Dynamic Response and Dispersion Analysis of a Damped Heterogeneous Coating over a Homogeneous Elastic Half-Space

Abstract

This study models the dynamic response of a damped heterogeneous coating layer over a homogeneous elastic half-space via the shear horizontal equation of motion. The so-called partial nonhomogeneous has been considered in the coating, where only the density of the material features the inhomogeneity parameter. This unusual consideration, motivated by the viscoelasticity setting, gives rise to the realization of Airy’s differential equation in the coating layer that poses Airy’s functions of the first and the second kinds, respectively. Moreover, the resulting dispersion relation has been utilized and analyzed, assessing the impact of the involved parameters. The study realized that an increase in both the damping coefficient and the inhomogeneity parameter accelerates the dispersion of waves in the media. Additionally, once the case of the doubly coated half-space is analyzed, as an extension of the earlier setup, it is noted that the case of a doubly coated half-space is more responsive to the excitations, which is pretty geared by the composition of different layers. In addition, more modes are noted when more coatings are wrapped over the half-space.

1. Introduction

The study of wave propagation in coated mediums is important to various fields of contemporary concern, including seismology, earthquake engineering, civil and structural engineering, and modern material science, to mention but a few [1]. Certainly, the pressing need for modern industrial processes and engineering applications has triggered much need for composite structures, which are characterized by various advantages [2]. Thus, many researchers lately skewed their attention to the study of waves in composite structures, which mainly comprise multilayered and coated mediums [3,4,5,6]; read also the work of Chattopadhyay and Raju [7] on the engineering relevance of polyurethane coatings on modern structures, and the submission by Padture et al. [8], which examined the state-of-art of thermal barrier coatings for turbine engines. Moreover, applications of coating processes can be seen in modern structures. See the recent study by Ebrahiminejad et al. [9] that proposed an effective non-destructive evaluation method for assessing thickness loss and mechanical properties of coated mediums, and the work by Zhang et al. [10], who presented a reliable testing technique for underwater coating layers with applications in marine engineering. In addition, Jin et al. [11] recently reviewed the trend of electromagnetic wave absorption coatings, which have vast applications in military gadgets and civil applications; Chirita and D’Apice [12] investigated the Rayleigh wave transmission pattern on an aluminum thermoelastic layer coating a thermoelastic copper half-space under Maxwell–Cattaneo thermoelasticity theory; exact and numerical simulation perspectives were supplied, with an emphasis on aluminum-copper multilayered bodies. In this regard, some of the recent examinations of wave propagation on layered and coated media by scientists include the study of wave dynamics by Rayleigh [13], which examined the wave movement over a semi-infinite elastic media, and Love’s work [14] that analyzed the propagation of waves on a layer overlying an elastic half-space. Additionally, one may read the good book by Stein and Wysession [15], which extensively examined the relevance of surface elastic waves in earthquake engineering and seismology very aptly. What is more, still in connection to Love wave propagation in a layer overlying a half-space, various researchers have explored dissimilar scenarios, mostly analytically, to obtain the respective displacement fields in the media. As an instance, Chattopadhyay [16] modeled the propagation of Love waves in a porous layer overlying a nonhomogeneous half-space, while Kielczynski [17] described the dispersion of Love waves in viscoelastic materials. In addition, Kuznetsov [18] examined how Love waves are transmitted over an elastic mediums via the use of shear horizontally polarized equations of motion; read also the work by Abd-Alla et al. [19], which modeled the dispersion of waves over an initially stressed nonhomogeneous orthotropic magneto-elastic plate over a semi-infinite body. Furthermore, Kundu et al. [20] modeled the dynamics of inhomogeneous elastic waves on a half-space when excited by a point source. Kumar et al. [21] studied the impact of material nonhomogeneity on the dispersion of Love-type waves on an isotropic layered body, while Venkatesan and Alam [22] deployed the famous Green’s function method to examine the effect of the impulsive point source on the propagation of Love-type waves in a multilayered poro-elastic medium with material inhomogeneity. See [23,24] for more on the significance of fractional-order on the propagation of nonlinear waves in dissimilar fluid-related mediums.

In particular, damping forces play a significant part in many engineering applications, including controlling progressions and processes. In fact, various physical processes require dampers to control the overall dynamics, as noted by Mohebbi et al. [25]. In addition, damping mechanisms are essential to modeling seismic structures; in this regard, the analysis of wave dispersion amidst the damping term remains paramount at the present time; refer to the good works of Kumari et al. [26] and Saha et al. [27] for some related studies about the damping characteristics in the propagation of Love waves on orthotropic viscoelastic material and initially stressed material, respectively. Certainly, based on the aforementioned literature about the dynamics of Love-type waves with material nonlinearity in the media, the nonhomogeneity parameter appears in both the rigidity and the density of the governing material; this sort of nonhomogeneity is what we refer to in this study as the “full nonhomogeneity”; see [28,29,30,31] for various forms of such inhomogeneities. However, motivated by the viscoelasticity property [32], which acts like material nonhomogeneity but is only present in the rigidity of the material (Lame’s constants), without its presence in the density $\rho$, a “partial nonhomogeneity” born out of curiosity to assess its significance in the propagation of Love-type waves on a coated half-space, is considered in the present study. Moreover, Ahmad and Zaman [33] proposed a semi-analytical technique to study the propagation of waves on a finite elastic plate when the Young’s modulus of the material is a space-dependent variable—a case of partial nonhomogeneity; refer to a somewhat similar consideration of a vibrating elastic beam resting on so-called “partial” elastic foundations [34]. More precisely, this study models and further examines the dynamic response of a damped heterogeneous coating layer over a homogeneous elastic half-space. Perfect interfacial conditions are assumed to hold between the upper coating layer and the half-space, while a traction-free condition is assumed on the upper face of the coating, in addition to the imposition of a boundedness condition in the half-space for deeper thickness. As highlighted above, the inhomogeneity in the coating layer is considered only in the density, which is taken to be space-dependent. In addition, the harmonic wave solution method will be deployed for the acquisition of the exact analytical expressions in the respective layers of the structure, which thereafter reveals the overall dispersion relation. Moreover, the effect of the damping force and the so-called partial inhomogeneity will be numerically studied in the end by considering some physical data of aluminum and copper materials; besides, the numerical analysis of the governing model will graphically be portrayed using the adopted physical data only, with no experimental analysis. Moreover, the recapitulation of the present study goes by the unusual discovery and solution of Airy’s differential equation in the coating layer, which presents Airy’s functions of the first and the second kinds as its solutions. What is more, the resulting dispersion relation equally portrays the presence of Airy’s functions; this is the first of its kind to our knowledge. Numerical simulation of the formulated model, through the choice of chosen physical data, shows that an increase in both the damping coefficient and the inhomogeneity parameter accelerates the dispersion of waves in the media. In this regard, one typically sees the action of the damping force in the present study as an over-damping phenomenon. Furthermore, as an extension of the earlier setup, the case of the doubly coated half-space has been noted to be more responsive to the presence of partial material inhomogeneity and the damping force. Besides, this study has various potentialities in contemporary fields of science and engineering applications, including its application to seismic safety analysis and coating optimization, to mention a few. Lastly, we arrange the presentation of the study as follows: Section 2 gives the governing equations. Section 3 presents the model formulation. Section 4 derives the exact solution. Section 5 determines the resulting dispersion relation, and Section 6 gives a numerical discussion of the results. Section 7 gives the extension of the previous model to a doubly coated structure, while Section 8 gives certain concluding notes.

