Document Type : Research Article
Authors
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, 603203, India
Abstract
Keywords
Main Subjects
Anandakrishnan Akshaya, Santosh Kumar ^{*}^{ } , Kulandhaivel Hemalatha
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, 603203, India
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20231212 Revised: 20240524 Accepted: 20240705 

This study delves into the investigation of shear horizontal transverse waves within a structure composed of a layer and a semiinfinite medium. The layer considered in this work is made of functionally graded magnetoelectroelastic material, and the semiinfinite medium is comprised of functionally graded piezoelectric material with a linear gradedness parameter. Additionally, the layer and semiinfinite medium has a mechanical imperfect interface. Furthermore, the solutions for the layer and semiinfinite medium are derived utilizing the WKB approximation technique. This involves initially transforming the partial differential equation into an ordinary differential equation through basic variable separation and the ordinary differential equation is solved by WKB approximation technique. The obtained results lead to dispersion relations presented in determinant form for two cases: electrically open and short. To explore the influences of various parameters on shear horizontal transverse waves, a specific model is considered. This model is composed of a layer made of magnetoelectroelastic material (BaTiO_{3}CoFe_{2}O_{4}) and a semiinfinite medium consisting of piezoelectric material (BaTiO_{3}). Also, the graphs are plotted to visualize the variation in shear horizontal transverse waves in relation to both phase velocity and wave number. 



Keywords: Shear horizontal transverse wave; Functionally graded material; WentzelKramersBrillouin asymptotic approach; Imperfect interface; Gradedness parameter. 


