Vibrations analysis of propagation of SH-type wave influenced by a point source in a porous piezoelectric layered structure by Green’s function approach

Abstract

The principal objective of this paper is to investigate the propagation behaviour of SH-type waves incurred due to a point source in a porous piezoelectric layer resting over a heterogeneous viscoelastic substrate. The Green’s function technique is used to determine the frequency equation which ultimately reduces to the classical Love wave equation as a special case of the present problem. Further, expressions for the particle displacement in the layer and substrate have been obtained. Some numerical computations and comparative analysis have been performed to demonstrate the behavioural characteristics of SH-type wave propagation under the influence of some of the critical factors such as heterogeneity parameter associated with rigidity, internal friction and, density. Other affecting factors include dissipation factor, attenuation coefficient, piezoelectric constant, the width of the PPM layer and dielectric constants, which have been unravelled and displayed graphically.

1 Introduction

In a world full of sophisticated mathematical techniques, Green’s function technique stands out, given its practical applicability and its ability in determining the mathematical solutions to a large number of families of differential equations. It is very helpful in numerical and analytical techniques including singular integral equation methods, boundary element methods and dislocation methods. To be more precise, it creates visual interpretations of the operation associated to a point source. It is a known fact that the motions created in the material medium are not only caused due to internal disturbances but also due to external interactions. The external disturbing forces are functions of time and space. Therefore, there is one particular type of force that originate from unit impulsive force which can be defined with the help of Dirac’s delta function. A class of elastodynamic problems involve such disturbances due to a point source which is a function of spatial coordinates and time. The Green’s function plays a pivotal role in determining the mathematical solutions to these boundary value problems. Further, it has the ability to be defined according to the mechanical interactions occurring in the medium due to an impulsive excitation force. Many researchers have their study on Green’s function such as Covert (1958) established particle displacement for Love-type waves using Green’s function. Followed by this, Ghosh (1963) studied the displacement of Love waves derived through a point source. Aki and Richards (1980) and Ben-Menahem and Singh (1981) have also made efforts to study Green’s function extensively. Chattopadhyay et al. (2012) studied the effect of point source and heterogeneity on the propagation of SH waves in a viscoelastic layer over a viscoelastic half-space. In recent years, Singh et al. (2017, 2018), Karmakar et al. (2020), Kundu et al. (2020), Mahanty et al. (2021); and Kumari and Singh (2021) worked on point source as well as Green’s function.

Piezoelectric crystals and ceramics exhibit distinctive electromechanical coupling characteristics which enable them to produce electrical fields under the application of mechanical loads and also display elastic deformation on applying electrical loads. Nowadays piezoelectric composite materials which consist of two or more piezoelectric materials have some drawbacks due to the failure of the mechanism under electrical or mechanical loading. The other limitation, higher acoustic impedance restricts the utilization of the device in under-sea application up to a certain depth in the field of earth science. The acoustic impedance is a significant prerequisite in designing sonar and other underwater object detection devices. Also, Love wave sensors in piezoelectric materials find applications in principal detectors, sensors, and actuators as well as in signal transmission. Generally, they are brittle in nature and attempts have often been made (Taunaumang et al. 1994; Chan et al. 1999) to develop composite piezoelectric structures to extenuate their brittleness and to reinforce the coupling effects between electrical and mechanical activities. Due to its brittleness and some other possible defects such as impurity, cavities, micro-voids, layer separations, inclusions and micro-cracks, the devices often have the tendency to fail. These problems can be mitigated by the introduction of controlled porosity into the piezoelectric material. The mismatching of impedance and failure rate of the piezoelectric composite device can be overcome altogether by taking the discrepancy in porosity of the piezoelectric material. Such porous piezoelectric materials (PPM) are extensively used in many applications such as miniature hydrophones, vibratory sensors, accelerometers and contact microphones. These PPMs with low acoustic impedance can improve the sensitivity of acoustic detectors by studying the propagation characteristics and localization behaviour of the wave at the surface of such piezoelectric configuration. Arai et al. (1991) studied the properties of hydrophones with porous piezoelectric ceramics. Hayashi et al. (1991) discussed the processing of Porous 3-3 PZT Ceramics using Capsule-Free O₂-HIP. Piezoelectric materials have one significant disadvantage, their low impedance value. Mizumura et al. (1991) studied the porous piezoelectric ceramic as transducer. The discussion talks about the low acoustic impedance of porous piezoelectric ceramics. To design efficient sonar devices, acoustic impedance is an essential requirement. The piezoelectric effect depends on a number of factors such as elastic constant, porosity, density and so on. Porosity variation can control the acoustic impedance remarkably and can reduce the issues such as device failure. Some recent works on this material have been mentioned below; Ezzin et al. (2016) studied the Love wave propagation in a transversely isotropic piezoelectric layer on a piezomagnetic half-space. Ezzin et al. (2017) discussed the propagation behaviour of SH waves in a layered piezoelectric/piezomagnetic plate. Singhal et al. (2018a, b) published a research paper on the approximation of surface wave frequency in piezocomposite structures. They also worked on Liouville–Green approximation: An analytical approach to study the elastic wave vibrations in the composite structure of piezo material. Ebrahimi and Barati (2017) did the vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundation. Ebrahimi and Dabbagh (2017) also did the wave propagation analysis of smart rotating porous heterogeneous piezoelectric nanobeams. Ebrahimi et al. (2017) discussed the damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate-based non-local strain gradient theory. Some other research works are: Nasedkin et al. (2017) did finite element simulation on effective properties of microporous piezoceramic material with metallized pore surfaces. Kudimova et al. (2017) emphasized the computer design of porous and ceramic piezocomposites in the finite element package. Sahu studied the Love-type waves in functionally graded piezoelectric material (FGPM) sandwiched between an initially stressed layer and elastic substrate. Some more eminent authors who have made their experimental studies that relate to porous piezoelectric materials are Xia et al. (2003), Paizza et al. (2005), Huang et al. (2006), Zeng et al. (2006), and Wang et al. (2008).

The present study delves into the propagation behaviour of SH-type wave originated from a point source in a porous piezoelectric material (PPM) layer lying over a heterogeneous viscoelastic substrate. Green’s function technique has been effectively used to find the particle displacement of the SH-type wave. Significant efforts have been made to align the numerical calculations and graphical analysis with the assistance of two-dimensional plots. These plots contain variations of real phase velocity and damped velocity against real wave number with some of the affecting parameters such as heterogeneity parameter associated with rigidity, internal friction and density. Other factors include dissipation factor, attenuation coefficient, piezoelectric constant, width of the layer, dielectric coupling between two phases of the porous aggregate and dielectric constant. Eventually, they provide a visual analysis of the present paper for better interpretation. The outcomes provide a theoretical framework for designing and developing of underwater acoustic devices for sensing and non-destructive testing.

2 Mathematical formulation of the problem

Let us consider a porous piezoelectric material (PPM) layer of finite width $H$ lying over a heterogeneous viscoelastic substrate. The y-axis is towards the direction of wave propagation, while the x-axis has been considered positive vertically downwards as shown in figure 1. The source of disturbance $S$ has been taken at the point of intersection of the separation interface and x-axis. The polarization direction of porous piezoelectric material layer is parallel to z-axis and the material properties are assumed to be constant in the layer.

Figure 1

Schematic diagram of the problem.

It is assumed that the displacement components for the PPM layer and heterogeneous viscoelastic substrate are given by $\left({u}_{\mathrm{ppm}},{v}_{\mathrm{ppm}},{w}_{\mathrm{ppm}}\right)$ and $\left({u}_{s},{v}_{s},{w}_{s}\right)$, respectively. Therefore, for the propagation of SH-type wave in the y direction, it may be written that

$$u_{{{\text{ppm}}}} = 0,v_{{{\text{ppm}}}} = 0,w_{{{\text{ppm}}}} = w_{{{\text{ppm}}}} \left( {x,y,t} \right),u_{s} = 0,v_{s} = 0,w_{s} = w_{s} \left( {x,y,t} \right),\partial_{z} \equiv 0.$$

(1)

2.1 Dynamics of upper PPM layer under the influence of a point source

The constitutive equations for porous piezoelectric medium are (Gupta and Venkatesh 2006)

$$\begin{aligned}& \sigma_{ij} = F_{ijkl} \varepsilon_{kl} + m_{ij} \varepsilon^{*} - e_{kij} E_{k} - \zeta_{kij} E_{k}^{*} , \hfill \\ & \sigma^{*} = m_{ij} \varepsilon_{ij} + R\varepsilon^{*} - \zeta_{k} E_{k} - e_{k}^{*} E_{k}^{*} , \hfill \\ & D_{i} = e_{ikl} \varepsilon_{kl} + \zeta_{i} \varepsilon^{*} + T_{il} E_{l} + A_{il} E_{l}^{*} , \hfill \\ & D_{i}^{*} = \zeta_{ikl} \varepsilon_{kl} + e_{i}^{*} \varepsilon^{*} + A_{il} E_{l} + T_{il}^{*} E_{l}^{*}, { } \hfill \\ \end{aligned}$$

(2)

where ${\sigma }_{ij}\left({\sigma }^{*}\right)$ and ${\varepsilon }_{ij}\left({\varepsilon }^{*}\right)$ are the stress and strain tensor components for the solid (fluid) phase in the upper PPM layer, respectively. ${D}_{i}\left({D}_{i}^{*}\right)$ and ${E}_{i}\left({E}_{i}^{*}\right)$ are the electric displacement and electric field vector for the solid (fluid) phase in the upper PPM layer, respectively. Also ${F}_{ijkl},{m}_{ij},R$ are elastic constants, ${e}_{ijk},{e}_{i}^{*},{\zeta }_{ijk},{\zeta }_{i}$ are the piezoelectric constants, and ${T}_{ij},{T}_{ij}^{*},{A}_{ij}$ are dielectric constants.