2. Governing Equations

The generalized equation of elasticity amidst the attendance of body ${\overrightarrow{F}}_i$ and damping $\gamma {\displaystyle \frac{\partial {\omega }_i}{\partial t}}$ forces in an isotropic medium is defined as follows [28,29,32]:

$${\displaystyle \frac{\partial {\sigma }_{ij}}{\partial x_j}}+{\overrightarrow{F}}_i=\overline{\rho }{\displaystyle \frac{{\partial }^2{\omega }_i}{\partial t^2}}+\overline{\gamma }{\displaystyle \frac{\partial {\omega }_i}{\partial t}},i=j=1,2,3,$$

(1)

where ${\omega }_i$ and ${\sigma }_{ij}$ are the related displacements and stresses for $i=j=1,2,3,$ $\gamma$ is the damping coefficient, and $\overline{\rho }$ is the density of the medium. Moreover, symmetric stress–displacement relation ${\sigma }_{ij}$ for $i=j=1,2,3,$ in an isotropic medium is further expressed as follows:

$$\begin{array}{cc}\hfill {\sigma }_{11}& =\lambda \left({\displaystyle \frac{\partial {\omega }_2}{\partial x_2}}+{\displaystyle \frac{\partial {\omega }_3}{\partial x_3}}\right)+(\lambda +2\mu ){\displaystyle \frac{\partial {\omega }_1}{\partial x_1}},{\sigma }_{12}=\mu \left({\displaystyle \frac{\partial {\omega }_1}{\partial x_2}}+{\displaystyle \frac{\partial {\omega }_2}{\partial x_1}}\right),\hfill \\ \hfill {\sigma }_{22}& =\lambda \left({\displaystyle \frac{\partial {\omega }_1}{\partial x_1}}+{\displaystyle \frac{\partial {\omega }_3}{\partial x_3}}\right)+(\lambda +2\mu ){\displaystyle \frac{\partial {\omega }_2}{\partial x_2}},{\sigma }_{13}=\mu \left({\displaystyle \frac{\partial {\omega }_1}{\partial x_3}}+{\displaystyle \frac{\partial {\omega }_3}{\partial x_1}}\right),\hfill \\ \hfill {\sigma }_{33}& =\lambda \left({\displaystyle \frac{\partial {\omega }_1}{\partial x_1}}+{\displaystyle \frac{\partial {\omega }_2}{\partial x_2}}\right)+(\lambda +2\mu ){\displaystyle \frac{\partial {\omega }_3}{\partial x_3}},{\sigma }_{23}=\mu \left({\displaystyle \frac{\partial {\omega }_2}{\partial x_3}}+{\displaystyle \frac{\partial {\omega }_3}{\partial x_2}}\right),\hfill \end{array}$$

(2)

where $\lambda$ and $\mu$ are the respective rigidities in the isotropic medium, also known as the Lam$\acute{e}$’s constants.

In this regard, a relatively simple, yet very efficient model that models the propagation of waves in elastic media is the simplified version of the plane equation of motion expressed in (1), which is referred to as the horizontally polarized shear (SH) or antiplane equation of motion [35,36,37,38]. Therefore, upon considering the SH equation with $x_1{x}_2$ as the plane of interest, the displacement vector thus takes the following form: $\mathit{\omega }(0,0,{\omega }_3),$ where ${\omega }_3$ is the only surviving displacement component, presiding over the antiplane motion. In light of the above development, and upon using x instead of $x_1,$ y instead of $x_2,$ F instead of $F_3,$ and $\omega (x_{,}y,t)$ instead of ${\omega }_3(x_1,x_2,t)$ for brevity, one obtains the equation of SH motion amidst the attendance of body and damping forces from (1) as follows:

$${\displaystyle \frac{\partial {\sigma }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yz}}{\partial y}}+F=\overline{\rho }{\displaystyle \frac{{\partial }^2\omega }{\partial t^2}}+\overline{\gamma }{\displaystyle \frac{\partial \omega }{\partial t}},$$

(3)

where the shear stresses ${\sigma }_{j3}$ for $j=x,y$ are obtained from (2) as follows

$${\sigma }_{xz}=\mu {\displaystyle \frac{\partial \omega }{\partial x}},{\sigma }_{yz}=\mu {\displaystyle \frac{\partial \omega }{\partial y}}.$$

(4)

3. Statement of the Problem

As depicted in the geometrical representation of a nonhomogeneously coated elastic half-space in Figure 1, the study considers a nonhomogeneous coating layer ($0\le y\le H,-\infty <x<\infty$) of uniform thickness H above a homogeneous half-space ($y\ge H,-\infty <x<\infty$). The y-axis is taken vertically downward, which starts at $y=0.$ Further, the respective displacements in the upper nonhomogeneous coating layer $(0\le y\le H)$ and lower half-space $(y\ge H)$ are respectively denoted by ${\omega }_1(x,y,t)$ and ${\omega }_2(x,y,t).$

