$u_i$ components of displacement vector, m

T absolute temperature, K

C mass concentration of diffusive material in the elastic body, $\text{Kg}/{\text{m}}^3$

$\mu ,\lambda$ Lame's constant, $\text{N}/{\text{m}}^2$

$\rho$ density of medium, $\text{Kg}/{\text{m}}^3$

${\beta }_i$ thermal constants, $\text{N}/{\text{m}}^2\text{K}$

${\sigma }_{ij}$ stress tensor, $\text{N}/{\text{m}}^2$

${\alpha }_t$ linear thermal expansion's coefficient, ${\text{K}}^{-1}$

${\alpha }_c$ diffusion expansion's coefficient, ${\text{m}}^3/\text{KgK}$

$e_{ij}$ strain tensor, $\text{N}/{\text{m}}^2$

$q_i$ jeat flux vector, $\text{W}/{\text{m}}^2$

${\delta }_{ij}$ kronecker delta

S entropy per unit mass, $\text{J}/\text{KgK}$

$T_0$ medium's temperature in its natural state, K

a,b measures of thermo diffusive effects, ${\text{m}}^2/{\text{Ks}}^2$

$D^{*}$ thermodiffusion coefficient, ${\text{m}}^2/\text{s}$

$\tau$ relaxation time, sec

P chemical potential per unit mass, J/Kg

$c_e$ specific heat, J/KgK

$K_{ij}^{*}$ material constant characteristic, W/mKs

$K_{ij}$ thermal conductivity constant, W/mK

${\tau }_v$ phase lag in thermal displacement gradient

${\tau }_q$ phase lag in heat flux vector

${\tau }_T$ phase lag in temperature gradient

$\Theta$ conductive temperature, K

$a^{*}$ two temperature parameter $(a^{*}>0)$

$\chi$ frequency, Hz

c phase velocity, m/s

$\eta$ wave number

B propagation speed of wave, m/s

F attenuation coefficient, ${\text{m}}^{-1}$

Z penetration depth, m

W wpecific Loss

$c_1$ longitudinal wave speed, m/s

$c_2$ transverse wave speed, m/s

It is well established that the heat conduction equations in classical coupled and uncoupled thermoelastic theories are of diffusion type. The major disadvantage of these theories is the parabolic nature of its temperature governing equations that predicts infinite speed of thermal disturbances noticed far away from their source. Many researchers have attempted to generalise and amend the classical thermoelastic theories in order to resolve the contradiction of infinite propagation speed of heat waves.

Lord and Shulman [1] proposed a generalised theory of thermoelasticity, which replaces the classical fourier law with a modified fourier law that incorporates relaxation time parameter and heat flux vector. Green and Lindsay [2] developed another generalization of thermoelasticity, where restrictions had been imposed on governing equations with the assistance of entropy inequality. Dhaliwal and Sherief [3] deduced the equations for generalised thermoelasticity in anisotropic medium and obtained the variational principle for governing equations. Green and Nagdhi [4], [5], [6] proposed a new theory that incorporate energy dissipation in thermal wave propagation. Tzou [7] developed the dual phase lag in heat flux and temperature gradient in phonon-electron interactions. Roy [8] developed the Three Phase Lag (TPL) theory in which fourier law with three different phase lag in temperature gradient, thermal gradient and heat flux vector employed in the heat conduction equation.

Here $K_{ij}^{*}$ is material constant characteristic and $K_{ij}$ is thermal conductivity constant. The delay time $\left({\tau }_T\right)$ is measured by micro-structural interactions (small scale effects of heat transfer in space such as phonon-electron interaction or phonon scattering). The second delay time $\left({\tau }_q\right)$ is influenced by thermal inertia's fast-transient effects or time-dependent heat transfer. The thermal displacement gradient lag $\left({\tau }_v\right)$ interpreted as the third delay time. ${\tau }_q$ and ${\tau }_T$ are considered to be positive and small depends upon the intrinsic property of the medium satisfying the condition $0\le {\tau }_v<{\tau }_T<{\tau }_q$ . Three Phase Lag (TPL) model is extremely useful for studying exothermic catalytic reactions, nuclear boiling, multi phase model and bio-heat conduction model for laser heating of living tissues etc.