© 2025 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an open access article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
The investigation into the propagation of shear horizontal transverse waves within a structure, comprising magnetoelectroelastic layers and piezoelectric material, constitutes an intriguing research area with diverse realworld applications such as surface acoustic wave devices [1], ultrasound imaging [2], and so on. Shear horizontal transverse waves arise from disturbances within a medium capable of storing and transmitting energy. The origin of these disturbances can be attributed to various mechanisms contingent upon the characteristics of the medium. The analyzed layer, comprising magnetoelectroelastic materials, serves as a smart composite with the ability to convert mechanical energy to electric or magnetic energy, and vice versa. The electromagnetic coupling effect occurs in magnetoelectro elastic materials, involving the interaction between electric and magnetic fields within the material, which gives rise to their piezoelectric and piezomagnetic properties. The piezomagnetic substance undergoes a change in its magnetic properties when subjected to external mechanical stress. It is possible to convert energy between electric and magnetic fields with the use of composite materials that include both piezomagnetic and piezoelectric. This study focuses specifically on piezoelectric materials [3], recognized for their widespread use in everyday items such as touchsensitive lighters, as well as in applications like microphones, sensors, motors, and various other technological devices.
One may produce a spontaneous magnetic moment in a piezomagnetic material by subjecting it to mechanical stress, and one can induce a physical deformation by subjecting it to a magnetic field. Li and Wei [4] studied the impact of short and open circuits as well as piezoelectric and piezomagnetic effects, on the speed of surface waves in a halfinfinite magnetoelectroelastic material. Harmonic wave propagation on magnetoelectroelastic multilayered plates was the subject of an analytical analysis published by Chen et al. [5], who also expanded on their previous work on multilayered plates with nonlocal effects [6]. Akshaya et al. [7] analyzed the transference of SHwave in two different functionally graded halfspaces bonded imperfectly. Liu et al. [8] investigated the Love wave propagation across layered structures, looking at the situation in which a piezomagnetic (PM) layer lies above a piezoelectric halfspace. Akshaya et al. [9] investigated the behavior of transverse waves at an imperfectly corrugated interface within a functionally graded structure. Yang et al. [10] studied the phenomenon of SH waves being scattered by splits in infinite magnetoelectroelastic materials. Li et al. [11] studied the SH wave at the stressfree surface of a functionally graded magnetoelectroelastic material subjected to initial stress. Hemalatha et al. [12] explored SH waves in a prestressed anisotropic magnetoelastic layer interacting with heterogeneous halfspaces.
The functional gradient material is a new kind of material whose characteristics vary with the variation in space. The designed piezoelectric material is crafted to exhibit specific features for a particular application, regularly modified in an appropriate direction with respect to both the structure and the material’s properties. Several new studies [1316] have been carried out on how Love waves move through a layered composite FGPM system. There have also been efforts to draw attention to the effect that material gradients have on the frequency curves of the examined surface waves. The behavior of Love waves as they travel through a semiinfinite, quadratically changing, functionally graded piezoelectric material is discussed by Eskandari and Shodja [17]. Akshaya et al. investigated the reflection and transmission coefficients of SH waves in a functionally graded piezoelectric structure. The behavior of Love waves during propagation in a threelayered composite structure subjected to initial stress was analyzed by Saroj et al. [19]. In Singh et al. [20], Lovetype wave propagation in a corrugated piezoelectric layer over an isotropic elastic halfspace is examined. Majhi et al. [21] established that the Love waves in a multilayer functionally graded piezoelectric system are affected by initial stress. The propagation of shear waves with a horizontal polarisation is examined by Sahu et al. [22] in a composite material with layers. The model under consideration has a functionally graded layer of piezoelectric material interceding between a corrugated layer of piezomagnetic material and a halfspace. Functionally gradient material plates stimulated by plane pressure wavelets were the subject of research by Liu et al. [23] on elastic waves. In a piezoelectric layer with a halfinfinite elastic layer, Liu et al. [24] examined the possibility of surface acoustic waves. Cao et al. [25] conducted a study on the behavior of Lamb waves in a plate made of functionally graded piezoelectricpiezomagnetic material. The material parameters were found to vary continuously along the thickness direction of the plate. Hemalatha et. al. [26] investigated the Rayleigh wave propagation in an elastic substrate and functionally graded piezoelectric layer. Sadab and Kundu [27] examined the characteristics of SHwave propagation in a heterogeneous dry sandy halfspace, which was bonded by a piezoelectric layer and abutting the vacuum.