3.1. Equation of SH Motions in the Layers

In the upper damped nonhomogeneous coating $(0\le y\le H,-\infty <x<\infty ),$ the equation of SH motion for the displacement ${\omega }_1(x,y,t)$ in the absence of body force is given from (3) and (4) as follows:

$${\displaystyle \frac{\partial {\sigma }_{xz}^1}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yz}^1}{\partial y}}={\overline{\rho }}_1\left(y\right){\displaystyle \frac{{\partial }^2{\omega }_1}{\partial t^2}}+\overline{\gamma }{\displaystyle \frac{\partial {\omega }_1}{\partial t}},$$

(5)

where ${\omega }_1={\omega }_1(x,y,t)$ is the out-of-plane displacement in the upper damped nonhomogeneous layer, ${\sigma }_{xz}^1$ and ${\sigma }_{yz}^1$ are the related shear stresses in the coating layer from (4) as follows:

$${\sigma }_{xz}^1={\mu }_1{\displaystyle \frac{\partial {\omega }_1}{\partial x}},{\sigma }_{yz}^1={\mu }_1{\displaystyle \frac{\partial {\omega }_1}{\partial y}},$$

(6)

where ${\mu }_1,$ is the rigidity, while ${\overline{\rho }}_1\left(y\right)$ is a spatial-dependent density in the coating layer, while $\overline{\gamma }$ is the damping parameter. Additionally, as the coating layer happens to be nonhomogeneous, which the material inhomogeneity is basically characterized by admitting spatial- or temporal-dependent material properties, the rigidity ${\overline{\mu }}_1,$ and the density ${\overline{\rho }}_1.$ In this regard, however, motivated Ahmad and Zaman [33], and subsequently Doyle [34], who examined the vibration trend of beams resting over partial elastic foundations, a partial nonhomogeneity is considered in favor of the involving density ${\overline{\rho }}_1\left(y\right)$ to admit the following form

$${\overline{\rho }}_1\left(y\right)={\rho }_1+ϵy,$$

(7)

which is a linear spatial-dependent inhomogeneity in the density ${\overline{\rho }}_1\left(y\right),$ where ${\rho }_1$ is constant density in the nonhomogeneous coating layer, while $ϵ$ is the inhomogeneity parameter.

In the same way, upon utilizing the equation of SH motion outlined in (3), and the constitutive equations given by the shear stresses in (4), the aiming equation of motion in the homogeneous semi-infinite lower medium $(y\ge H,-\infty <x<\infty )$ is expressed more explicitly as follows:

$${\displaystyle \frac{{\partial }^2{\omega }_2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{\omega }_2}{\partial y^2}}={\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\partial }^2{\omega }_2}{\partial t^2}},$$

(8)

where ${\omega }_2={\omega }_2(x,y,t)$ is the out-of-plane displacement in the lower homogeneous layer, and $c_2$ is the transverse shear speed in the layer expressed as follows:

$$c_2=\sqrt{{\displaystyle \frac{{\mu }_2}{{\rho }_2}}},$$

(9)

where ${\mu }_2$ and ${\rho }_2$ are the rigidity and density, respectively, in the half-space.

3.2. Boundary and Interfacial Conditions

Now, considering the geometrical illustration of the problem, the non-homogeneously coated elastic half-space, the following boundary and interfacial conditions are imposed:

I.

Traction-free coating surface, at $y=0$

$${\displaystyle \frac{\partial {\omega }_1}{\partial y}}=0,\mathrm{at}y=0,-\infty <x<\infty .$$

(10)

II.

Bounded displacement field in the half-space as $y\to \infty$

$${\omega }_2\to 0,\mathrm{as}y\to \infty .$$

(11)

III.

Perfect interfacial conditions, at $y=H$

$${\omega }_1={\omega }_2,\mathrm{at}y=H,-\infty <x<\infty ,$$

(12)

and

$${\mu }_1{\displaystyle \frac{\partial {\omega }_1}{\partial y}}={\mu }_2{\displaystyle \frac{\partial {\omega }_2}{\partial y}},\mathrm{at}y=H,-\infty <x<\infty .$$

(13)

Certainly, the prescribed conditions in the latter equations are necessary for the existence of legitimate displacement fields in the respective layers of the governing structure. Thus, Equation (10) imposes a traction-free or rather a free surface condition on the coating layer. Equation (11) ensures that the displacement field in the homogeneous half-space is bounded as $y\to \infty ;$ while Equations (12) and (13) impose perfect continuity conditions on the interface of the coated structure, imposing perfect equality between the related stresses and displacement fields.

4. Acquisition of Exact Analytical Solution

To acquire the resulting exact analytical solutions in both the nonhomogeneous coating and the half-space, the following harmonic wave solutions are beseeched in the respective layers of the structure [39,40,41,42]

$${\omega }_j(x,y,t)=U_j\left(y\right)e^{i(kx-\xi t)},j=1,2,$$

(14)

where the indices 1 and 2 represent the fields in the coating and half-space, respectively, k is the dimensional wavenumber, $\xi$ is the dimensional frequency, and $U_j\left(y\right)$ for $j=1,2$ are the respective amplitudes in the layers, while i is the imaginary number. Therefore, in what follows, the respective displacement fields in the structure are obtained via the presumed harmonic wave solution in (14).