Wave propagation in thermoelastic solids has been extensively studied in the literature. It has major relevance for its usage in non-destructive testing of composite structures used in aviation, spacecraft and other technical applications. Since the nineteenth century, several researchers have explored wave propagation in the context of various thermoelastic theories. Rayleigh [9] discovered a new form of surface wave that propagates slightly faster thanthe shear wave speed and include both longitudinal as well as transverse motions. Rayleigh's work considered the surface as an infinite plane and found that wave's velocity do not depend upon wavelength in the absence of gravity. Abd-Alla et al. [10] analyzed the influence of initial stress and gravity on the proliferation of Rayleigh waves in magneto-thermoelastic medium. Abo-Dahab [11] analyzed the effect of initial stress and gravity on surface wave propagation in an orthotropic medium. Biswas and Abo-Dahab [12] examined the influence of magnetic field and initial stress on Rayleigh wave progression in thermoelastic medium. Biswas et al. [13] examined the Rayleigh wave proliferation in orthotropic half apace with Three Phase Lag (TPL) model. Biswas and Mukhopadhayay [14] solved the basic equations of Three Phase Lag (TPL) model for homogeneous orthotropic thermoelastic medium with eigen expansion method. Kaur and Lata [15] investigated the behaviour of circular plate loaded with ring in isotropic medium with and without dissipation of energy at two temperature. Sheokand et al. [16] demonstrated the influence of rotation, phase lag and fibre reinforcement on the propagation of plane wave in the context of Three Phase Lag (TPL) model. Singh et al. [17] used eigen value and Honig-Hirdes methods to solve the basic equations of Three Phase Lag (TPL) model. Sharma and Kumari [18] studied the Rayleigh wave propagation in context of generalized thermoelasticity with two temperature. Abd-Alla et al. [19] explored the rotational and gravitational effect on magneto thermoelastic solid having heat source on its boundry. Biswas [20] compared the effects of thermal shock on an infinite thermoelastic body between Lord-Shulman (L-S) model, Green-Nagdhi (GN-III) model and Three Phase Lag (TPL) model using normal mode analysis. The authors observed that thermoelastic deformation is more compatible with Lord-Shulman (L-S) than Three Phase Lag (TPL) model. Singh et al. [21] studied the propagation of Rayleigh wave in reference to Three phase lag (TPL) model with two temperature. Singh and Kumari [22] analyzed the influence of gravity and initial stress on Rayleigh wave propagation in purview of Three Phase Lag (TPL) model with two temperature theory. Abo-Dahab et al. [23] studied the effect of electro-magnetic field on deformation of micro-polar thermoelastic medium in terms of four thermoelastic models. Singh et al. [24] solved the basic equations of micro-polar thermoelasticity with impedance boundry conditions in specilized plane for the reflection of plane waves. Sharma [25] demonstrated the influence of initial stress and gravity on polarization and phase velocity of Rayleigh wave propagation in orthotropic solid.

Diffusion is referred to random movement of particles from high concentration zone to low concentration zone. The occurrence of this phenomenon caused by second law of thermodynamics, asserts that the entropy in any system must always rise with time. Heat conduction in a solid body is a prominent example of diffusion. Conduction occurs as a result of molecule collisions, which transports the heat through kinetic energy without the need for macroscopic material movement. Thermoelastic diffusion occurs in an elastic solid due to coupling of temperature, strain fields and mass diffusion. In the absence of diffusion phenomena, the theory reduces to thermoelasticity. For research purposes, thermal diffusion is widely used procedure for separating light isotopes of noble gases. Podstrigach and Pavlina [26], [27] focused on the relationship between temperature, deformation and mass concentration using irreversible thermodynamics process. Müller [28] discussed the thermal diffusive characteristics of Maxwellian gases in a mixture. Naerlovi-Veljkovic and Plavsic [29] investigated thermodiffusion in an elastic material with micro-structures and also discussed the non-linear theory of coupled mechanical and thermo-diffusional effects in elastic materials. Nowacki [30], [31], [32] established thermoelastic diffusion theory using coupled thermoelastic model. Crank [33] discussed several boundary problems involving heat flow and mass diffusion in his works. Wyrwal [34] established the variational concept in thermoelastic diffusion solids. Olesiak [35] explored the thermodiffusive characteristics of solids. Sherief et al. [36], [37] derived the basic equations of thermoelastic diffusive theory and solve the half space problem to demonstrate the variation of stress and temperature distribution.