The equation of motion in the absence of body forces and surface charge density for PPM layer are as follows:

$$\begin{aligned} \sigma_{ij,j}& = \left( {\rho_{11} } \right)_{ij} \ddot{u}_{j} + \left( {\rho_{12} } \right)_{ij} \ddot{U}_{j}^{*} \hfill \\ \sigma_{,i}^{*}& = \left( {\rho_{12} } \right)_{ij} \ddot{u}_{j} + \left( {\rho_{22} } \right)_{ij} \ddot{U}_{j}^{*} \hfill \\ D_{i,i}& = \, 0, D_{i,i}^{*} = 0.{ } \hfill \\ \end{aligned}$$

(3)

Further, we have

$$\varepsilon_{ij} = \frac{1}{2}\left( {u_{i,j} + u_{j,i} } \right), \varepsilon^{*} = U_{i,i}^{*} ,{ }E_{i} = - \phi_{,i} , E_{i}^{*} = - \phi_{,i}^{*} .$$

(4)

Here ${u}_{i}$ and ${U}_{i}^{*}$ are the mechanical displacement components of PPM layer. ${\left({\rho }_{11}\right)}_{ij}$, ${\left({\rho }_{12}\right)}_{ij}$ and ${\left({\rho }_{22}\right)}_{ij}$ are dynamical mass coefficients. $\phi \left({\phi }^{*}\right)$ is the electric potential function for the solid (fluid) phase for ${M}_{1}$.

If ${\rho }_{s}$ is the mass density of solid part and ${\rho }_{l}$ is the mass density of liquid part of the porous part, then we have the relation

$$\rho_{11} + \rho_{12} = \left( {1 - f} \right)\rho_{s} , \quad \rho_{11} + \rho_{12} = f\rho_{l} .$$

So that the total mass density is

$$\rho_{1} = \rho_{11} + 2\rho_{12} + \rho_{22}.$$

Moreover, these mass coefficients obey the following inequalities

$$\rho_{11} > 0, \rho_{12} \le 0, \rho_{22} > 0\quad {\text{and}}\quad \rho_{11} \rho_{22} - \rho_{12}^{2} > 0.$$

Also, $\gamma_{11} = \frac{{\rho_{11} }}{{\rho_{1} }},\gamma_{12} = \frac{{\rho_{12} }}{{\rho_{1} }}$ and $\gamma_{22} = \frac{{\rho_{22} }}{{\rho_{1} }}$ are the non-dimensional material parameter of porous medium suggested by Karmakar et al. (2020).

Thus we conclude

• If the layer is non-porous solid, then $d\to 1$.

• If the layer is fluid, then $d\to 0$.

• If the layer is poroelastic, then $0<d<1,$

where $d = \gamma_{11} - \frac{{\gamma_{12}^{2} }}{{\gamma_{22} }}$.

The constitutive equations for the PPM layer medium can be written as:

$$\sigma_{11} = F_{11} \varepsilon_{11} + F_{12} \varepsilon_{22} + F_{13} \varepsilon_{33} + m_{11} \varepsilon^{*} - e_{31} E_{3}$$

$$\sigma_{22} = F_{12} \varepsilon_{11} + F_{11} \varepsilon_{22} + F_{13} \varepsilon_{33} + m_{11} \varepsilon^{*} - e_{31} E_{3} { }$$

$$\sigma_{33} = F_{13} \varepsilon_{11} + F_{13} \varepsilon_{22} + F_{33} \varepsilon_{33} + m_{33} \varepsilon^{*} - e_{33} E_{3} { }$$

$$\sigma_{32} = 2F_{44} \varepsilon_{32} - e_{15} E_{2} { }$$

$$\sigma_{31} = 2F_{44} \varepsilon_{31} - e_{15} E_{1}$$

$$\sigma_{12} = 2F_{66} \varepsilon_{12}$$

$$\sigma^{*} = m_{11} \varepsilon_{11} + m_{11} \varepsilon_{22} + m_{33} \varepsilon_{33} + R\varepsilon^{*}$$

$$D_{1} = 2e_{15} \varepsilon_{13} + a_{11} E_{1} + A_{11} E_{1}^{*}$$

$$D_{2} = 2e_{15} \varepsilon_{23} + a_{11} E_{2} + A_{11} E_{2}^{*}$$

$$D_{3} = e_{31} \varepsilon_{11} + e_{31} \varepsilon_{22} + e_{33} \varepsilon_{33} + \zeta_{3} \varepsilon^{*} + a_{33} E_{3} + A_{33} E_{3}^{*}$$

$$D_{1}^{*} = A_{11} E_{1} + a_{11}^{*} E_{1}^{*}$$

$$D_{2}^{*} = A_{11} E_{2} + a_{11}^{*} E_{2}^{*}$$

$$D_{3}^{*} = e_{3}^{*} \varepsilon^{*} + A_{33} E_{3} + a_{33}^{*} E_{3}^{*} .{ }$$

(5)

where $F_{66} = \frac{{\left( {F_{11} - F_{12} } \right)}}{2}.$

The constitutive relations and governing equations of transversely isotropic porous piezoelectric materials under anti-plane deformation are:

$$F_{44} \nabla^{2} w_{{{\text{ppm}}}} + e_{15} \nabla^{2} \phi_{{{\text{ppm}}}} = \rho^{p} \ddot{w}_{{{\text{ppm}}}}$$

(6)

and

$$e_{15} \nabla^{2} w_{{{\text{ppm}}}} - T_{11}^{p} \nabla^{2} \phi_{{{\text{ppm}}}} = 0$$

(7)

where

$$\nabla^{2} = \left( {\partial_{xx} + \partial_{yy} } \right), \rho^{p} = \rho_{1} \left( {\gamma_{11} - \frac{{\left( {\gamma_{12} } \right)^{2} }}{{\gamma_{22} }}} \right) = \rho_{1} d, T_{11}^{p} = T_{11} - \frac{{\left( {A_{11} } \right)^{2} }}{{T_{11}^{*} }},$$

and

$$\nabla^{2} \phi^{*} = - \frac{{A_{11} }}{{T_{11}^{*} }}\nabla^{2} \phi.$$

If the source is introduced in the form of Dirac-delta function, the expression for the displacement at a point on the surface is the same as the value of Green’s function for the whole system at that point.

Let $\sigma_{1} \left( {r,t} \right)$ be considered as the force density distribution function for the upper layer due to the point source, then the equation of motion for SH-type wave propagating along the y-axis becomes:

$$F_{44} \nabla^{2} w_{{{\text{ppm}}}} + e_{15} \nabla^{2} \phi_{{{\text{ppm}}}} - \rho^{p} \ddot{w}_{{{\text{ppm}}}} = 4\pi \sigma_{1} \left( {r,t} \right),$$

(8)

$$e_{15} \nabla^{2} w_{{{\text{ppm}}}} - T_{11}^{p} \nabla^{2} \phi_{m} = 0$$

(9)

where $r$ is the distance from the origin at which the force was applied to a point and $t$ is the time.

Now using the substitutions $w_{{{\text{ppm}}}} \left( {x,y,t} \right) = w_{{{\text{ppm}}}} \left( {x,y} \right)e^{i\omega t}$, $\phi_{{{\text{ppm}}}} \left( {x,y,t} \right) = \phi_{{{\text{ppm}}}} \left( {x,y} \right)e^{i\omega t}$ and $\sigma_{1} \left( {r,t} \right) = \sigma_{1} \left( r \right)e^{i\omega t}$ in the above equation, we obtain

$$F_{44} \nabla^{2} w_{{{\text{ppm}}}} + e_{15} \nabla^{2} \phi_{{{\text{ppm}}}} + \rho^{p} \omega^{2} w_{{{\text{ppm}}}} = 4\pi \sigma_{1} \left( r \right),$$

(10)

$$e_{15} \nabla^{2} w_{{{\text{ppm}}}} - T_{11}^{p} \nabla^{2} \phi_{{{\text{ppm}}}} = 0$$

(11)

where $\omega =kc$ is the angular frequency. Here, $k$ and $c$ are the wave number and phase velocity, respectively. Therefore, the disturbances caused by the impulsive force ${\sigma }_{1}\left(r\right)$ may be represented in terms of Dirac-delta function at the point source as ${\sigma }_{1}\left(r\right)=\delta \left(y\right)\delta \left(x-H\right)$.