Hemalatha et al. [28] delved into the propagation of Rayleigh waves in a structure comprising a functionally graded piezoelectric material (FGPM) layer over an elastic substrate. Singh [29,30] discussed the problem of Love waves in a layered medium bounded by an irregular bounded surface and also examined the propagation of SH waves by adopting Rayleigh's method of approximation. Singh et al. [31] studied the horizontally polarised shear wave (SHwave) propagating in an elastic halfspace having a corrugated and loosely connected common surface that has been initially stressed. An imperfect interface between a layer and two solids is studied by Lavrentyev and Rokhlin [32] using ultrasonic spectroscopy. Sahu et al. [33] encountered the problem of reflection and refraction of the incident plane waves at the loosely bonded interface of an isotropic elastic medium and a piezothermoelastic medium. Wang et al. [34] conducted research on the dispersion of an antiplane shear wave by a circular piezoelectric cylinder with an imperfect connection. Singh et al. [35] considered the propagation of torsional surface waves with corrugated as well as loosely bonded boundary surfaces.
The WKB Approximation is a way to get the solution of a timeindependent linear differential equation, it is named after the scientists who developed it, WentzelKramersBrillouin. In quantum physics, a semiclassical calculation usually involves rewriting the wavefunction as an exponential function, semiclassically expanding it, and then assuming that either the amplitude or phase is changing slowly. This method also makes the Froman and Froman [36] link between classical mechanics and quantum mechanics. The Love waves propagation in the functionally graded multilayer piezoelectric device is investigated by Liu and Wang [37] using the WKB approach. Jones [38] discovered the WKB approximation's uses in the computation of radio wave scattering from dense meteor trains. With the help of the WKB method, they were able to get approximate analytic solutions for Love waves. The comprehensive description of elastic wave propagation across layered, inhomogeneous media by Cerveny and Ravindra [39] is based on direct asymptotic expansion. Love wave behavior under initial stress is examined by Qian et al. [40] used the WKB technique in a functionally graded material layered nonpiezoelectric halfspace. Analysis of shear wave propagation in a piezoelectric structure comprised of an imperfectly bonded piezoelectric material layer was studied by Kumar et al. [41] using the WKB approach. In an FGPM layer coupled between a piezomagnetic plate and a prestressed piezoelectric halfspace, Singhal et al. [42] used the WKB approach to explore the propagation of a Lovetype wave.
Functionally graded materials (FGMs) have attracted intense research interest due to their ability to tailor material properties for specific applications, leading to advancements in everything from aircraft structures to medical implants. The widespread utilization of FGM in these applications serves as strong motivation for further exploration and investigation in this field.
This paper describes the propagation of polarized shear horizontal transverse waves in a layered composite structure consisting of a functionally graded magnetoelectroelastic layer imperfectly bonded to a piezoelectric semiinfinite medium, which is yet unaddressed for study as far as the known available literature. Previous studies primarily focused on investigating the behavior of horizontally polarized shear (SH) waves propagating through a FGPM layer that is imperfectly bonded to the functionally graded porous piezoelectric (FGPPM) halfspace, employing the WKB method. The material coefficients of both the FGPM layer and the FGPPM halfspace were assumed to exhibit functional gradedness, following a linear function of depth. In this work, the piezoelectric halfspace and the magnetoelectroelastic layer’s functionally graded material coefficients are both taken into consideration by the linear function of depth. Additionally, the solutions for layer and semiinfinite medium are found by reducing the partial differential equation to the ordinary differential equation using the fundamental variableseparable approach. An analytical method based on the WKB asymptotic method was used to solve an ordinary differential equation and to obtain a solution for both the layer and the semiinfinite medium. A dispersion relation is achieved for electrically open and electrically short scenarios in the form of a determinant. A specific model made up of a (BaTiO_{3}CoFe_{2}O_{4}) magnetoelectroelastic material layer and a (BaTiO_{3}) piezoelectric semiinfinite medium has been used to determine the effects of the various parameters. Findings are represented using graphs with the help of Mathematica 7.
Consider the magnetoelectroelastic structured layer depicted in Fig 1. It comprises a functionally graded piezoelectric semiinfinite medium and a functionally graded magnetoelectroelastic layer with a finite thickness of h and an imperfect bonding interface. The system’s Cartesian coordinates are taken so that the axis is vertically downward and the axis is in the direction of shear horizontal transverse wave propagation along the interface between the functionally graded magnetoelectroelastic layer and functionally graded piezoelectric semiinfinite medium.
Fig. 1. Geometry of the problem
The magnetoelectroelastic layer's field equations may be represented as