To begin with, the explicit expression for the related displacement field in the damped nonhomogeneous coating layer is determined here. In this scenario, the equation for SH motion for the damped coating layer $(0\le y\le H,-\infty <x<\infty )$ expressed in (5) and (6) with the density expressed in (7) takes the following expression

$${\displaystyle \frac{{\partial }^2{\omega }_1}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{\omega }_1}{\partial y^2}}=\left({\displaystyle \frac{1}{c_1^2}}+{\displaystyle \frac{ϵ}{{\mu }_1}}y\right){\displaystyle \frac{{\partial }^2{\omega }_1}{\partial t^2}}+{\displaystyle \frac{\overline{\gamma }}{{\mu }_1}}{\displaystyle \frac{\partial {\omega }_1}{\partial t}},$$

(15)

where ${\omega }_1={\omega }_1(x,y,t)$ is the out-of-plane displacement in the damped nonhomogeneous coating, and $c_1$ is the transverse shear speed in the layer expressed as follows

$$c_1=\sqrt{{\displaystyle \frac{{\mu }_1}{{\rho }_1}}},$$

(16)

where ${\mu }_1$ and ${\rho }_1$ are constant rigidity and density, respectively, in the coating layer, while $ϵ$ is the inhomogeneity parameter. Furthermore, with the deployment of the harmonic solution in (14) on the present equation expressed in (15), one gets the following ordinary differential equation (ODE)

$${\displaystyle \frac{d^2{U}_1}{d{y}^2}}-\left(b_1-b_{2}y\right)U_1=0,$$

(17)

where

$$b_1=k^2-{\displaystyle \frac{{\xi }^2}{c_1^2}}-i{\displaystyle \frac{\xi \overline{\gamma }}{{\mu }_1}},\mathrm{and}{b}_2={\displaystyle \frac{ϵ{\xi }^2}{{\mu }_1}}.$$

(18)

Furthermore, (17) is a well known ODE with a non-constant coefficient that is termed as the Airy’s equation. Accordingly, Equation (17) admits the following solution:

$$U_1\left(y\right)=A_1\mathrm{Ai}\left({\displaystyle \frac{b_1+b_{2}y}{{(-b_2)}^{2/3}}}\right)+A_2\mathrm{Bi}\left({\displaystyle \frac{b_1+b_{2}y}{{(-b_2)}^{2/3}}}\right),$$

(19)

where $A_1$ and $A_2$ are constants that will be determined later from the prescribed boundary and interfacial data; the functions $\mathrm{Ai}(.)$ and $\mathrm{Bi}(.)$ are the Airy’s functions of the first and second kinds, sequentially.

In the same fashion, we proceed to determine the overall displacement field in the homogeneous half-space. Consequently, upon deploying the presumed harmonic wave solution in (14) in the SH equation for the homogeneous half-space $(y\ge H,-\infty <x<\infty )$ expressed in (8), the following reduced ODE is thus obtained

$${\displaystyle \frac{d^2{U}_2}{d{y}^2}}-\left(k^2-{\displaystyle \frac{{\xi }^2}{c_2^2}}\right)U_2=0,$$

(20)

Further, Equation (20) admits the following exact analytical solution:

$$U_2\left(y\right)=B_1{e}^{-b_{3}y}+B_2{e}^{b_{3}y},$$

(21)

where

$$b_3=\sqrt{k^2-{\displaystyle \frac{{\xi }^2}{c_2^2}}},$$

(22)

where $B_1$ and $B_2$ are constants to be determined. In addition, the boundedness condition outlined in (11) as $y\to \infty ,$ further requires that $B_2$ should vanish. Thus, the solution acquired in (21) reduces to the following:

$$U_2\left(y\right)=B_1{e}^{-b_{3}y},H\le y<\infty ,$$

(23)

characterizing a valid displacement solution in the homogeneous half-space.

5. Resultant Dispersion Relation

This section determines the consequential dispersion relation of the governing two-layered structure that is made up of a damped nonhomogeneous coating layer underlying a homogeneous half-space. Moreover, the partial material inhomogeneity under consideration in the coating layer will explicitly be examined concerning the existence of a valid dispersion relation. In addition, some interesting special cases will be deduced. In the same vein, the section makes use of the following assumption: $\overline{\gamma }=i\gamma ,$ (i is the imaginary number) for a handy analysis of the damping effect on the wave propagation.

Consequently, with the help of the given boundary and interfacial conditions, the following dimensionless dispersion matrix is obtained for the governing coated structure.

$$A_1=\left(\begin{array}{ccc}{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)& {\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)& 0\\ \mathrm{Ai}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& \mathrm{Bi}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& -e^{-{\alpha }_4}\\ {\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& {\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& -{\alpha }_5{e}^{-{\alpha }_4}\end{array}\right),$$

(24)

where ${\alpha }_j$ for $j=1,2,3,4$ and 5 are dimensionless quantities that are explicitly expressed as follows:

$$\begin{array}{cc}\hfill {\alpha }_1& =K^2-{\mathrm{\Omega }}^2+\delta ,{\alpha }_2={\alpha }_1+\tau {\mathrm{\Omega }}^2,{\alpha }_3={(-\tau {\mathrm{\Omega }}^2)}^{2/3},\hfill \\ \hfill & {\alpha }_4=\sqrt{K^2-{\chi }^2{\mathrm{\Omega }}^2},{\alpha }_5=\mu {\displaystyle \frac{{\alpha }_4}{\sqrt{{\alpha }_3}}},\hfill \end{array}$$

(25)

where

$$K=kH,\mathrm{\Omega }={\displaystyle \frac{\xi H}{c_1}},\mu ={\displaystyle \frac{{\mu }_2}{{\mu }_1}},\rho ={\displaystyle \frac{{\rho }_2}{{\rho }_1}},\chi ={\displaystyle \frac{c_1}{c_2}},$$

(26)

with K standing for the dimensionless wavenumber, $\mathrm{\Omega }$ denoting the dimensionless frequency, and $\mu$ dimensionless Lame’s constant; together with the following

$$\tau ={\displaystyle \frac{ϵH}{{\rho }_1}},\delta ={\displaystyle \frac{\gamma \xi H^2}{{\mu }_1}},$$

(27)

where $\tau$ and $\delta$ are scaled material inhomogeneity and damping coefficient, sequentially.

Therefore, the resultant dispersion relation is thus obtained, having set the determinant of the dispersion matrix in Equation (24) to zero as follows

$$\begin{array}{cc}\hfill & {\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)-{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)+\hfill \\ \hfill & {\alpha }_5\left({\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)-{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)\right)=0.\hfill \end{array}$$

(28)

Notably, with the acquisition of the above resultant dispersion relation, several examinations can be carried out, including obtaining the equations for cut-off frequency and the static equation for the wavenumber. In addition, some sorts of special cases for the latter dispersion relation can also be obtained when either the scaled damping coefficient $\delta$ or the inhomogeneity parameter $\epsilon$ or both vanish. However, when the inhomogeneity parameter vanishes, the obtained Airy’s function solutions will no longer be visible in the upper layer; rather, exponential function solutions or hyperbolic function solutions, since the reduced differential equation will be a normal constant-coefficients second-order differential equation. In this regard, several special dispersion relations can be realized, including that of a coated half-space with no inhomogeneity and that of a single-layer coating with either fixed or free faces, among others.