Sharma et al. [38] examined the proliferation of Rayleigh wave in an isotropic thermo-diffusive elastic half plane. Kumar and Kansal [39], [40], [41] examined the progression of Rayleigh wave in a thermoelastic half-plane with mass diffusion. Aouadi [42] examined the stability issues associated with a non-simple thermoelastic diffusive problem. Othman et al. [43] examined the impact of thermal relaxation and magnetic field on the two-dimensional problem of generalised thermoelastic diffusion in terms both coupled and Lord-Shulman theories. Kumar et al. [44] explored the Rayleigh wave propagation in generalized thermoelastic diffusive solids. Kumar and Gupta [45] utilized the harmonic wave solution to obtain three coupled dilatational waves in the context of Dual Phase Lag Diffusion (DPLD) model. Kumar and Gupta [46] derived the frequency equation of Rayleigh surface waves with mass diffusion in an isotropic thermodiffusive half plane. Xiong and Guo [47] analyzed the half space with variable material properties in the domain of fractional order thermoelastic diffusive theory. Othman and Said [48] analyzed the impact of internal source and diffusion on thermoelastic medium in the context of Three Phase Lag (TPL) model. Bhattacharya et al. [49] used finite element technique to determine the solution of generalised elastic-thermo-diffusion problems in an isotropic medium using Three Phase Lag (TPL) model. Kanoria and Sur [50] examined the effect of thermo-diffusive interactions in an isotropic thermoelastic medium with one and two relaxation parameters. The researchers observed that diffusion has significant impact on thermo-physical characteristics of the material. Yadav [51] analyzed the effect of diffusion and impedance boundary condition on the reflection of plane waves in rotating magneto thermoelastic medium. Yadav [52] studied the propagation of plane wave in an initially stressed thermodiffusive medium in context of generalized fractional order derivative thermoelasticity. Mabrouk et al. [53] studied the thermal variable conductivity of rotating semiconductor medium in the presence of external magnetic field during process of diffusion. Davydov and Zemskov [54] analyzed the impact of phase lag in continuum layer within the framework of thermodiffusive phase lag model. Bezzina and Zenkour [55] applied the generalized diffusive model on isotropic cylinder in the presence of thermal flux and chemical potential. Zenkour [56] employed Green-Nagdhi (GN-II) and Green-Nadghi (GN-III) model to study the thermodiffusive isotropic half space. The author solved the governing equations of thermoelastic diffusive theory using initial and boundary conditions. Kaushal et al. [57] studied the influence of impedance parameter, diffusion and relaxation time on wave propagation in the context Green-Lindsay (G-L) and Coupled theory of thermoelasticity.

Chen et al. [58], [59], [60] developed the two temperature theory of thermoelasticity depending upon conductive temperature ($\Theta$ ) and thermodynamic temperature ( T) involving two temperature parameter ($a^{*}$ ). If $a^{*}$ tends to zero; this theory transformed into classical theory of heat conduction. Youssef [61] developed the theory of thermoelasticity that relates the difference between thermodynamic temperature and conductive temperature with heat supply in elastic bodies. Ezzat et al. [62] constructed a two temperature magneto thermoelastic model of fractional order within the framework of Dual Phase Lag (DPL) model. Shivay and Mukhopahyay [63] established the temperature rate dependent two temperature thermoelastic theory (TRDTT). This theory relies on the conductive and thermodynamic temperature dependence on temperature rate. Bajpai et al. [64] studied the effect of two temperature on wave propagation in isotropic uniform plate and used the concept of thermo-mechanical loading to obtain the expression for thermodynamic and conductive temperature. Sharma et al. [65] explored the dynamic characteristics of thermoelastic plate in the context of diffusion model at two temperatures with variable diffusivity and conductivity.