Defining the Fourier transform $\overline{w}_{{{\text{ppm}}}} \left( {\eta ,x} \right)$ of $w_{{{\text{ppm}}}} \left( {x,y} \right)$ as

$$\overline{w}_{{{\text{ppm}}}} \left( {\eta ,x} \right) = \frac{1}{2\pi }\mathop \int \limits_{ - \infty }^{\infty } w_{{{\text{ppm}}}} \left( {x,y} \right)e^{i\eta y} dy$$

(12)

Then the inverse Fourier transform can be given as:

$$w_{{{\text{ppm}}}} \left( {x,y} \right) = \mathop \int \limits_{ - \infty }^{\infty } \overline{w}_{{{\text{ppm}}}} \left( {\eta ,x} \right)e^{ - i\eta y} d\eta$$

(13)

Now, applying the Fourier transform in equations (10 and 11), the following differential equations are obtained:

$$F_{44} \left( {D_{xx} \overline{w}_{{{\text{ppm}}}} - \eta^{2} \overline{w}_{{{\text{ppm}}}} } \right) \quad + e_{15} \left( {D_{xx} \overline{\phi }_{{{\text{ppm}}}} - \eta^{2} \overline{\phi }_{{{\text{ppm}}}} } \right) + \rho^{p} \omega^{2} \overline{w}_{{{\text{ppm}}}} = 2\delta \left( {x - H} \right),$$

(14)

$$e_{15} \left( {D_{xx} \overline{w}_{{{\text{ppm}}}} - \eta^{2} \overline{w}_{{{\text{ppm}}}} } \right) - T_{11}^{p} \left( {D_{xx} \overline{\phi }_{{{\text{ppm}}}} - \eta^{2} \overline{\phi }_{{{\text{ppm}}}} } \right) = 0,$$

(15)

Putting equation (15) in equation (14) yields

$$D_{xx} \overline{w}_{{{\text{ppm}}}} - \alpha^{2} \overline{w}_{{{\text{ppm}}}} = 4\pi \sigma_{2} \left( x \right)$$

(16)

where $\alpha^{2} = \eta^{2} - \frac{{\rho^{p} \omega^{2} }}{{\overline{F}_{44} }}$, $\sigma_{2} \left( x \right) = \frac{{\delta \left( {x - H} \right)}}{{2\pi \overline{F}_{44} }}$ and $\overline{F}_{44} = \left( {F_{44} + \frac{{e_{15}^{2} }}{{T_{11}^{p} }}} \right)$.

2.2 Dynamics of lower heterogeneous viscoelastic substrate

The stress–strain relations for homogeneous viscoelastic substrate are given by Borcherdt (2009)

$$\tau_{ij} = \overline{\lambda }\nabla \delta_{ij} + 2\overline{\mu }e_{ij} ,{ } \quad \left( {i,j = 1,2,3} \right)$$

(17)

where $\overline{\mu } = \mu + \mu ^{\prime}\partial_{t} ,\overline{ \lambda } = \lambda + \lambda ^{\prime}\partial_{t}$.

The non-vanishing strain components $e_{ij}$ in terms of displacement components may be obtained as:

$$e_{ij} = \frac{1}{2}\left( {u_{i,j} + u_{j,i} } \right).$$

(18)

Therefore, in light of equations (1, 18 and 17) give the components of stresses for the viscoelastic substrate are

$$\tau_{11} = \left( {\overline{\lambda } + 2\overline{\mu }} \right)\partial_{x} u_{s} + \overline{\lambda }\partial_{y} v_{s} ,{ }\tau_{12} = \overline{\mu }\left( {\partial_{y} u_{s} + \partial_{y} v_{s} } \right),{ }\tau_{13} = \overline{\mu }\partial_{x} w_{s} , \tau_{32} = \overline{\mu }\partial_{y} w_{s} .{ }$$

(19)

For the propagation of SH-type wave in y-direction, the only non-vanishing equation of motion can be written as:

$$\partial_{x} \left[ {\left( {\mu + \mu ^{\prime}\partial_{t} } \right)\partial_{x} w_{s} } \right] + \partial_{y} \left[ {\left( {\mu + \mu ^{\prime}\partial_{t} } \right)\partial_{y} w_{s} } \right] = \rho_{2} \ddot{w}_{s} .$$

(20)

Now, the heterogeneity of the lower heterogeneous viscoelastic substrate has been considered in the form

$$\begin{aligned} \mu &= \mu_{0} + \chi_{1} \left( {x - H} \right)^{n} , \\ \mu^{\prime}& = \mu^{\prime}_{0} + \chi_{2} \left( {x - H} \right)^{n} , \\ \rho_{2}& = \rho_{2}^{0} + \chi_{3} \left( {x - H} \right)^{n}. \end{aligned}$$

(21)

where $\chi_{1} ,\chi_{2}$ and $\chi_{3}$ are small positive real constants having dimensions ${\text{ML}}^{ - 2} {\text{T}}^{ - 2}$, ${\text{MT}}^{ - 1}$ and ${\text{ML}}^{ - 4}$ are termed as frequency of inhomogeneity associated with rigidity, internal friction and density, respectively.

Therefore, substitution of equation (21) into equation (20) yields:

$$\begin{gathered} \partial_{x} \left[ {\left( {\mu_{0} + \mu ^{\prime}_{0} \partial_{t} } \right)\partial_{x} w_{s} + \left( {x - H} \right)^{n} \left( {\chi_{1} + \chi_{2} \partial_{t} } \right)\partial_{x} w_{s} } \right] \hfill \\ \quad + \partial_{y} \left[ {\left( {\mu_{0} + \mu ^{\prime}_{0} \partial_{t} } \right)\partial_{y} w_{s} \quad + \left( {x - H} \right)^{n} \left( {\chi_{1} + \chi_{2} \partial_{t} } \right)\partial_{y} w_{s} } \right] \\ \quad = \left[ {\rho_{2}^{0} + \chi_{3} \left( {x - H} \right)^{n} } \right]\ddot{w}_{s} . \hfill \\ \end{gathered}$$

(22)

Now using the substitution ${w}_{s}\left(x,y,t\right)={w}_{s}\left(x,y\right){e}^{i\omega t}$ in the above equation, we get:

$$\left[{\alpha }_{1}+{\alpha }_{2}{\left(x-H\right)}^{n}\right]{\nabla }^{2}{w}_{s}+n{\left(x-H\right)}^{n-1}{\alpha }_{2}{\partial }_{x}{w}_{s}+{\omega }^{2}\left[{\rho }_{2}^{0}+{\chi }_{3}{\left(x-H\right)}^{n}\right]{w}_{s}=0.$$

(23)

where $\alpha_{1} = \left( {\mu_{0} + i\omega \mu ^{\prime}_{0} } \right)$ and $\alpha_{2} = \left( {\chi_{1} + i\omega \chi_{2} } \right)$.

Defining the Fourier transform ${\overline{w}}_{s} \left( {\eta ,x} \right)$ of $w_{s} \left( {x,y} \right)$:

$${\overline{w}_{s}} \left( {\eta ,x} \right) = \frac{1}{2\pi }\mathop \int \limits_{ - \infty }^{\infty } w_{s} \left( {x,y} \right)e^{i\eta y} dy.$$

(24)

Then the inverse transform can be given as:

$$w_{s} \left( {x,y} \right) = \mathop \int \limits_{ - \infty }^{\infty } {\overline{w}_{s}} \left( {\eta ,x} \right)e^{ - i\eta y} d\eta .$$

(25)

Using Fourier transformation in equation (23), we have

$$D_{xx} \overline{w}_{s} - \beta^{2} \overline{w}_{s} = 4\pi \sigma_{3} \left( x \right)$$

(26)

where

$$\beta^{2} = \frac{{\left[ {\alpha_{1} \eta^{2} - \rho_{2}^{0} \omega^{2} } \right]}}{{\alpha_{1} }},$$

$$4\pi \sigma_{3} \left( x \right) = - 1/\alpha_{1} \Big[ {\alpha_{2} \left( {x - H} \right)^{n} D_{xx} \overline{w}_{s} + n\left( {x - H} \right)^{n - 1} \times \alpha_{2} D_{x} \overline{w}_{s} - \left( {x - H} \right)^{n} \left\{ {\alpha_{2} \eta^{2} - \chi_{3} \omega^{2} } \right\}\overline{w}_{s} } \Big].$$

2.3 Boundary conditions

Some of the boundary conditions that need to be satisfied by the geometry of the problem are as follows:

a) The uppermost surface must be free from shear stress, electrical displacement and electrical potential function at the interface $x = 0$.

$$\sigma_{13} = 0.$$

(27)

$$\overline{\phi }_{{{\text{ppm}}}} \left( {x,y} \right) = 0.$$

(28)

$$D_{x} \left( {x,y} \right) + D_{x}^{*} \left( {x,y} \right) = 0.$$

(29)

b) The periodicity of the components of the stress, mechanical displacement and electrical potential function at $x = H$ leads to the following conditions:

$$\sigma_{13} = \tau_{13} .$$

(30)

$$\overline{w}_{{{\text{ppm}}}} \left( {x,y} \right) = {\overline{w}_{s}} \left( {x,y} \right).$$

(31)

$$\overline{\phi }_{{{\text{ppm}}}} \left( {x,y} \right) = 0.$$

(32)

Now, to deduce the expression for displacements in the layer as well as in the substrate, boundary conditions equation (27) with equation (29) and equation (30) with equation (32) are combined to obtain

$$D_{x} \overline{w}_{{{\text{ppm}}}} = 0,{\text{ at }}x = 0$$

(33)

and

$$F_{44} D_{x} \overline{w}_{{{\text{ppm}}}} = \alpha_{1} D_{x} \overline{w}_{s}, \;\; \text{at} \; x = H.$$

(34)

respectively.