6. Numerical Consideration and Analysis

The current section of the manuscript numerically examines the acquired exact derived dispersion relation in Equation (28) and graphically assesses the significance of the partial inhomogeneity $\tau$ in the coated medium. In addition, the impact of the damping coefficient $\delta$ will be securitized on the dispersion of waves in inhomogeneous composite medium. Moreover, in simulating the acquired dispersion relation, stiff materials are considered to form the layers of the coated half-space, with a stiffer material being the coating. In particular, only the numerical analysis approach is adopted only in this study, and it is expected to provide the much-needed insights into the role being played by the involving excitations—without the need for experimental analysis. Thus, the physical data of the copper and aluminum materials are considered; see

Table 1. Physical data of the coating elastic substrate.

Layer	Material	${\mathit{\mu }}_{\mathit{j}}$ ($\times {10}^{10}$ N/${\mathbf{m}}^2$)	${\mathit{\rho }}_{\mathit{j}}$ $(\times {10}^3$ kg/${\mathbf{m}}^3$)
Coating	Copper	$3.860$	$8.954$
Half-space	Aluminum	$2.643$	$2.700$

. Certainly, the physical date a copper material was once utilized in Abo-Dahab et al. [39] to simulate the propagation dynamics of a magneto-elastic layer lying over a diffusive thermoelastic half-space, while Anya et al. [40] computationally simulated the reflection of SV-waves at a free surface under the action of certain external excitation using the same physical data of aluminum.

Accordingly, Figure 2 and Figure 3 examine the relation between the dimensionless wavenumber K and dimensionless frequency $\mathrm{\Omega }$, while Figure 4 and Figure 5 examine the relation between the dimensionless wavenumber K and dimensionless phase speed. To begin with, we analyze Figure 2, where harmonic curves are seen to comply with low-frequency propagation, with the fundamental model propagating close to $\mathrm{\Omega }=0$. Besides, it is a well-known fact in the literature that propagation under the assumption of long-wave low-frequency is very expensive; one may read [37,38] among others for the flavor of such propagations. Thus, one can easily observe that only the first harmonic curve begins in the low-frequency range and transcends to the high-frequency range, while other harmonic curves propagate in the high-frequency spectrum.

What is more, Figure 3 examines the relation between the dimensionless wavenumber K and dimensionless frequency $\mathrm{\Omega }$, with the variation of the scaled damping coefficient $\delta$ in Figure 3a, and that of material inhomogeneity $\tau$ in Figure 3b. In fact, these figures examine the significance of these parameters in the characterization of the first harmonic curve. Thus, from Figure 3a, the damping coefficient has been noted to decrease the dependency of K on $\mathrm{\Omega },$ with the harmonic curves seen moving away from the low-frequency range to the high-frequency range while increasing in $\delta .$ In addition, Figure 3b specifically examines the state of the affairs of variation of $\tau ,$ the dimensionless inhomogeneity parameter on the dependency of K over $\mathrm{\Omega }.$ Notably, an increase in the material inhomogeneity parameter $\tau$ is observed to decrease the harmonic curve, while at the same time maintaining the origin with only slide movement for higher values of $\tau .$

Furthermore, Figure 4 captures the dependency of the wavenumber K on the phase speed, with various harmonic curves observed while fixing both the inhomogeneity parameter and the damping coefficient. In addition, Figure 5 graphically portrays the significance of varying the dimensionless damping coefficient $\delta ,$ and the dimensionless inhomogeneity parameter $\tau ,$ in Figure 5a and Figure 5b, respectively. Remarkably, it is noted from Figure 5a that an increase in $\delta$ increases the harmonic curves in the coated medium, The same trend has equally been noted from Figure 5b while varying $\tau ,$ the inhomogeneity parameter, such that an increase in $\tau$ increases the dispersion in the media.

7. Extension to Doubly Coated Half-Space

This section attempts to extend the previous study of a single-layered coated half-space by wrapping an additional coating layer around the single coating to have a half-space coated with two coating layers; one may just add an additional coating on top of the existing coating layer. This will, however, give a sense of the dynamic response of the propagation of waves in nonhomogeneous multilayered composite structures. No wonder, this study will open up new directions for the design and analysis of contemporary structures emerging in the diverse fields of science and technological areas.

7.1. Model Formulation

We consider a doubly coated elastic half-space. The two coating layers are assumed to be damped and inhomogeneous, while the half-space is homogeneous and undamped. In addition, all the layers are considered to be made of isotropic materials. Thus, in the damped nonhomogeneous coating layers $(0\le y\le H_1,-\infty <x<\infty ),$ and $(H_1\le y\le H_2,-\infty <x<\infty )$, the equations of SH motion for displacement ${\omega }_1^{\left(p\right)}(x,y,t)$ for $p=1,2$ in the absence of body force are given from (3) and (4) as follows:

$${\displaystyle \frac{\partial {\sigma }_{xz}^{\left(p\right)}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yz}^{\left(p\right)}}{\partial y}}={\overline{\rho }}_1^{\left(p\right)}\left(y\right){\displaystyle \frac{{\partial }^2{\omega }_1^{\left(p\right)}}{\partial t^2}}+\overline{\gamma }{\displaystyle \frac{\partial {\omega }_1^{\left(p\right)}}{\partial t}},p=1,2,$$

(29)

where ${\omega }_1^{\left(p\right)}={\omega }_1^{\left(p\right)}(x,y,t)$ for $p=1,2,$ are the out-of-plane displacements in the upper and lower damped nonhomogeneous layers, while ${\sigma }_{xz}^{\left(p\right)}$ and ${\sigma }_{yz}^{\left(p\right)}$ are the related shear stresses in the coating layers from (4), expressed as follows:

$${\sigma }_{xz}^{\left(p\right)}={\mu }^{\left(p\right)}{\displaystyle \frac{\partial {\omega }_1^{\left(p\right)}}{\partial x}},{\sigma }_{yz}^{\left(p\right)}={\mu }^{\left(p\right)}{\displaystyle \frac{\partial {\omega }_1^{\left(p\right)}}{\partial y}},p=1,2,$$

