mathematics Article Effect of Initial Stress on an SH Wave in a Monoclinic Layer over a Heterogeneous Monoclinic Half-Space Ambreen Afsar Khan 1 , Anum Dilshad 1 , Mohammad Rahimi-Gorji 2, * and Mohammad Mahtab Alam 3   Citation: Khan, A.A.; Dilshad, A.; Rahimi-Gorji, M.; Alam, M.M. Effect of Initial Stress on an SH Wave in a Monoclinic Layer over a Heterogeneous Monoclinic Half-Space. Mathematics 2021, 9, 3243. https://doi.org/10.3390/ math9243243 Academic Editor: Xiangmin Jiao Received: 15 September 2021 Accepted: 11 December 2021 Published: 14 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan; ambreen.afsar@iiu.edu.pk (A.A.K.); anumdilshad3@gmail.com (A.D.) 2 Faculty of Medicine and Health Sciences, Ghent University, 9000 Ghent, Belgium 3 Department of Basic Medical Sciences, College of Applied Medical Science, King Khalid University, Abha 62529, Saudi Arabia; mmalam@kku.edu.sa * Correspondence: mohammad.rahimigorji@ugent.be Abstract: Considering the propagation of an SH wave at a corrugated interface between a monoclinic layer and heterogeneous half-space in the presence of initial stress. The inhomogeneity in the half- space is the causation of an exponential function of depth. Whittaker’s function is employed to find the half-space solution. The dispersion relation has been established in closed form. The special cases are discussed, and the classical Love wave equation is one of the special cases. The influence of nonhomogeneity parameter, coupling parameter, and depth of irregularity on the phase velocity was studied. Keywords: SH-wave; monoclinic; heterogeneous; undulatory; initial stress 1. Introduction Many experimental and theoretical studies predicted that the Earth is a convo- luted model in nature. Therefore, a more natural presentation of the medium through which seismic waves propagate is required. The propagation of a wave in an inho- mogeneous medium is of keen interest due to the continuous variation in the elastic properties of the material. The heterogeneity in the material is produced by a change in rigidity and density. Various authors have taken various forms of the variation, like linear exponential, quadratic, etc., for simulating the variation in different geo- logical parameters inside the Earth. Jeffrey [1] deliberated the impact of heterogene- ity on the Love wave. Bullen [2] mentioned that density varies inside the Earth at different rates. Wilson [3] examined the propagation of the Rayleigh wave in a het- erogeneous medium. Dhua and Chattopadhyay [4] discussed wave propagation in heterogeneous layers of the Earth. Alam et al. [5] examined the propagation of an SH-wave in an anisotropic crustal layer over a heterogeneous half-space. Taking into account the structure and characteristics of the Earth, a variety of crustal forms are possible. Crystals are solid in nature and bounded by faces or plane surfaces. The monoclinic form is one of them. It has three unequal axes, two intersecting at an oblique angle, and the third is transverse to them. Several authors worked in this direction [6–12]. Initial stress develops in the body due to many physical causes. The initial stress has a great impact on the rigidity of the elastic structure and produces a me- chanical fault called buckling. Biot [13] firstly examined the propagation of light under initial stress. A detailed study on the propagation of Love, Rayleigh, and SH-waves in a pre-stress heterogeneous half-space was made by Chaterjee et al. [14]. Singh et al. [15] analyzed the impact of stress and irregularity on the Love wave in a heterogeneous medium. They have shown that horizontal stress has a favor- able impact on the phase velocity. Abd-Alla et al. [16] discussed the impact of Mathematics 2021, 9, 3243. https://doi.org/10.3390/math9243243 https://www.mdpi.com/journal/mathematics