The introduction of the Three Phase Lag (TPL) model provides a general theoretical heat conduction model that considered various microstructural considerations, enabling scientists working in the field of heat conduction to accurately predict the thermal behaviour of structures using a multi-scale model. Despite these advances, not much research has been conducted on Rayleigh wave propagation in thermodiffusive medium with Three Phase Lag (TPL) model at two temperature. In this present study, the frequency equation has been derived for Rayleigh wave in isotropic half-space with mass diffusion in context of Three Phase Lag (TPL) model with two temperature. The variation of various properties like attenuation coefficient, non-dimensional speed, penetration depth and specific loss are represented graphically corresponding to thermo-diffusion parameter, relaxation time, wave number and frequency at different values two temperature parameter.

We assume uniform transversely isotropic generalised thermodiffusive elastic half space with mass concentration $C(x_1,x_3,t)$ at an initial temperature $T_0$ . The coordinate system's origin $(x_1,x_2,x_3)$ is considered to be any point on planar horizontal surface. We extend the $x_3$ -axis vertically downward along the axis of material symmetry in the half-space designated by $x_3\ge 0$ . The half space $x_1-x_3$ is considered to be stress-free with relevant thermally insulated and isothermal boundary conditions. We choose the Rayleigh wave propagation along $x_1$ -axis such that the particles on a line parallel to $x_2$ -axis are equally displaced. As a result, no field quantity is dependent on $x_2$ coordinate.

According to Sokolnikoff [66], the strain displacement relation given as:

Following Atkins [67], the entropy equation:

Sherief et al. [36], [37] established the equation of motion in the absence of external forces for thermodiffusive elastic medium as:

Here ${\beta }_1=(2\mu +3\lambda ){\alpha }_t$ and ${\beta }_2=(2\mu +3\lambda ){\alpha }_c$

Using strain-displacement relation, entropy equation and equation of motion, Sherief et al. [36], [37] formulated the diffusion equation:

Using the relations (1)-(4), Sherief et al. [36], [37] established the constitutive relations of thermodiffusive problem:

The heat conduction equation of Three Phase Lag (TPL) model following Othman and Said [48], We have the relation:

where $\frac{\partial T_{i,j}}{\partial t}=v_{i,j}$ at the condition $|\frac{T}{T_0}|<<1$ .

Following Chen et al. [58], [59], [60], the two temperature relation:

In two dimensional problem $\overrightarrow{u}(x_1,x_3,t)=(u_1,0,u_3)$ are considered as displacement vector. According to Helmohltz decomposition theorem, displacement vector $\overrightarrow{u}$ in terms of scalar potential $\varphi$ and vector potential $\psi$ are expressed as $\overrightarrow{u}=\nabla \varphi +\nabla \times \overrightarrow{\psi }$ . Hence displacement components $u_1$ and $u_3$ in terms of potentials can be written as:

Using Eqs. (7) and (8) in equation of motion (3), we have the following relation:

Using Eqs. (7) and (8) in Three Phase Lag (TPL) model heat conduction Eq. (6), we have the following relation:

Using Eqs. (7) and (8) in diffusion Eq. (4), we obtained the following relation:

where, ${\tau }_n=1+\tau {\displaystyle \frac{\partial }{\partial t}}$

The boundary conditions for stress free thermally insulated surface $x_3=0$ are considered as follows:

1. Normal stress component vanished ${\sigma }_{33}=0$

2. Tangential stress component vanished ${\sigma }_{13}=0$

3. Thermal conditions $q_3+hT=0$ Here normal component of heat flux vector $q_3$ related to absolute temperature T as:

4. Chemical potential per unit mass $P=0$

Generally Fourier and Laplace transformations can be used to solve generalised thermoelastic problems in the $x_1-x_3$ plane but these methods are time consuming and yield an approximate solution. The major problems of these approaches is that they produce discrete and truncation errors during inverse transformation. To eradicate this problem, Normal mode analysis approach has been adopted in which all components of oscillating system move at the same frequency and with a fixed phase relationship. We assumes the solution in the following form as harmonic waves propagate along the $x_1$ axis:

Here ${\varphi }^{*},C^{*},{\psi }^{*},{\Theta }^{*}$ represent unknown functions describing depth dependent amplitudes. As $(\chi =\eta c)$ represents angular frequency. In general both $\eta$ and c are considered as complex quantities. Therefore $c=\text{Real}\left(c\right)+i\text{Img}\left(c\right)$ assumed as complex constant such that Real $\left(c\right)\ge 0$ and $\exp \left(\text{Img}\right(c\left)t\right)$ provides the damping term in time t. Hence we consider that $\text{Img}\left(c\right)\le 0$ , If $\text{Img}\left(c\right)=0$ or completely vanish, then wave motion will be undamped and for $\text{Img}\left(c\right)<0$ , the wave motion will be damped.