(30)

where ${\mu }_1^{\left(p\right)},$ for $p=1,2,$ are the rigidities, while ${\overline{\rho }}_1^{\left(p\right)}\left(y\right)$ for $p=1,2$ are the spatial-dependent densities in the coating layers defined as follows:

$${\overline{\rho }}_1^{\left(p\right)}\left(y\right)={\rho }_1^{\left(p\right)}+ϵy,p=1,2,$$

(31)

which are linear spatial-dependent densities, where ${\rho }_1^{\left(p\right)}$ for $p=1,2$ are constant densities for the nonhomogeneous coatings, while $ϵ$ is the inhomogeneity parameter. In addition, $\overline{\gamma }$ is the damping coefficient.

In the same way, the equation of motion in the homogeneous semi-infinite lower medium $(y\ge H_2,-\infty <x<\infty )$ is expressed more explicitly as follows:

$${\displaystyle \frac{{\partial }^2{\omega }_2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{\omega }_2}{\partial y^2}}={\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\partial }^2{\omega }_2}{\partial t^2}},$$

(32)

where ${\omega }_2={\omega }_2(x,y,t)$ is the out-of-plane displacement in the homogeneous half-space, and $c_2$ is the transverse shear speed in the layer defined as $c_2=\sqrt{{\displaystyle \frac{{\mu }_2}{{\rho }_2}}},$ where ${\mu }_2$ and ${\rho }_2$ are the rigidity, and density in the layer, respectively.

In addition, considering the geometry of the damped doubly coated nonhomogeneous elastic half-space, the following boundary and interfacial conditions are endowed

I.

Traction-free coating surface (imposed on the upper coating layer ${\omega }_1^{\left(1\right)}$) at $y=0,$ when

$${\displaystyle \frac{\partial {\omega }_1^{\left(1\right)}}{\partial y}}=0,\mathrm{at}y=0,-\infty <x<\infty .$$

(33)

II.

Bounded displacement field in the half-space as $y\to \infty ,$ that is,

$${\omega }_2\to 0,\mathrm{as}y\to \infty .$$

(34)

III.

Perfect interfacial conditions (between the upper and the lower coating layers), at $y=H_1$

$$\begin{array}{cc}\hfill {\omega }_1^{\left(1\right)}& ={\omega }_1^{\left(2\right)},\mathrm{at}y=H_1,-\infty <x<\infty ,\hfill \\ \hfill {\mu }_1^{\left(1\right)}{\displaystyle \frac{\partial {\omega }_1^{\left(1\right)}}{\partial y}}& ={\mu }_1^{\left(2\right)}{\displaystyle \frac{\partial {\omega }_1^{\left(2\right)}}{\partial y}},\mathrm{at}y=H_1,-\infty <x<\infty .\hfill \end{array}$$

(35)

IV.

Perfect interfacial conditions (between the lower coating and the half-space), at $y=H_2$

$$\begin{array}{cc}\hfill {\omega }_1^{\left(2\right)}& ={\omega }_2,\mathrm{at}y=H_2,-\infty <x<\infty ,\hfill \\ \hfill {\mu }_1^{\left(2\right)}{\displaystyle \frac{\partial {\omega }_1^{\left(2\right)}}{\partial y}}& ={\mu }_2{\displaystyle \frac{\partial {\omega }_2}{\partial y}},\mathrm{at}y=H_2,-\infty <x<\infty .\hfill \end{array}$$

(36)

What is more, Equation (33) prescribed a traction-free on the upper coating layer, while the condition in (34) ensures that the displacement field ${\omega }_2$ in the homogeneous half-space is bounded for larger $y.$ In addition, the conditions in (35) and (36) ensure perfect continuity in between the two interfaces of the doubly coated structure.

7.2. Acquisition of Exact Analytical Solution

To acquire the resulting exact analytical solutions in both the two nonhomogeneous coatings ($p=1,2$) and the half-space, the following harmonic wave solutions are beseeched in the respective layers of the structure [39,40,41,42]

$$\begin{array}{cc}\hfill {\omega }_1^{\left(p\right)}(x,y,t)& =U_1^{\left(p\right)}\left(y\right)e^{i(kx-\xi t)},p=1,2,\hfill \\ \hfill {\omega }_2(x,y,t)& =U_2\left(y\right)e^{i(kx-\xi t)},\hfill \end{array}$$

(37)

with ${\omega }_1^{\left(p\right)}(x,y,t)$ for $p=1,2,$ denoting the fields in the two coatings, while ${\omega }_2(x,y,t)$ the field in the half-space. In addition, k is the dimensional wavenumber, $\xi$ is the dimensional frequency, with i as an imaginary number, while $U_1^{\left(p\right)}\left(y\right)$ for $p=1,2,$ and $U_2\left(y\right)$ are the respective amplitudes in the layers.

Accordingly, the application of the assumed solution in (37) yields the following displacement fields in all the three layers of the structure

$$\begin{array}{cc}\hfill U_1^{\left(1\right)}\left(y\right)& =A_1^{\left(1\right)}\mathrm{Ai}\left({\displaystyle \frac{b_1^{\left(1\right)}+b_2^{\left(1\right)}y}{{(-b_2^{\left(1\right)})}^{2/3}}}\right)+A_2^{\left(1\right)}\mathrm{Bi}\left({\displaystyle \frac{b_1^{\left(1\right)}+b_2^{\left(1\right)}y}{{(-b_2^{\left(1\right)})}^{2/3}}}\right),0\le y<H_1,\hfill \\ \hfill U_1^{\left(2\right)}\left(y\right)& =A_1^{\left(2\right)}\mathrm{Ai}\left({\displaystyle \frac{b_1^{\left(2\right)}+b_2^{\left(2\right)}y}{{(-b_2^{\left(2\right)})}^{2/3}}}\right)+A_2^{\left(2\right)}\mathrm{Bi}\left({\displaystyle \frac{b_1^{\left(2\right)}+b_2^{\left(2\right)}y}{{(-b_2^{\left(2\right)})}^{2/3}}}\right),H_1\le y<H_2,\hfill \\ \hfill U_2\left(y\right)& =B_1{e}^{-b_{3}y}+B_2{e}^{b_{3}y},H_2\le y<\infty ,\hfill \end{array}$$