Now, let $G_{{{\text{ppm}}}} \left( {x/x_{0} } \right)$ be the Green’s function for the PPM layer satisfying the condition $D_{x} G_{{{\text{ppm}}}} = 0$ at $x = 0,H$ and $x_{0}$ is an arbitrary point in the medium, then the equation satisfied by $G_{{{\text{ppm}}}} \left( {x/x_{0} } \right)$ is

$$D_{xx} G_{{{\text{ppm}}}} \left( {x/x_{0} } \right) - \alpha^{2} G_{{{\text{ppm}}}} \left( {x/x_{0} } \right) = \delta \left( {x - x_{0} } \right)$$

(35)

$G_{{{\text{ppm}}}} \left( {x/x_{0} } \right)$ and $\overline{w}_{{{\text{ppm}}}}$ when multiplied with equations (16) and (35), respectively and then resulting equations subtracted and further integrated with respect to $x$ from $x=0$ to $H$ yield

$$\begin{aligned} & \mathop \int \limits_{x = 0}^{H} \left[ {G_{{{\text{ppm}}}} \left( {x/x_{0} } \right)D_{xx} \overline{w}_{{{\text{ppm}}}} - \overline{w}_{{{\text{ppm}}}} \left( {x/x_{0} } \right)D_{xx} G_{{{\text{ppm}}}} } \right]dx \hfill \\ & \quad = \mathop \int \limits_{x = 0}^{H} 4\pi \sigma_{2} \left( x \right)G_{{{\text{ppm}}}} \left( {x/x_{0} } \right)dx \\ & \quad - \mathop \int \limits_{x = 0}^{H} \delta \left( {x - x_{0} } \right)\overline{w}_{{{\text{ppm}}}} dx \hfill \\ \end{aligned}$$

(36)

with the help of the conditions $D_{x} G_{{{\text{ppm}}}} \left( {x/x_{0} } \right) = 0$ at $x = 0$ and $H$, equation (36) results in

$$G_{{{\text{ppm}}}} \left( {H/x_{0} } \right)\left[ {D_{x} \overline{w}_{{{\text{ppm}}}} } \right]_{x = H} \, = \, \frac{2}{{\overline{F}_{44} }}G_{{{\text{ppm}}}} \left( {H/x_{0} } \right) - \overline{w}_{{{\text{ppm}}}} \left( {x_{0} } \right).$$

(37)

Thus, the expression for displacement in the PPM layer $\overline{w}_{{{\text{ppm}}}}$ at any point, $x$ is obtained as

$$\overline{w}_{{{\text{ppm}}}} \left( x \right) = \frac{2}{{\overline{F}_{44} }}G_{{{\text{ppm}}}} \left( {x/H} \right) - G_{{{\text{ppm}}}} \left( {H/x} \right)\left[ {D_{x} \overline{w}_{{{\text{ppm}}}} } \right]_{x = H}$$

(38)

where ${x}_{0}$ is replaced by $x$ and noting that ${G}_{\mathrm{ppm}}\left(H/x\right)={G}_{\mathrm{ppm}}\left(x/H\right)$.

Let ${G}_{s}\left(x/{x}_{0}\right)$ be the Green’s function for the substrate satisfying the condition ${D}_{x}{G}_{s}=0$ at $x=H$ and it also approaches zero as $x$ tends to infinity.

The equation which is satisfied by ${G}_{s}\left(x/{x}_{0}\right)$ may be written as:

$$D_{xx} G_{s} \left( {x/x_{0} } \right) - \beta^{2} G_{s} \left( {x/x_{0} } \right) = \delta \left( {x - x_{0} } \right)$$

(39)

${G}_{s}\left(x/{x}_{0}\right)$ and ${\overline{w} }_{s}$ when multiplied to equations (26 and 39), respectively and then resulting equations are subtracted and further integrated with respect to $x$ from $x=H$ to $\infty$ yield

$$\begin{gathered} \mathop \int \limits_{x = H}^{\infty } \left[ {G_{s} \left( {x/x_{0} } \right)D_{xx} \overline{w}_{s} - {\overline{w}_{s}} \left( {x/x_{0} } \right)D_{xx} G_{s} } \right]dx \hfill \\ \quad = \mathop \int \limits_{x = H}^{\infty } 4\pi \sigma_{3} \left( x \right)G_{s} \left( {x/x_{0} } \right)dx - \mathop \int \limits_{x = H}^{\infty } \delta \left( {x - x_{0} } \right)\overline{w}_{s} dx. \hfill \\ \end{gathered}$$

(40)

Replacing ${x}_{0}$ by $x$ in equation (40), and keeping in mind that ${G}_{s}\left(H/x\right)={G}_{s}\left(x/H\right)$ and ${G}_{s}\left(x/{x}_{0}\right)={G}_{s}\left({x}_{0}/x\right)$, the expression for displacement ${\overline{w} }_{s}\left(x\right)$ at any point $x$ in substrate is obtained as:

$${\overline{w}_{s}} \left( x \right) = G_{s} \left( {x/H} \right)\left[ {D_{x} \overline{w}_{s} } \right]_{x = H} + \mathop \int \limits_{x = H}^{\infty } 4\pi \sigma_{3} \left( {x_{0} } \right)G_{s} \left( {x/x_{0} } \right)dx_{0} .$$

(41)

With the help of equation (34), the above equation gives

$$\left[ {D_{x} \overline{w}_{{{\text{ppm}}}} } \right]_{x = H} \, = \, \frac{{\left( {2/\overline{F}_{44} } \right)G_{{{\text{ppm}}}} \left( {H/H} \right)}}{{\left( {F_{44} /\alpha_{1} } \right)G_{s} \left( {H/H} \right) + G_{{{\text{ppm}}}} \left( {H/H} \right)}} - \frac{{\mathop \int \nolimits_{x = H}^{\infty } 4\pi \sigma_{3} \left( {x_{0} } \right)G_{s} \left( {{H}/{{x_{0} }}} \right)dx_{0} }}{{\left( {{{F_{44} }}/{{\alpha_{1} }}} \right)G_{s} \left( {H}/{H} \right) + G_{{{\text{ppm}}}} \left( {H}/{H} \right)}}.$$

(42)

Substituting the value of $\left[ {D_{x} \overline{w}_{{{\text{ppm}}}} } \right]_{x = H}$ from equation (42) into equation (38), we obtain

$$\overline{w}_{{{\text{ppm}}}} = \frac{{\frac{{2F_{44} }}{{\overline{F}_{44} }}G_{{{\text{ppm}}}} \left( {x/H} \right)G_{s} \left( {H/H} \right)}}{{\left\{ {F_{44} G_{s} \left( {H/H} \right) + \alpha_{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}} - \frac{{G_{{{\text{ppm}}}} \left( {x/H} \right)\mathop \int \nolimits_{x = H}^{\infty } 4\pi \sigma_{3} \left( {x_{0} } \right)G_{s} \left( {H/x_{0} } \right)dx_{0} }}{{\frac{1}{{\alpha_{1} }}\left\{ {F_{44} G_{s} \left( {H/H} \right) + \alpha_{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}}.$$

(43)

Moreover, employing the expression of $4\pi \sigma_{3} \left( {x_{0} } \right)$, the equation (43) may be rewritten as:

$${\bar{w}}_{\text{ppm}} = \frac{{\frac{{2F_{{44}} }}{{\bar{F}_{{44}} }}G_{\text{ppm}} \left( {x/H} \right)G_{s} \left( {H/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{\text{ppm}} \left( {H/H} \right)} \right\}}} - \frac{{G_{\text{ppm}} \left( {x/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{m} \left( {H/H} \right)} \right\}}} \quad \times \mathop \int \limits_{{x = H}}^{\infty } \left[ {\alpha _{2} \left( {x_{0} - H} \right)^{n} D_{{x_{0} x_{0} }} {\bar{w}}_{s} \quad + n\left( {x_{0} - H} \right)^{{n - 1}} D_{{x_{0} }} {\bar{w}}_{s} \quad - \left( {x_{0} - H} \right)^{n} \left\{ {\alpha _{2} \eta ^{2} - \chi _{3} \omega ^{2} } \right\} {\bar{w}}_{s} } \right]G_{s} \left( {H/x_{0} } \right)dx_{0}.$$

(44)

Again using equations (34 and 41) and proceeding as mentioned previously, the value of ${\overline{w} }_{s}$ is obtained as:

$${\bar{w}}_{s} = \frac{{\frac{{2F_{{44}} }}{{\bar{F}_{{44}} }}G_{{{\text{ppm}}}} \left( {H/H} \right)G_{s} \left( {x/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}} + \frac{{\left( {F_{{44}} } \right)G_{s} \left( {x/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}} \times \mathop \int \limits_{H}^{\infty } \left[ {\alpha _{2} \left( {x_{0} - H} \right)^{n} D_{{x_{0} x_{0} }} {\bar{w}}_{s} + n\left( {x_{0} - H} \right)^{{n - 1}} \alpha _{2} D_{{x_{0} }} - \left( {x_{0} - H} \right)^{n} \times \left\{ {\alpha _{2} \eta ^{2} - \chi _{3} \omega ^{2} } \right\}{\bar{w}}_{s} } \right]G_{s} \left( {H/x_{0} } \right)dx_{0} - \left( {1/\alpha _{1} } \right)\times \mathop \int \limits_{H}^{\infty } \left[ {\alpha _{2} \left( {x_{0} - H} \right)^{n} D_{{x_{0} x_{0} }} {\bar{w}}_{s} + n\left( {x_{0} - H} \right)^{{n - 1}} \alpha _{2} D_{{x_{0} }} {\bar{w}}_{s} - \left( {x_{0} - H} \right)^{n} \left\{ {\alpha _{2} \eta ^{2} - \chi _{3} \omega ^{2} } \right\}{\bar{w}}_{s} } \right]G_{s} \left( {H/x_{0} } \right)dx_{0}.$$

(45)