(1) 
here i, j = , mass density is represented by and and indicate the electric, mechanical, and magnetic displacements in the direction, correspondingly, the stress tensor is denoted by . The dot shows differentiation in time, the comma shows differentiation in space, and the repeating index in the subscript shows summation.
The following is the relation between displacement and strain components is obtained from [41]:

(2) 
Quasistatic approximation of the Maxwell equations yields:

(3) 
here, denotes the electrostatic potential and denotes the magnetic potential.
For a transversely isotropic medium with axis as the symmetric axis and the poling axis of the magnetoelectroelastic material, the constitutive equations can be written in terms of components, which are obtained from [11].

(4) 
which is a function of the elastic constants , the dielectric constants and the magnetic permittivity ( = ). Also , and are the piezoelectric, piezomagnetic, and electromagnetic constants functions of only. i.e., , , , , , .
Shear horizontal transverse wave propagation involves the shearing motion of waves within a material. Assuming the propagation of shear horizontal transverse waves in the direction, they induce displacements in the direction. The corresponding mechanical displacement, scalar electrical ( ), and magnetic potential function ( ) components are given as follows:

(5) 
and independent of coordinate.
The mechanical displacement is represented by , whereas the electric and magnetic potential function is represented by and in the region . Substituting condition (5) and Equations (4) into Equations (1), the equation of motion for shear horizontal transverse wave propagation is obtained as

(6) 

(7) 

(8) 
The constitutive equations for a transversely isotropic magnetoelectroelastic material, with x3axis being the symmetric axis of the material, are given by the following relationships between the stress, electric displacement, and magnetic displacement components, expressed in terms of mechanical and potential displacements:

(9) 

(10) 

(11) 

(12) 
Equations (6), (7), (8), and (9)  (12) give us the field equations in the magnetoelectroelastic layer:

(13) 

(14) 

(15) 
where " " denote the differentiation with respect to .
For propagation of shear horizontal transverse waves in a semiinfinite medium, the solution to the equation of motion is as follows:

(16) 

(17) 

(18) 
The stress, electric and magnetic displacement components are characterized as follows in terms of mechanical and potential displacement:

(19) 
,

(20) 
,

(21) 

(22) 
Equations (16)(18) and (19)(22) give us the field equations for the piezoelectric layer:

(23) 

(24) 

(25) 
where " " denote the differentiation with respect to .
Air's dielectric constant and varies substantially from the dielectric constant of piezoelectric and magnetoelectroelastic materials. As a result, the magnetoelectroelastic layer's upper surface is often exposed to air. Therefore, electric and magnetic potential function and may be considered a vacuum for air in the region .

