematical Physics Volume 2017, Article ID 5236898, 8 pages https://doi.org/10.1155/2017/5236898