Neglecting the higher powers of ${\chi }_{1},{\chi }_{2}$ and ${\chi }_{3}$, the expression for displacement in the heterogeneous viscoelastic substrate, as the first-order approximation is

$${\overline{w}_{s}} \left( x \right) = \frac{{\frac{{2F_{44} }}{{\overline{F}_{44} }}G_{{{\text{ppm}}}} \left( {H/H} \right)G_{s} \left( {x/H} \right)}}{{\left\{ {F_{44} G_{s} \left( {H/H} \right) + \alpha_{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}}$$

(46)

which gives the displacement at any point in the heterogeneous viscoelastic substrate, if it is taken as homogeneous, putting this value of ${\overline{w}_{s}} \left( x \right)$ in equation (44), we get:

$$\bar{w}_{{{\text{ppm}}}} = \frac{{\frac{{2F_{{44}} }}{{\bar{F}_{{44}} }}G_{{{\text{ppm}}}} \left( {x/H} \right)G_{s} \left( {H/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}}} - \frac{{\left( {2F_{{44}} /\bar{F}_{{44}} } \right)G_{{{\text{ppm}}}} \left( {x/H} \right)G_{{{\text{ppm}}}} \left( {H/H} \right)}}{{\left\{ {F_{{44}} G_{s} \left( {H/H} \right) + \alpha _{1} G_{{{\text{ppm}}}} \left( {H/H} \right)} \right\}^{2} }} \times \mathop \int \limits_{H}^{\infty } \left[ {( {x_{0} - H})^{n} \alpha _{2} D_{{x_{0} x_{0} }} G_{s} \left( {x_{0} /H} \right) + n\left( {x_{0} - H} \right)^{{n - 1}} \times \alpha _{2} D_{{x_{0} }} G_{s} \left( {x_{0} /H} \right)} \right. \left. { - \left( {x_{0} - H} \right)^{n} \left\{ {\alpha _{2} \eta ^{2} - \chi _{3} \omega ^{2} } \right\} \times G_{s} \left( {x_{0} /H} \right)} \right]G_{s} \left( {H/x_{0} } \right)dx_{0} .$$

(47)

The solution of equation (47) represents the elastic displacements due to a unit impulsive force in space and time.

To evaluate ${\overline{w} }_{\mathrm{ppm}}$, the values of ${G}_{\mathrm{ppm}}\left(x/H\right)$ is required and must be determined.

Now, in order to obtain the solution of equation (16), which is satisfied by ${G}_{\mathrm{ppm}}\left(x/{x}_{0}\right)$, let the two independent solutions of the equation

$$D_{xx} Q - \alpha^{2} Q = 0,$$

(48)

vanishing at $x=-\infty$ and $x=\infty$ be ${Q}_{1}\left(x\right)={e}^{\alpha x}$ and ${Q}_{2}\left(x\right)={e}^{-\alpha x}$.

Therefore, the solution of equation (48) for an infinite medium is $- \frac{{e^{{ - \alpha \left| {x - x_{0} } \right|}} }}{2\alpha }$.

${G}_{\text{ppm}}\left(x/{x}_{0}\right)$ should satisfy the equation ${D}_{x}{G}_{\text{ppm}}\left(x/{x}_{0}\right)=0$ at both $x=0$ and $H$, it may be written as:

$$G_{{{\text{ppm}}}} \left( {x/x_{0} } \right) = - \frac{{e^{{ - \alpha \left| {x - x_{0} } \right|}} }}{2\alpha } + Ae^{\alpha x} + Be^{ - \alpha x}$$

(49)

where $A$ and $B$ are arbitrary constants. With the help of the aforementioned stated conditions, ${G}_{\mathrm{ppm}}\left(x/{x}_{0}\right)$ is obtained as

$$G_{{{\text{ppm}}}} \left( {\frac{x}{{x_{0} }}} \right) = \, \frac{ - 1}{{2\alpha }}\left[ {e^{{ - \alpha \left| {x - x_{0} } \right|}} + \frac{{e^{\alpha x} \left( {e^{{\alpha \left| {x - x_{0} } \right|}} + e^{{ - \alpha \left| {x - x_{0} } \right|}} } \right)}}{{e^{\alpha x} - e^{ - \alpha x} }} + \frac{{e^{ - \alpha x} \left( {e^{{\alpha \left| {x - x_{0} } \right|}} + e^{{ - \alpha \left| {x - x_{0} } \right|}} } \right)}}{{e^{\alpha x} - e^{ - \alpha x} }}} \right].$$

(50)

Proceeding in a similar way and given equation (26), ${G}_{s}\left(x/{x}_{0}\right)$ may be obtained as:

$$G_{s} \left( {x/x_{0} } \right) = \frac{ - 1}{{2\beta }}\left[ {e^{{ - \beta \left| {x - x_{0} } \right|}} + e^{{ - \beta \left| {x + x_{0} - 2H} \right|}} } \right].$$

(51)

Moreover, the following values for further usage may be easily derived from equations (49, 50 and 51):

$$G_{{{\text{ppm}}}} \left( {x}/{H} \right) = - \frac{{e^{\alpha x} + e^{ - \alpha x} }}{{\alpha \left( {e^{\alpha H} - e^{ - \alpha H} } \right)}},$$

(52)

$$G_{{{\text{ppm}}}} \left( {H/H} \right) = - \frac{{e^{\alpha H} + e^{ - \alpha H} }}{{\alpha \left( {e^{\alpha H} - e^{ - \alpha H} } \right)}},$$

(53)

$$G_{s} \left( {H/x_{0} } \right) = - \frac{{e^{{\alpha \left( {x_{0} - H} \right)}} }}{\beta },$$

(54)

$$G_{s} \left( {H/H} \right) = - \frac{1}{\beta }.$$

(55)

3 Frequency equation

Substitution of the values from equations (52, 53, 54 and 55) in equation (47) leads to

$$\overline{w}_{{{\text{ppm}}}} \left( x \right) = - \frac{{2F_{44} }}{{\overline{F}_{44} }} \times \frac{{\left( {e^{\alpha x} + e^{ - \alpha x} } \right)}}{{\eta_{3} - \eta_{4} }} \left[ {1 + \frac{{\left( {e^{\alpha H} + e^{ - \alpha H} } \right)}}{{\eta_{3} - \eta_{4} }}\chi_{4} } \right]$$

(56)

where

$$\eta_{3} = e^{\alpha H} \left( {\alpha F_{44} + \beta \alpha_{1} } \right), \;\; \eta_{4} = e^{ - \alpha H} \left( {\alpha F_{44} - \beta \alpha_{1} } \right)$$

and

$$\chi_{4} = \frac{{{\Gamma }\left( {n + 1} \right)}}{{2^{n + 1} \beta^{n + 1} }}\left[ { - \alpha_{2} - \frac{1}{{\beta^{2} }}\left\{ {\alpha_{2} \eta^{2} - \chi_{3} \omega^{2} } \right\}} \right].$$

Equation (56) is further simplified to yield

$$\overline{w}_{{{\text{ppm}}}} \left( {\eta ,x} \right) = \frac{{ - \frac{{2F_{44} }}{{\overline{F}_{44} }}\left( {e^{\alpha x} + e^{ - \alpha x} } \right)}}{{\left( {\eta_{3} - \eta_{4} } \right) - \left( {e^{\alpha H} + e^{ - \alpha H} } \right)\chi_{4} }}.$$

(57)

Taking the inverse Fourier transform of ${\overline{w} }_{\mathrm{ppm}}$ in equation (57), the displacement in the PPM layer may be obtained as:

$$w_{{{\text{ppm}}}} \left( {x,y} \right) = \mathop \int \limits_{ - \infty }^{\infty } \frac{{ - \frac{{2F_{44} }}{{\overline{F}_{44} }}\left( {e^{\alpha x} + e^{ - \alpha x} } \right)e^{ - i\eta y} d\eta }}{{\left( {\eta_{3} - \eta_{4} } \right) - \chi_{4} \left( {e^{\alpha H} + e^{ - \alpha H} } \right)}}.$$

(58)

The frequency equation of SH-type waves will be obtained by equating to zero the denominator of the above integral, i.e.,

$$e^{\alpha H} \left( {\alpha F_{44} + \beta \alpha_{1} } \right) - e^{ - \alpha H} \left( {\alpha F_{44} - \beta \alpha_{1} } \right)- (e^{\alpha H}+e^{ - \alpha H}) \times \frac{{{\Gamma }\left( {n + 1} \right)}}{{2^{n + 1} \beta^{n + 1} }}\times \left[ { - \alpha_{2} - \frac{1}{{\beta^{2} }}\left\{ {\alpha_{2} \eta^{2} - \chi_{3} \omega^{2} } \right\}} \right] = 0.$$

(59)

Replacing $\eta$ by $k$, $\omega$ by $kc$ and $\alpha$ by $i{\alpha }_{0}$, then the closed form of velocity (equation 59) is obtained as:

$${\text{tan}}\, \alpha_{0} H = \frac{{\beta \alpha_{1} }}{{\alpha_{0} F_{44} }} - \frac{1}{{\alpha_{0} F_{44} }} \quad \times \left[ {\frac{{{\Gamma }\left( {n + 1} \right)}}{{2^{n + 1} \beta^{n + 1} }}\left[ { - \alpha_{2} - \frac{1}{{\beta^{2} }}\left\{ {\alpha_{2} \eta^{2} - \chi_{3} \omega^{2} } \right\}} \right]} \right]$$

(60)

where

$$\alpha_{1} = \left( {\mu_{0} + ikc\mu ^{\prime}_{0} } \right),$$

$$\alpha_{2} = \left( {\gamma_{1} + ikc\gamma_{2} } \right),$$

$$\alpha_{0}^{2} = \frac{{\rho^{p} k^{2} c^{2} }}{{\overline{F}_{44} }} - k^{2}$$

and

$$\beta^{2} = \frac{{\left[ {\alpha_{1} k^{2} - \rho_{2}^{0} \left( {kc} \right)^{2} } \right]}}{{\alpha_{1} }}.$$

Equation (60) is the shear wave period equation for porous piezoelectric layer lying over a heterogeneous viscoelastic substrate due to the point source $S$.