(26) 
The electric and magnetic displacement components of a vacuum are described as follows

(27) 
Equations (13), (14) and (15) can be solved by assuming the following solution:

(28) 
denoting wave number as and wavelength as , as well as wave propagation phase velocity as and .
By using equation (28), equation (13), (14) and (15) reduces to

(29) 

(30) 

(31) 
The functionally graded magnetoelectroelastic layer has linear function distributions in the direction of axis because functionally gradedness enhances the magnetoelectroelastic material layer's properties. As a consequence, the functionally graded magnetoelectroelastic layer's material attributes are classified as

(32) 

where , , , , , and are the homogenous values of , , , , , and at . The inverse of the length of the gradient material parameter is represented by the .
Substituting equation (32) in equation (29), (30) and (31), we get

(33) 

(34) 

(35) 
Rearranging Equations (33), (34), and (35), and let
, ,
and

(36) 

(37) 

(38) 
This is a secondorder differential equation with variable coefficients. The asymptotic solutions to these equations can be determined using the WKB approach, which is a highfrequency and shortwave technique with a wave number of k>>1. Equation (36) may be approximated asymptotically using the WKB approach by following this strategy. Using the phase integral approach, we may propose the following transformation.

(39) 
Equation (36) can be converted into the following form using equation (39)

(40) 
For which the asymptotic series expansion in terms of the inverse power of k may be found as follows

(41) 
We can create an infinite number of equations by replacing equation (41) with equation (40), comparing the coefficients of k, and then setting them equal to zero.

(42) 
The following are the solutions to the preceding system of equations

(43) 
Equation (43) solutions can be swapped into equation (41).

(44) 
Using equation (44), equation (39) can be written as

(45) 
where , are undetermined constants and
= –
with the aid equation (45), equation (39) yields

(46) 
Using the same asymptotic method and substituting equation (46) for equations (37) and (38), the entire solution of and can be represented as

(47) 

(48) 
where , , and are undetermined constants and = +
Equations (23), (24) and (25) can be solved by assuming the following solution:

(49) 
By using equation (49), equation (23), (24) and (25) reduces to

(50) 

(51) 

(52) 
The functionally graded piezoelectric semiinfinite medium has linear function distributions in the direction of axis because functionally gradedness enhances the piezoelectric material properties. As a consequence, the functionally graded piezoelectric semiinfinite medium material attributes are classified as

(53) 
where , , , and are the homogenous values of , , , and . The inverse of the length of the gradient material parameter is represented by the .
Substituting equation (53) in equation (50), (51) and (52) and letting and = + and we get

(54) 

(55) 

(56) 
The same asymptotic WKB method that is used to solve for a functionally graded magnetoelectroelastic layer can also be used to solve expression (54) for a functionally graded piezoelectric semiinfinite medium.
We're looking for a solution for mechanical displacement and electric potential expression in a semiinfinite medium where mechanical displacement and electric potential diminish as depth increases which means as , . The solution to this problem is thus an approximation, which is

(57) 
where is undetermined constants and
= –
Using the same asymptotic method and substituting equation (57) for equation (55), the entire solution of can be represented as

(58) 
The entire solution of can be represented as

(59) 
where and is undetermined constants and
= +
The solution to equation (26) is obtained by, assuming that the electric and magnetic potential function and diminishes as which results in

(60) 
where and is undetermined constants.
In this problem, both the functionally graded magnetoelectroelastic layer and the piezoelectric semiinfinite medium have a mechanically imperfect interface. Only the upper boundary is considered electrically open and shortcase. At the interface, the electrical displacement and scalar potential function are considered continuous. These conditions are illustrated mathematically below.
1. The mechanical tractionfree condition at the free surface of the functionally graded magnetoelectroelastic layer is given by
i. ⇒ =0 
(61) 
2. The electrical boundary condition at are
a) Electrically open case
i. ii. ⇒ = iii. iv. ⇒ = 
(62) 
b) Electrically short case
i. ii. 
(63) 
3. It is assumed that the interface of functionally graded magnetoelectroelastic layer and functionally graded piezoelectric semiinfinite medium is damaged mechanically. For the imperfect interfacial bonding, the stresses and electric displacements are continuous at
i. ⇒ = [ ii. ⇒[ iii. iv. v. ⇒ = vi. ⇒ = 
(64) 
where K is the mechanical interfacial imperfection bonding parameter which are uniform and nonnegative. Obviously, if K→0, the interface of the layer and semiinfinite medium is perfectly bonded and conducting. Otherwise, it is mechanically imperfect.
Using Equations (46), (47), (48), (57), (58), (59), and (60) into boundary conditions (61) to (64) homogeneous equations for the unknown variables to are obtained.