(38)

where

$$b_1^{\left(p\right)}=k^2-{\displaystyle \frac{{\xi }^2}{{\left(c_1^{\left(p\right)}\right)}^2}}-i{\displaystyle \frac{\xi \overline{\gamma }}{{\mu }_1^{\left(p\right)}}},\mathrm{and}{b}_2\left(p\right)={\displaystyle \frac{ϵ{\xi }^2}{{\mu }_1^{\left(p\right)}}},p=1,2,$$

(39)

while $b_3$ was earlier expressed in (22) as $b_3=\sqrt{k^2-{\displaystyle \frac{{\xi }^2}{c_2^2}}},$ where $A_1^{\left(p\right)}$ and $A_2^{\left(p\right)}$ for $p=1,2,$ and $B_1$ and $B_2$ are constants to be determined later from the prescribed boundary and interfacial data, while the functions $\mathrm{Ai}(.)$ and $\mathrm{Bi}(.)$ in the first two solutions are the Airy’s functions of the first and second kinds, sequentially.

7.3. Resultant Dispersion Relation

Certainly, with the help of the imposed boundary and interfacial conditions in the above subsection, the following dimensionless dispersion matrix is accordingly acquired for examining the doubly coated half-space

$$A_2=\left(\begin{array}{ccccc}{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)& {\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)& 0& 0& 0\\ \mathrm{Ai}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& \mathrm{Bi}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)& -\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right)& -\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right)& 0\\ {\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)& {\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)& -{\alpha }_{55}{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)& -{\alpha }_{55}{\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)& 0\\ 0& 0& \mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)& \mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)& -e^{-{\alpha }_{66}}\\ 0& 0& {\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)& {\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)& -{\alpha }_{77}{e}^{-{\alpha }_{66}}\end{array}\right),$$

(40)

where ${\alpha }_j$ for $j=1,2,3,$ and ${\alpha }_{jj}$ for $j=1,2,\dots ,7$ are dimensionless quantities that are explicitly expressed as follows:

$$\begin{array}{cc}\hfill {\alpha }_1& =K^2-{\mathrm{\Omega }}^2+\delta ,{\alpha }_2={\alpha }_1+\tau {\mathrm{\Omega }}^2,{\alpha }_3={(-\tau {\mathrm{\Omega }}^2)}^{2/3},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\alpha }_{11}& =K^2-{\chi }^2{\mathrm{\Omega }}^2+{\displaystyle \frac{\delta }{\mu }},{\alpha }_{22}={\alpha }_{11}+{\displaystyle \frac{\tau }{\mu }}{\mathrm{\Omega }}^2,{\alpha }_{33}={\left(-{\displaystyle \frac{\tau }{\mu }}{\mathrm{\Omega }}^2\right)}^{2/3},{\alpha }_{44}={\alpha }_{11}+{\displaystyle \frac{\tau }{\mu H}}{\mathrm{\Omega }}^2,\hfill \end{array}$$

(42)

$$\begin{array}{c}\hfill {\alpha }_{55}=\mu \sqrt{{\displaystyle \frac{{\alpha }_{33}}{{\alpha }_3}}},{\alpha }_{66}={\displaystyle \frac{1}{H}}\sqrt{K^2-{\mathrm{\Omega }}^2},{\alpha }_{77}={\displaystyle \frac{H}{\mu }}{\displaystyle \frac{{\alpha }_{66}}{\sqrt{{\alpha }_{33}}}},\end{array}$$

(43)

where

$$K=k{H}_1,\mathrm{\Omega }={\displaystyle \frac{\xi H_1}{c_1}},\mu ={\displaystyle \frac{{\mu }_2}{{\mu }_1}},\rho ={\displaystyle \frac{{\rho }_2}{{\rho }_1}},H={\displaystyle \frac{H_1}{H_2}},\chi ={\displaystyle \frac{c_1}{c_2}},$$

(44)

with K equally standing for the dimensionless wavenumber and $\mathrm{\Omega }$ denoting the dimensionless frequency; $\mu ,$ $\rho$ and H are the dimensionless material constants density and thickness ratio, respectively, together with the scaled dimensionless material inhomogeneity ($\tau$) and damping coefficient ($\delta$) as follows:

$$\tau ={\displaystyle \frac{ϵH_1}{{\rho }_1}},\delta ={\displaystyle \frac{\gamma \xi H_1^2}{{\mu }_1}}.$$

(45)

Remarkably, the dispersion matrix above was attained upon considering a realistic assumption of alternating layered structure [43]. That is, the material properties in the upper coating and the half-space layers are assumed to be the same, specifically, ${\mu }_1^{\left(1\right)}={\mu }_2$ and ${\rho }_1^{\left(1\right)}={\rho }_2.$ Additionally, the assumption imposed in the single-coated half-space is equally applied here concerning the complex-valued damping coefficient for computational simplicity.

Consequently, setting the determinant of the resultant dispersion matrix (40) to zero yields the analytically obtained dispersion relation as follows:

$$\begin{array}{cc}\hfill & \left({\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)-{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)\right)f_1+\hfill \\ \hfill & {\alpha }_{55}\left({\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right)-\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_2}{{\alpha }_3}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_3}}\right)\right)f_2=0,\hfill \end{array}$$

(46)

where $f_1$ and $f_2$ are expressed as follows:

$$\begin{array}{cc}\hfill f_1& ={\alpha }_{77}\left(\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)-\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right)\right)-\hfill \\ \hfill & \mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)+{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{11}}{{\alpha }_{33}}}\right),\hfill \\ \hfill f_2& ={\alpha }_{77}\left({\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)\mathrm{Bi}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right)-\mathrm{Ai}\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)\right)+\hfill \\ \hfill & {\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right)-{\mathrm{Ai}}^{\prime }\left({\displaystyle \frac{{\alpha }_{22}}{{\alpha }_{33}}}\right){\mathrm{Bi}}^{\prime }\left({\displaystyle \frac{{\alpha }_{44}}{{\alpha }_{33}}}\right).\hfill \end{array}$$

(47)

Moreover, the dispersion relation determined in (46) is the overall analytically obtained dispersion relation for the doubly coated half-space under the effect of material inhomogeneity in the two coating layers and the damping force. In addition, one obtains the results of the previous section upon disregarding the second coating layer. Besides, various special cases can be deduced from this exact expression.