Let the complex wave number $k$ be expressed by:

$$k = k_{1} + ik_{2} = k_{1} \left( {1 + i\delta } \right)$$

(61)

where $\delta = \frac{{k_{2} }}{{k_{1} }}$ is the attenuation coefficient. The dimensionless dissipation factor is given by

$$Q_{0}^{ - 1} = \frac{{\mu ^{\prime}_{0} \omega^{*} }}{{\mu_{0} }}$$

(62)

where ${\omega }^{*}={\text{Re}}\left(k\right)c$. Equation (60) is complex due to the presence of dissipation, attenuation and complex wave number, thus on separating equation (60) into real and imaginary parts yields two real non-linear equations

$${\text{Re}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = 0,$$

(63)

$${\text{Im}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = 0.$$

(64)

Equation (63) is the frequency equation of SH-type wave propagation due to a disturbing source point which originates the dispersion curve, i.e., phase velocity against real wave number; equation (64) is the damping equation of SH-type waves which produces the damping characteristics of a wave.

After simplification of equations (63 and 64), we get:

$${\text{Re}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = \frac{{{\text{sin}}2t_{1} }}{{{\text{cos}}2t_{1} + {\text{cosh}}2t_{2} }} = \frac{{q_{1} q_{3} + q_{2} q_{4} }}{{q_{3}^{2} + q_{4}^{2} }}$$

(65)

$${\text{Im}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = \frac{{{\text{sinh}}2t_{2} }}{{{\text{cos}}2t_{1} + {\text{cosh}}2t_{2} }} = \frac{{q_{2} q_{3} - q_{1} q_{4} }}{{q_{3}^{2} + q_{4}^{2} }}$$

(66)

where ${q}_{1},{q}_{2},{q}_{3}$ and ${q}_{4}$ are given in Appendix A.

So far, only the displacements of the layer and substrate have been taken into consideration. Now, the purpose is to find the expression for scalar potential $\phi \left(x\right)$ equation (11) suggests that the special solution of it is given by:

$$\phi_{\text{ppm}}^{\left( s \right)} = \frac{{e_{15} }}{{T_{11}^{p} }}w_{{{\text{ppm}}}} ,$$

(67)

and solution of homogeneous equation corresponding to equation (16) is given as:

$$\phi_{{{\text{ppm}}}}^{\left( h \right)} = Ce^{ - \eta x} + De^{\eta x}$$

(68)

where $C$ and $D$ are arbitrary constants, ‘$s$’ and ‘$h$’ are superscripts and correspond to special and homogeneous solution, respectively. Therefore, combining equations (67 and 68), the complete solution of $\overline{\phi }\left(x\right)$ may be written as:

$$\overline{\phi }_{{{\text{ppm}}}} \left( x \right) = Ce^{ - \eta x} + De^{\eta x} + \frac{{e_{15} }}{{T_{11}^{p} }}\overline{w}_{{{\text{ppm}}}} .$$

(69)

In light of boundary conditions equations (28, 32 and 15), equation (69) results in

$$\begin{aligned} \phi_{{{\text{ppm}}}} \left( {x,y} \right) & = \mathop \int \limits_{ - \infty }^{\infty } \frac{{\frac{{2c_{44}^{m} h_{15} }}{{\overline{c}_{44}^{m} \mu_{11} }}\left[ {\left( {e^{{\eta \left( {H - y} \right)}} + e^{{ - \eta \left( {H - y} \right)}} } \right) - \left( {e^{\eta y} + e^{ - \eta y} } \right)\left( {e^{\alpha H} + e^{ - \alpha H} } \right) - \left( {e^{\alpha y} + e^{ - \alpha y} } \right)\left( {e^{\eta H} + e^{ - \eta H} } \right)} \right]}}{{\left( {\eta_{3} - \eta_{4} } \right) - \chi \left( {e^{\alpha H} + e^{ - \alpha H} } \right)\left\{ {\frac{1}{2\beta } + \frac{{\left[ {4\beta^{3} - \omega^{2} + \frac{3}{2}\eta^{2} } \right]}}{{4\beta^{3} }}} \right\}}} \\ & \quad \times e^{ - i\eta y} d\eta .\\ \end{aligned}$$

(70)

On solving the integrand appearing in equation (70) for poles, we arrive at equation (59) which is the frequency equation of SH-type wave propagating in a PPM layer under the influence of a point source and lying over a heterogeneous viscoelastic substrate.

4 Special cases

Case 1: When porous piezoelectric material becomes non-piezoelectric porous material, i.e., ${e}_{15}={T}_{11}={T}_{11}^{*}={A}_{11}=0$, then we have:

$${\text{Re}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = \frac{{{\text{sin}}2t_{1} }}{{{\text{cos}}2t_{1} + {\text{cosh}}2t_{2} }} = \frac{{q_{11} q_{13} + q_{12} q_{14} }}{{q_{13}^{2} + q_{14}^{2} }}$$

(71)

$${\text{Im}}\left[ {\gamma_{n} \left( {k,c} \right)} \right] = \frac{{{\text{sinh}}2t_{2} }}{{{\text{cos}}2t_{1} + {\text{cosh}}2t_{2} }} = \frac{{q_{12} q_{13} - q_{11} q_{14} }}{{q_{13}^{2} + q_{14}^{2} }}$$

(72)

where ${q}_{11},{q}_{12},{q}_{13}$ and ${q}_{14}$ are given in Appendix B.

Equation (71) is the frequency equation for SH-type wave propagation in a non-piezoelectric layer lying over a heterogeneous viscoelastic substrate. Equation (72) is the damping equation for SH-type wave propagation in a non-piezoelectric layer lying over a heterogeneous viscoelastic substrate.

Case 2: When porous piezoelectric material becomes isotropic material, i.e., ${e}_{15}={T}_{11}={T}_{11}^{*}={A}_{11}=0,{F}_{44}={\mu }_{1}$ and $d\to 1$ and lower heterogeneous viscoelastic medium become isotropic, i.e., $\delta =0,{ Q}_{0}^{-1}=0,{\chi }_{1}=0,{\chi}_{2}=0$ and ${\chi }_{3}=0$, then we have:

$$\tan kH\left[ {\frac{{c^{2} }}{{c_{1}^{2} }} - 1} \right]^{\frac{1}{2}} = \frac{{\mu_{0} \left[ {1 - \frac{{c^{2} }}{{c_{2}^{2} }}} \right]^{\frac{1}{2}} }}{{\mu_{1} \left[ {\frac{{c^{2} }}{{c_{1}^{2} }} - 1} \right]^{\frac{1}{2}} }}$$

(73)

where

$$c_{1} = \sqrt {\frac{{\mu_{1} }}{\rho_1}} \quad \text{and} \quad c_{2} = \sqrt {\frac{{\mu_{0} }}{{\rho_{2}^{0} }}}.$$

Equation (73) is the well-known frequency equation given by Ewing et al. (1957), Qian et al. (2010) and Kumari et al. (2016) for Love wave propagation in an isotropic layer lying over an isotropic half-space which validates the current problem.

5 Numerical calculation and discussion

With a view to give better perception of the present study, i.e., the analysis of propagation of SH-type wave in PPM layer resting over a heterogeneous viscoelastic substrate, the authors have endeavoured with the help of two-dimensional plots. These plots refer to the variation of phase and damped velocities drawn against real wave number $\left( {{\text{Re}}\left[ {k_{1} H} \right]} \right)$ under the influence of parameters such as heterogeneity parameter associated with rigidity $\left( {{\Pi }_{1} = \frac{{\chi_{1} H^{n} }}{{\mu_{0} }}} \right)$, internal friction $\left( {{\Pi }_{2} = \frac{{\chi_{2} H^{n} }}{{\mu ^{\prime}_{0} }}} \right)$ and density $\left( {{\Pi }_{3} = \frac{{\chi_{3} H^{n} }}{{\rho_{0} }}} \right)$. Other plots include variation of phase and damped velocities against real wave number under the influence of dissipation factor $({Q}_{0}^{-1})$, attenuation coefficient $(\delta )$, piezoelectric constant $({e}_{15})$, width of the PPM layer $(H)$ and dielectric constants $({A}_{11}$ and ${T}_{11})$.

For calculation purposes, we have taken some elastic constants of one of the porous piezoelectric materials (PZT-5H), which are given in table 1.

Table 1 Elastic constants for porous piezoelectric material (Gupta and Venkatesh 2006).

Additionally, some other data that have been used for viscoelastic substrate are from Gubbins (1990)

$$\mu_{0} = 7.1 \times 10^{10} \, {\text{N}}/{\text{m}}^{2} ,$$

$$\rho_{0} = 3321{\text{ kg}}/{\text{m}}^{3}$$

$$\mu_{0} /\mu^{{\prime}} _{0}= 30{\text{ s}}^{ - 1}.$$

Figures 2 and 3 represent the variation of real phase velocity and damped velocity against real wave number, respectively under the influence of heterogeneity parameter associated with rigidity $\left( {{\Pi }_{1} = \frac{{\chi_{1} H^{n} }}{{\mu_{0} }}} \right)$. It is observed that the phase velocity has an increasing behaviour under the influence of the considered parameter. While on the other hand, the damped velocity has a decreasing effect. The effect of ${\Pi }_{1}$ on damped velocity is more for smaller values of wave number and as the wave number increases this effect decreases gradually until it becomes almost negligible.