(65) 

(66) 

(67) 

(68) 

(69) 

(70) 
0 
(71) 

(72) 

(73) 

(74) 

(75) 

(76) 

(77) 
where , , , , , , , , , , , , and are defined in Appendix A.
The dispersion relation of the shear horizontal transverse wave for a functionally graded magnetoelectroelastic layer covering a piezoelectric semiinfinite medium subjected to the boundary condition specified in section 4 is extracted in eleventhorder determinant form by eliminating from the following Equations (65) to (69) and (71) to (77).

(78) 
where defined in Appendix B.
Excluding and from the following Equations (65), (70) to (77) we arrive at the ninthorder determinant form of the dispersion relation for the shear horizontal functionally graded magnetoelectroelastic layer covering a piezoelectric semiinfinite medium subject to the boundary condition discussed in section 4.

(79) 
Here defined in Appendix C.

(80) 
and

(81) 
where and are defined in Appendix D.

(82) 
and

(83) 
where and are defined in Appendix E. The dispersion relations (equations (82) and (83)) are in good agreement with the results obtained by Li et al. [11].
In this investigation, we consider two cases, i.e. electrically open case and electrically short case.
Fig. 2. For varying values of h, the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically open case
A function of wave number is used to describe the influence on the phase velocity of the initial stress. For this study, we utilize the piezoelectric material BaTiO_{3, }as referenced in the work of Cao et al. [25], and the magnetoelectroelastic material BaTiO_{3}CoFe_{2}O_{4}, as studied by Li et al. [11]. The following Table 1 summarizes all of the material constants used in this article. F/m and are the dielectric constant and magnetic permittivity of vacuum, respectively.
The nondimensional velocities of shear horizontal transverse waves in the functionally graded magnetoelectroelastic material layer and piezoelectric semiinfinite medium are represented as and , respectively, and are expressed as follows:
and
Graphical representations have been depicted to illustrate the influence of several key parameters, including the functional gradient parameters and associated with the layer and semiinfinite medium and the mechanical imperfection parameter ( ), on the dimensionless phase velocity ( ) of the propagating SHwave relative to the dimensionless wave number (kh), for both electrically open and electrically short cases, as depicted in the corresponding figures.
In this section, we visually represent the effects of the inhomogeneity parameters ( , ), depth (h), and mechanical interfacial imperfection bonding parameter ( ) on phase velocity for the electrically open situation.
Fig. 3. For varying values of K, the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically open case
The electrically open case is pertinent only at the layervacuum interface. The electrically open situation refers to a specific boundary condition or scenario in which electromagnetic waves or electrical signals are not constrained or confined within a closed system or structure. In this case, there are no impedance mismatches or reflections of the waves at the boundaries, enabling the free transmission of electromagnetic energy. The variations in wave number and phase velocity are depicted in Figures 2 to 5. The material used in this study consists of a magnetoelectroelastic material BaTiO_{3}CoFe_{2}O_{4} and a piezoelectric material BaTiO_{3}. The wave number and phase velocity play significant roles in influencing the propagation of shear horizontal transverse waves in the considered structure. Figure 2 depicts the variation of phase velocity versus wave number for different depths, h (=1000, 1500, and 2000). It has been observed that, as the depth increases, the phase velocity increases.
Table 1. Material constants for BaTiO_{3}CoFe_{2}O_{4} [26] and BaTiO_{3} [11]
Materials 
Elastic constant 
Piezoelectric constant 
Piezomagnetic constant 
Magnetic Constant 
Dielectric constant 
Magnetoelectric constant 
Mass density 
BaTiO_{3} 
4.8 
0.08 
238 
−258 
0.19 
0.005 
7.50 
BaTiO_{3} 
4.4 
11.4 
− 
5 
9.86 
− 
5.7 
This observed behavior carries significant physical implications, suggesting that an increase in the depth of the material or structure is associated with a higher speed of wave propagation.
Fig. 4. For varying values of , the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically open case
Such a phenomenon might be linked to changes in material properties, potentially indicating that thinner layers exhibit enhanced stiffness or density. Figure 3 depicts the impact of different interfacial imperfection bonding parameters (= 0.1, 1, 10) GPa on the relationship between phase velocity and wave number. The mechanically interfacial imperfection bonding parameter plays a crucial role in wave transmission, reflection, and scattering phenomena at the interface, influencing various propagation characteristics, including wave amplitude, phase, and dispersion properties at the material or layer interface. The graph clearly demonstrates that the phase velocity increases proportionally with an increase in .
Fig. 5. For varying values of , the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically open case
Figures 4 and 5 provide insights into the impact of the inhomogeneity parameters, represented by parameters for layer and for semiinfinite medium, on the relationship between phase velocity and wave number. The curves in Figure 4 show that raising the values of inhomogeneity parameter increases the phase velocity of shear horizontal transverse type wave. In Figure 5, increasing the value of inhomogeneity parameter decreases the phase velocity of a shear horizontal transverse type wave. By manipulating the inhomogeneity parameter, researchers can analyze and model the impact of material variations on wave propagation, facilitating a more comprehensive understanding of wave behavior in inhomogeneous media.
This knowledge is essential for customizing materials and comprehending how waves behave, especially in fields like materials engineering and structural design
Similar to the electrically open case, this study also examined a configuration involving the magnetoelectroelastic material BaTiO_{3}CoFe_{2}O_{4} and the piezoelectric material BaTiO_{3 }for the electrically short case. This section investigates how the depth (h), mechanical interfacial imperfection bonding parameter ( ), and inhomogeneity parameters ( and ) impact the phase velocity in the electrically short scenario. In an electrically short case, the system or structure being studied is characterized by its small physical size compared to the wavelength of the electromagnetic waves or electrical signals with which it interacts.
This condition has implications for understanding the behavior of electrical signals and making appropriate considerations for impedance matching, signal transmission, and electromagnetic wave propagation. Figures 6 to 9 provide insights into how these parameters influence the characteristics of wave propagation.
Figure 6 illustrates the observed variations for different values of h (= 1000, 1500, and 2000), portraying the relationship between phase velocity and wave number. The graph unambiguously illustrates that the phase velocity shows a proportional increase with an increment indepth h.
Subsequently, Figure 7 illustrates the influence of various values of on the relationship between phase velocity and wave number. The mechanical interfacial imperfection bonding parameter plays a critical role in wave transmission, reflection, and scattering phenomena at the interface, impacting several propagation characteristics such as wave amplitude, phase, and dispersion properties at the material or layer interface. The graph demonstrates that the phase velocity increases with an increase in .
Fig. 6. For varying values of h, the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically short case
Fig. 7. For varying values of K, the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically short case
Table 2. Summary of the effect of parameters on phase velocity
Parameters 
Effect on phase velocity 
Effect on phase velocity 
Depth (h) 
Direct proportion 
Direct proportion 
Mechanically interfacial 
Direct proportion 
Direct proportion 
Material gradient parameter 
Direct proportion 
Direct proportion 
Material gradient parameter 
Inverse proportion 
Inverse proportion 
Figures 8 and 9 show the phase velocity versus wave number for various values of the material gradient parameters for layer (= 0.1, 0.5, 0.9) and for semiinfinite medium (= 0.1, 0.5, 0.9). As seen in the open case, the phase velocity increases as the value of the material gradient parameter increases and the phase velocity decreases as the value of the inhomogeneity parameter increases. Grasping this connection is vital for customizing materials with precise gradient profiles and enhancing their effectiveness in fields such as materials engineering and structural design. The effect of these parameters on phase velocity is summarised in Table 2.
Fig. 8. For varying values of , the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically short case
Fig. 9. For varying values of , the relationship between the nondimensional phase velocity and the nondimensional wave number for an electrically short case
This study has offered valuable insights into the propagation behavior of shear horizontal transverse waves in a functionally graded magnetoelectroelastic layer and a piezoelectric semiinfinite medium. By considering various parameters such as depth h, interfacial imperfection bonding parameter , and inhomogeneity parameters and , we have gained a comprehensive understanding of the factors influencing the phase velocity of shear horizontal waves in both electrically open and short scenarios.
One of the authors gratefully acknowledges SRMIST, Kattankulathur, India for facilitating with best research facility and providing a research fellowship to carry out our research.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
The author declares that there is no conflict of interest regarding the publication of this article.
Appendix A
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Appendix B
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Appendix C
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Appendix D
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Appendix E
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