Using Eq. (13) in (9)-(12), we have:

where $r={\displaystyle \frac{\rho }{\mu }},{\tau }_1=\frac{{\tau }_3}{{\tau }_5},{\tau }_2=\frac{{\tau }_4}{{\tau }_5},{\tau }_3=1-i\chi {\tau }_T,{\tau }_4=1-i\chi {\tau }_v,{\tau }_5=1-i\chi {\tau }_q-\frac{{\chi }^2{\tau }_q^2}{2}$

Eliminating ${\psi }^{*},{\varphi }^{*},{\Theta }^{*}$ and $C^{*}$ from (14) to (17). Hence we get

where,

$\begin{array}{ccc}\hfill a_1& =& 1+a^{*}{\eta }^2{a}_2=\lambda +2\mu a_3=\rho {\chi }^2{a}_4=i{K}_1{\eta }^2\chi {\tau }_1+K_1^{*}{\eta }^2{\tau }_2\hfill \\ \hfill a_5& =& \rho c_e{\chi }^2{a}_6=K_3^{*}{\tau }_2{a}_7=i{K}_3\chi {\tau }_1{a}_8=T_0{\beta }_3{\chi }^2\hfill \\ \hfill a_9& =& T_0{\beta }_1{\eta }^2{\chi }^2{a}_{10}=T_0(i\eta {\beta }_3-i\eta {\beta }_1){\chi }^2{a}_{11}=a{T}_0{\chi }^2{a}_{12}=D^{*}{\beta }_2\hfill \\ \hfill a_{13}& =& D^{*}a{a}_{14}=D^{*}b{a}_{15}=i\chi (1-\tau i\chi )a_{16}=-{\eta }^2+r{\chi }^2\hfill \\ \hfill a_{17}& =& a_6-a_7\hfill \end{array}$ $\begin{array}{ccc}\hfill K& =& {\displaystyle \frac{a_2\left[a_{14}(a_4+a_{16}{a}_{17}+a_{11}{a}_{13})\right]+(a_{17}-a^{*}{a}_5)[a_2(a_5-2{\eta }^2+a_{15})+a_3{a}_{14}+{\beta }_2{a}_{12}(2{\eta }^2+a_5)]}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{+{\beta }_1(a_{11}{a}_{12}+a_8{a}_{14})-{\beta }_2[a_{12}(a_{16}{a}_{17}+a_4)-a_8{a}_{13}]}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ \hfill L& =& {\displaystyle \frac{(a_{17}-a^{*}{a}_5)[(a_3{a}_{14}+a_2{a}_5)(a_{16}-{\eta }^2)+{\eta }^4(a_2{a}_{14}+{\beta }_2{a}_{12})]+(a_4+a_1{a}_5)[a_{16}({\beta }_1{a}_{12}+a_2{a}_{14})}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{+a_3{a}_{14}+2{\beta }_2{a}_{12}{\eta }^2]+(a_3+a_2{a}_{16})(a_{11}{a}_{13}+a_{15}{a}_{17})+(a_{16}-{\eta }^2)(2a^{*}{\beta }_2{a}_5{a}_{12}+{\beta }_1{a}_8{a}_{14})-2a_2(a_1}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{a_5{a}_{14}+a_{11}{a}_{13}+a_4{a}_{14}){\eta }^2-{\beta }_2[a_{13}(a_9-a_8)+a_4{a}_{12}{a}_{16}]+{\beta }_1(a_8{a}_{15}-a_9{a}_{14}-2a_{11}{a}_{12}{\eta }^2)}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ \hfill M& =& {\displaystyle \frac{(a_{17}-a^{*}{a}_5)[a_{16}(a_3-a_2{\eta }^2)(a_{14}{\eta }^2+a_{15})-{\beta }_2{a}_2{a}_{16}{\eta }^4]-(a_4+a_1{a}_5)[2a_2{a}_{14}{a}_{16}{\eta }^2+a_2{a}_5({\eta }^2}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{-a_{16})+a_3(a_4{a}_{16}+a_{15})]+a_{12}{\eta }^2(2a_{16}+{\eta }^2)({\beta }_1{a}_{11}-{\beta }_2{a}_4)+{\eta }^2(a_3+a_2{\eta }^2)(a_{11}{a}_{13}+a_4{a}_{14})+a_9}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{({\eta }^2-a_{16})({\beta }_1{a}_{14}-{\beta }_2{a}_{13})+a_1{a}_5{\eta }^2(1+{\eta }^2)(a_2{a}_{14}-{\beta }_2{a}_{15})-a_{13}{a}_{16}{\eta }^2(a_2{a}_{11}+{\beta }_2{a}_8)-{\beta }_1{a}_{15}(a_9}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{-a_8{a}_{16})+a_{16}(a_3{a}_{11}{a}_{13}+2{\beta }_2{a}_1{a}_5{a}_{12})}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ \hfill N& =& {\displaystyle \frac{a_{16}(a_4+a_1{a}_5)[{\eta }^4(a_2{a}_{14}-{\beta }_2{a}_{12})+a_3(a_{15}-a_{14}{\eta }^2)-a_2{a}_{15}{\eta }^2]+a_{11}{a}_{16}{\eta }^4(a_2{a}_{13}+{\beta }_1{a}_{12})+}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \\ & & {\displaystyle \frac{+{\beta }_1{a}_9{a}_{16}(a_{15}-a_{14}{\eta }^2)+a_3{a}_3{a}_{16}{\eta }^2({\beta }_2{a}_9-a_3{a}_{11})}{a_2{a}_{14}(a_{17}-a^{*}{a}_5)-{\beta }_2{a}_{12}{a}_{17}}}\hfill \end{array}$