Figure 2

Variation of phase velocity against real wave number for different values of heterogeneity parameter associated with rigidity $({\Pi }_{1}=\frac{{\chi }_{1}{H}^{n}}{{\mu }_{0}})$.

Figure 3

Variation of damped velocity against real wave number for different values of heterogeneity parameter associated with rigidity $({\Pi }_{1}=\frac{{\chi }_{1}{H}^{n}}{{\mu }_{0}})$.

Figures 4 and 5 have been plotted to demonstrate the effect of heterogeneity parameter associated with internal friction $\left( {{\Pi }_{2} = \frac{{\chi_{2} H^{n} }}{{\mu ^{\prime}_{0} }}} \right)$ on phase and damped velocities, respectively. Both the phase and damped velocities exhibit mixed characteristics under the influence of ${\Pi }_{2},$ i.e., initially, they get decreased but after some point, the behaviour changes and the velocities are increased.

Figure 4

Variation of phase velocity against real wave number for different values of heterogeneity parameter associated with internal friction $({\Pi }_{2}=\frac{{\chi }_{2}{H}^{n}}{\mu^{\prime}_{0}})$.

Figure 5

Variation of damped velocity against real wave number for different values of heterogeneity parameter associated with internal friction $({\Pi }_{2}=\frac{{\chi }_{2}{H}^{n}}{\mu^{\prime}_{0}})$.

Figures 6 and 7 portray the effect of heterogeneity parameter associated with density $\left( {{\Pi }_{3} = \frac{{\chi_{3} H^{n} }}{{\rho_{0} }}} \right)$. This parameter puts a constant increasing effect on the phase velocity while the damped velocity undergoes variable behaviour. However, looking at the curves it can be said that ${\Pi }_{3}$ puts positive influence on the damped velocity as well.

Figure 6

Variation of phase velocity against real wave number for different values of heterogeneity parameter associated with density $({\Pi }_{3}=\frac{{\chi }_{3}{H}^{n}}{{\rho }_{2}^{0}})$.

Figure 7

Variation of damped velocity against real wave number for different values of heterogeneity parameter associated with density $({\Pi }_{3}=\frac{{\chi }_{3}{H}^{n}}{{\rho }_{2}^{0}})$.

The effect of dissipation factor $({Q}_{0}^{-1})$ on the phase velocity and the damped velocity have been discussed in figures 8 and 9. Looking carefully at the curves, it is observed that the dissipation factor has a favourable influence on the phase velocity. On the other hand, the effect on damped velocity is more noticeable in the higher frequency region. Here as well, ${Q}_{0}^{-1}$ exerts a positive effect on the damped velocity.

Figure 8

Variation of phase velocity against real wave number for different values of dissipation factor $({Q}_{0}^{-1})$ associated with viscoelastic substrate.

Figure 9

Variation of damped velocity against real wave number for different values of dissipation factor $({Q}_{0}^{-1})$ associated with viscoelastic substrate.

The influence of attenuation coefficient $(\delta )$ on the phase and damped velocities have been presented in figures 10 and 11. It reveals that $\delta$ puts an unfavourable influence on the phase velocity. This effect is constant throughout. While in figure 11, it is observed that $\delta$ exerts a negative influence on the damped velocity as well. Here, the effect is more pronounced for smaller values of real wave number.

Figure 10

Variation of phase velocity against real wave number for different values of attenuation coefficient $(\delta )$.

Figure 11

Variation of damped velocity against real wave number for different values of attenuation coefficient $(\delta )$.

Figures 12 and 13 depict the variation of phase and damped velocities under the influence of piezoelectric constant $({e}_{15})$. Observing the curves, the phase velocity has an increasing characteristic with ${e}_{15}$ constantly, irrespective of the real wave number. While on the other curve, the damped velocity has a slight decrement with piezoelectric constant in the low-frequency region. At the high-frequency region, ${e}_{15}$ puts a positive influence on the damped velocity.

Figure 12

Variation of phase velocity against real wave number for different values of piezoelectric constant $({e}_{15})$.

Figure 13

Variation of damped velocity against real wave number for different values of piezoelectric constant $({e}_{15})$.

The purpose of figures 14 and 15 is to demonstrate the influence of different values of the width of the PPM layer $(H)$. The variation of phase and damped velocities are very similar to the variation discussed due to ${e}_{15}$.

Figure 14

Variation of phase velocity against real wave number for different widths of PPM layer $(H)$.

Figure 15

Variation of damped velocity against real wave number for different widths of PPM layer $(H)$.

Figures 16 and 17 exhibit the variation of phase and damped velocities under the influence of dielectric constant $({A}_{11})$. The phase, as well as damped velocity, have negligible effects due to ${A}_{11}$. However, the phase velocity decreases gradually with wave number. While damped velocity has a slight increase initially which then forms a peak and afterwards it decreases gradually.

Figure 16

Variation of phase velocity against real wave number for different values of dielectric constant $({A}_{11})$.

Figure 17

Variation of damped velocity against real wave number for different values of dielectric constant $({A}_{11})$.

The aim of figures 18 and 19 is to display the variation of phase and damped velocities under the influence of dielectric constant $({T}_{11})$. As observed for the effect of ${A}_{11}$, here as well, the constant ${T}_{11}$ has negligible influence on both the velocities. However, if closely looked upon the curves of damped velocity for both constants $({A}_{11}$ and ${T}_{11})$, it is seen that for ${T}_{11}$, the damped velocity starts at a much higher value. It then forms a plateau-like curve which then decreases gradually afterwards.

Figure 18

Variation of phase velocity against real wave number for different values of dielectric constant $({T}_{11})$.

Figure 19

Variation of damped velocity against real wave number for different values of dielectric constant $({T}_{11})$.

Lastly, a general attempt has been made in figure 20 to compare the porous piezoelectric material (PPM) with the non-porous piezoelectric material. It is noticed that for PPM, there occurs a sudden decline in phase velocity with wave number. However, for a non-porous piezoelectric material, there is a small increment in phase velocity and then decreases gradually.

Figure 20

Comparison of real phase velocity between a porous piezoelectric material and piezoelectric materials.

6 Concluding remarks

A thorough and complete analysis has been done to study the behaviour of SH-type wave in a porous piezoelectric material (PPM) layer resting over a heterogeneous viscoelastic substrate influenced by a point source. The calculation technique implemented here is Green’s function technique. The variation of phase and damped velocities of SH-type wave under the influence of parameters such as heterogeneity parameter associated with rigidity, associated with internal friction and associated with density, dissipation factor, attenuation coefficient, piezoelectric constant, width of the PPM layer and dielectric constants have been studied. Here are some of the conclusive points of the analysis:

• 1. Closed-form solutions of frequency equation have been obtained using Green’s function technique.

• 2. Greater the amplitude of heterogeneity associated with rigidity in the lower viscoelastic half-space, the greater the phase velocity but lower damped velocity. The phase and damped velocities are influenced more by heterogeneity associated with rigidity at the lower end of wave number.

• 3. The heterogeneity associated with internal friction shows mixed characteristics on the phase and damped velocities.

• 4. The heterogeneity associated with density and dissipation factor help in increment of both phase and damped velocities, i.e., both the velocities rise with the increase in value of the considered parameter.

• 5. The attenuation coefficient of the viscoelastic half-space reduces the phase and damped velocities of SH wave. The influence of this coefficient on damped velocity is predominantly more when the real wave number is small.

• 6. It is established that with increasing piezoelectric constant and width, the phase velocity gets amplified. On the other hand, damped velocity does not have much influence during lower real wave numbers. But, after a certain wave number such as 2.2 approx, the peizoelectric constant and the width have a considerable influence on damped velocity.

• 7. The dielectric constants (A₁₁ and T₁₁) have negligible influence on both the phase and damped velocities. The behaviour of phase and damped velocities remain the same irrespective of dielectric constants.

• 8. General outcome observed was that the phase velocity of SH-type wave was more in a porous piezoelectric material than the piezoelectric material.

As the development of any material and device is a never-ending process, therefore the current research paper has the potential to inject some useful insight into those developments. These developments can be valuable for materials used in ultrasonic imaging devices which can be extremely helpful in the medical sciences.