7.4. Numerical Consideration and Analysis

Having considered a multilayered structure with alternating layers in the derivation of the resultant dispersion relation, the numerical simulation in this section thus requires us to consider the upper coating layer to be of copper material, the lower coating layer to be of aluminum, while the half-space is assumed to be of copper material, as in the case of alternating layers [43]. In this regard, the physical material data reported in Table 1 will be fully utilized for the simulation purpose. Accordingly, Figure 6 and Figure 7 examine the relation between the dimensionless wavenumber K and dimensionless frequency $\mathrm{\Omega },$ while Figure 8 and Figure 9 examine the relation between the dimensionless wavenumber K and dimensionless phase speed. Therefore, Figure 6 portrays the harmonic curves via the dependency plot of the dimensionless wavenumber K versus the dimensionless frequency $\mathrm{\Omega },$ where multiple modes are noted to be more than that of the single coated half-space (of course, the same frequency range is considered in both the singly and doubly coated half-spaces). Thus, one may say that the doubly coated half-space responds more than the singly coated half-space. Further, Figure 7 demonstrates the influence of the variation of the scaled damping coefficient $\delta$ in Figure 7a, and that of the inhomogeneity parameter $\tau$ in Figure 7b. Certainly, one observes that an increase in both the damping coefficient and the material inhomogeneity decreases the propagation of waves in the media; however, the two parameters have different effects on the dynamic response of the multi-coated structure. Moreover, to supplement this result, one may read the work of Mubaraki et al. [31] that examined the case of a doubly coated half-space, with the half-space featuring a full exponential nonhomogeneity in the spatial variable. There, the nature of the exponential nonhomogeneity becomes obvious, in addition to the fact that the nonhomogeneity parameter appeared both in Lame’s constant and the density; one may equally read [28,29,30] for the influences of different forms of full nonhomogeneities in both the solid and fluid structures.

In the same fashion, Figure 8 demonstrates the dependency of the wavenumber K on the phase speed via harmonic curves. Notably, the same trend has been observed as in the case of the single-coated half-space; however, many modes are realized in the doubly coated media case, which signifies much sensitivity due to different materials. Therefore, having observed much sensitivity in the doubly coated media, we proceed to examine the dynamic response of the media with the variation of the involving effects in Figure 9. Thus, Figure 9a,b, graphically examine the significance of varying the dimensionless damping coefficient $\delta ,$ and the dimensionless inhomogeneity parameter $\tau ,$ respectively. Notably, it is equally noted from Figure 9a,b that an increase in $\delta ,$ and $\tau$ increases the dispersion of waves in the media. Indeed, the damping mechanism plays a vital role in model structures, including bridges, as an instance, where Qu et al. [44] has it that “damping not only affects the response of the bridge to dynamic loads but also determines the energy dissipation capacity of the bridge when subjected to external shocks”. The same trend has equally been noted from Figure 9b while varying $\tau ,$ the inhomogeneity parameter, such that an increase in $\tau$ increases the dispersion in the media as rightly noted in the single-coated half-space. However, the response by the doubly coated half-space is much noted owing to the realization of more modes. Moreover, having realized the thickness ratio H in the case of the doubly coated half-space, Figure 10 thus captures the dependency of the wavenumber K on the frequency $\mathrm{\Omega }$ in Figure 10a, and the dependency of the wavenumber K on the phase speed in Figure 10b. Notably, it is observed from Figure 10a that an increase in the thickness ratio draws the fundamental mode back to the origin, which thus guarantees low-frequency propagation. What is more, Figure 10b shows the variational effect of H on K versus (vs.) phase speed, where an increase in the ratio stretches the dependency of the two parameters. Besides, these plots were based on setting $\tau =\delta =0.3;$ it is therefore pertinent to state here that for smaller values of these parameters ($\tau$ and $\delta$), it was numerically discovered that the fundamental mode is non-responsive to the thickness ratio variation—only high modes are varied with the variation of $H;$ recall that only the fundamental mode is of interest in the graphical analysis for computational brevity.

8. Conclusions

In conclusion, the present study modeled and further examined the dynamic response of a damped heterogeneous coating layer over a homogeneous elastic half-space. The equation of SH motion has been considered based on its relative straightforwardness in the analysis. Perfect interfacial conditions have been imposed between the coating and the half-space, which were formed by isotropic materials, while a traction-free condition is assumed on the upper face of the coating. Furthermore, motivated by the viscoelasticity setting, where the related rigidity parameter(s) are assumed to be time-dependent, the present study thus made a consideration that the related density in the coating layer is space-dependent, thereby creating a nonhomogeneity in the layer; the so-called partial nonhomogeneous coating. Therefore, the study further deployed the harmonic wave solution method and thereafter acquired Airy’s differential equation in the coating layer, which gave rise to Airy’s functions of the first and the second kinds, respectively. In addition, the utilization of the imposed conditions resulted in the derivation of the resulting dispersion relation, which was subsequently analyzed numerically by considering physical data. In fact, the study realized that both the damping coefficient and the inhomogeneity parameter accelerate the dispersion of waves in the media. In addition, the study further extended the study to the doubly coated half-space cases and further reaffirmed the trend of a singly coated structure. However, the study noted that the case of a doubly coated half-space is more responsive to the excitations, which is pretty geared by the composition of different materials (layers) in the media. Lastly, this study is significant in the contemporary fields of seismic waves, material science, and structural engineering to mention a few, where various materials are composed to realize new materials for industrial needs, including their application to seismic safety analysis and coating optimization.