Appendices

Appendix A

$${t}_{01}={\left(1-\delta \right)}^{2},{ t}_{02}=2\delta ,{ b}_{1}=\left(\frac{{\rho }^{p}\omega^{*2}}{{\overline{F} }_{44}}-{k}_{1}^{2}\right),$$

$${t}_{03}=\sqrt{\frac{\sqrt{{t}_{01}^{2}+{t}_{02}^{2}}+{t}_{01}}{2}}, {t}_{04}=\sqrt{\frac{\sqrt{{t}_{01}^{2}+{t}_{02}^{2}}-{t}_{01}}{2}},$$

$${t}_{1}={\left(\frac{{\rho }^{p}\omega^{*2}}{{\overline{F} }_{44}}-{k}_{1}^{2}\right)}^{{1}/{2}}{t}_{03},$$

$${t}_{2}={\left(\frac{{\rho }^{p}\omega^{*2}}{{\overline{F} }_{44}}-{k}_{1}^{2}\right)}^{1/2}{t}_{04},$$

$${t}_{05}=\frac{\left(1-\delta {Q}_{0}^{-1}\right)}{{\left(1-\delta {Q}_{0}^{-1}\right)}^{2}+{\left({Q}_{0}^{-1}\right)}^{2}},$$

$${t}_{06}=\frac{{Q}_{0}^{-1}}{{\left(1-\delta {Q}_{0}^{-1}\right)}^{2}+{\left({Q}_{0}^{-1}\right)}^{2}},$$

$${t}_{07}=\left[\left(1-\frac{{c}^{2}}{{\beta }_{2}^{2}}{t}_{05}\right)\left(1-{\delta }^{2}\right)-2\delta \frac{{c}^{2}}{{\beta }_{2}^{2}}{t}_{06}\right]$$

$${t}_{08}=\left[\frac{{c}^{2}}{{\beta }_{2}^{2}}{t}_{06}\left(1-{\delta }^{2}\right)+2\delta \left(1-\frac{{c}^{2}}{{\beta }_{2}^{2}}{t}_{05}\right)\right]$$

$$t_{09} = \sqrt {\frac{{\sqrt {t_{07}^{2} + t_{08}^{2} } + t_{07} }}{2}} , t_{10} = \sqrt {\frac{{\sqrt {t_{07}^{2} + t_{08}^{2} } - t_{07} }}{2}} ,$$

$$t_{11} = k_{1}^{n} \left( {t_{09}^{2} + t_{10}^{2} } \right)^{n/2} {\text{cos}}\left[ {n {\text{arctan}}\left( {\frac{{t_{10} }}{{t_{09} }}} \right)} \right],$$

$$t_{12} = k_{1}^{n} \left( {t_{09}^{2} + t_{10}^{2} } \right)^{{n}/{2}} {\text{sin}}\left[ {n {\text{arctan}}\left( {\frac{{t_{10} }}{{t_{09} }}} \right)} \right],$$

$${\zeta }_{1}=\left[{t}_{09}\left(1-\delta {Q}_{0}^{-1}\right)-{t}_{10}{Q}_{0}^{-1}\right],$$

$${\zeta }_{2}=\left[{t}_{10}\left(1-\delta {Q}_{0}^{-1}\right)+{t}_{09}{Q}_{0}^{-1}\right],$$

$${\zeta }_{3}=\left[{t}_{07}\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\omega }^{*}{\chi }_{2}{t}_{08} \quad +\left(\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\chi }_{3}{c}^{2}\right)\left(1-\delta \right)-2\delta {\omega }^{*}{\chi }_{2}\right],$$

$${\zeta }_{4}=\left[{t}_{08}\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\omega }^{*}{\chi }_{2}{t}_{07}+{\omega }^{*}{\chi }_{2}\left(1-\delta \right) \quad +2\delta \left(\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\chi }_{3}{c}^{2}\right)\right],$$

$${\zeta }_{5}=\left[{t}_{07}\left({t}_{11}{t}_{09}-{t}_{10}{t}_{12}\right)-{t}_{08}\left({t}_{09}{t}_{12}+{t}_{10}{t}_{11}\right)\right],$$

$${\zeta }_{6}=\left[{t}_{07}\left({t}_{09}{t}_{12}+{t}_{10}{t}_{11}\right)+{t}_{08}\left({t}_{11}{t}_{09}-{t}_{10}{t}_{12}\right)\right],$$

$$\begin{aligned} & S = \frac{{{\Gamma }\left[ {n + 1} \right]}}{{2^{{\left( {n + 1} \right)}} k_{1} }} \hfill \\ & q_{1} = \mu_{0} k_{1} \left( {\zeta_{1} \zeta_{5} - \zeta_{2} \zeta_{6} } \right) + S\zeta_{3} , \hfill \\ & q_{2} = \mu_{0} k_{1} \left( {\zeta_{2} \zeta_{5} + \zeta_{1} \zeta_{6} } \right) + S\zeta_{4} , \hfill \\ & q_{3} = b_{1}^{1/2} F_{44} \left( {t_{03} \zeta_{5} - t_{04} \zeta_{6} } \right), \hfill \\ & q_{4} = b_{1}^{1/2} F_{44} \left( {t_{04} \zeta_{5} + t_{03} \zeta_{6} } \right). \hfill \\ \end{aligned}$$

Appendix B

$${t}_{01}={\left(1-\delta \right)}^{2},{t}_{02}=2\delta ,{b}_{1}=\left(\frac{{\rho }^{p}{{\omega }^{*}}^{2}}{{F}_{44}}-{k}_{1}^{2}\right),$$

$$t_{03} = \sqrt {\frac{{\sqrt {t_{01}^{2} + t_{02}^{2} } + t_{01} }}{2}} ,t_{04} = \sqrt {\frac{{\sqrt {t_{01}^{2} + t_{02}^{2} } - t_{01} }}{2}} ,$$

$${t}_{1}={\left(\frac{{\rho }^{p}{{\omega }^{*}}^{2}}{{F}_{44}}-{k}_{1}^{2}\right)}^{1/2}{t}_{03},$$

$${t}_{2}={\left(\frac{{\rho }^{p}{{\omega }^{*}}^{2}}{{F}_{44}}-{k}_{1}^{2}\right)}^{1/2}{t}_{04},$$

$$t_{05} = \frac{{\left( {1 - \delta Q_{0}^{ - 1} } \right)}}{{\left( {1 - \delta Q_{0}^{ - 1} } \right)^{2} + \left( {Q_{0}^{ - 1} } \right)^{2} }},$$

$$t_{06} = \frac{{Q_{0}^{ - 1} }}{{\left( {1 - \delta Q_{0}^{ - 1} } \right)^{2} + \left( {Q_{0}^{ - 1} } \right)^{2} }},$$

$$t_{07} = \left[ {\left( {1 - \frac{{c^{2} }}{{\beta_{2}^{2} }}t_{05} } \right)\left( {1 - \delta^{2} } \right) - 2\delta \frac{{c^{2} }}{{\beta_{2}^{2} }}t_{06} } \right]$$

$$t_{08} = \left[ {\frac{{c^{2} }}{{\beta_{2}^{2} }}t_{06} \left( {1 - \delta^{2} } \right) + 2\delta \left( {1 - \frac{{c^{2} }}{{\beta_{2}^{2} }}t_{05} } \right)} \right]$$

$$t_{09} = \sqrt {\frac{{\sqrt {t_{07}^{2} + t_{08}^{2} } + t_{07} }}{2}} ,t_{10} = \sqrt {\frac{{\sqrt {t_{07}^{2} + t_{08}^{2} } - t_{07} }}{2}} ,$$

$$t_{11} = k_{1}^{n} \left( {t_{09}^{2} + t_{10}^{2} } \right)^{n/2} {\text{cos}}\left[ {n{\text{arctan}}\left( {\frac{{t_{10} }}{{t_{09} }}} \right)} \right],$$

$$t_{12} = k_{1}^{n} \left( {t_{09}^{2} + t_{10}^{2} } \right)^{n/2} {\text{sin}}\left[ {n{\text{arctan}}\left( {\frac{{t_{10} }}{{t_{09} }}} \right)} \right],$$

$${\zeta }_{1}=\left[{t}_{09}\left(1-\delta {Q}_{0}^{-1}\right)-{t}_{10}{Q}_{0}^{-1}\right],$$

$${\zeta }_{2}=\left[{t}_{10}\left(1-\delta {Q}_{0}^{-1}\right)+{t}_{09}{Q}_{0}^{-1}\right],$$

$${\zeta }_{3}=\left[{t}_{07}\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\omega }^{*}{\chi }_{2}{t}_{08} \quad +\left(\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\chi }_{3}{c}^{2}\right)\left(1-\delta \right)-2\delta {\omega }^{*}{\chi }_{2}\right],$$

$${\zeta }_{4}=\left[{t}_{08}\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\omega }^{*}{\chi }_{2}{t}_{07}+{\omega }^{*}{\chi }_{2} \quad \left(1-\delta \right)+2\delta \left(\left({\chi }_{1}-\delta {\omega }^{*}{\chi }_{2}\right)-{\chi }_{3}{c}^{2}\right)\right],$$

$${\zeta }_{5}=\left[{t}_{07}\left({t}_{11}{t}_{09}-{t}_{10}{t}_{12}\right)-{t}_{08}\left({t}_{09}{t}_{12}+{t}_{10}{t}_{11}\right)\right],$$

$${\zeta }_{6}=\left[{t}_{07}\left({t}_{09}{t}_{12}+{t}_{10}{t}_{11}\right)+{t}_{08}\left({t}_{11}{t}_{09}-{t}_{10}{t}_{12}\right)\right],$$

$$S = \frac{{{\Gamma }\left[ {n + 1} \right]}}{{2^{{\left( {n + 1} \right)}} k_{1} }}$$

$$\begin{gathered} q_{11} = \mu_{0} k_{1} \left( {\zeta_{1} \zeta_{5} - \zeta_{2} \zeta_{6} } \right) + S\zeta_{3} , \hfill \\ q_{12} = \mu_{0} k_{1} \left( {\zeta_{2} \zeta_{5} + \zeta_{1} \zeta_{6} } \right) + S\zeta_{4} , \hfill \\ q_{13} = b_{1}^{1/2} F_{44} \left( {t_{03} \zeta_{5} - t_{04} \zeta_{6} } \right), \hfill \\ q_{14} = b_{1}^{1/2} F_{44} \left( {t_{04} \zeta_{5} + t_{03} \zeta_{6} } \right). \hfill \\ \end{gathered}$$