Eq. (18) can be summarized as follows:

It gives the four roots $m_i^2$ for $i=1$ to 4 for the positive solutions of (19)

where, ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are the roots of resolvent cubic equation are given as:

$\begin{array}{ccc}\hfill {\lambda }_1& =& \sqrt{\frac{1}{3}[2u\text{sin}\left(v\right)-f]}{\lambda }_2=\sqrt{\frac{1}{3}[-f-u(\sqrt{3}\cos v+\sin v)]}\hfill \\ \hfill {\lambda }_3& =& \sqrt{\frac{1}{3}[-f+u(\sqrt{3}\cos v-\sin v)]}\hfill \end{array}$ Here, $\begin{array}{ccc}\hfill f& =& 2[{\displaystyle \frac{3K^2}{8}}-{\displaystyle \frac{3K^2}{4}}+L]\hfill \\ \hfill g& =& {[{\displaystyle \frac{3K^2}{8}}-{\displaystyle \frac{3K^2}{4}}+L]}^2-4[{\displaystyle \frac{K^4}{256}}-{\displaystyle \frac{K^4}{64}}+{\displaystyle \frac{K^{2}L}{16}}-{\displaystyle \frac{KM}{4}}+N]\hfill \\ \hfill h& =& -{[-{\displaystyle \frac{K^3}{16}}+{\displaystyle \frac{3K^2}{16}}-{\displaystyle \frac{KL}{2}}+M]}^2\hfill \\ \hfill u& =& \sqrt{f^2-3g}v={\displaystyle \frac{{\sin }^{-1}w}{3}}\text{and}w={\displaystyle \frac{2f^3-9fg+27h}{2u^3}}\hfill \end{array}$

When, $x_3\to \infty$, the solutions of Eqs. (9)-(12) can be written as follows:

where, $A_i,B_i,C_i$ and $D_i$ are constants for i=1,2,3,4

Here, the value of $b_i,p_i$ and $d_i$ given as:

and

Using the boundary conditions for stress free surface, the following relations has been obtained:

Here, $h\to 0$ signifies thermally insulated surface and $h\to \infty$ signifies isothermal surface.

Using the expression (20) for boundary conditions (21)-(24), the homogeneous system of linear equations are obtained in terms of $A_1,A_2$ , $A_3$ and $A_4$ as:

where, $\alpha ={\displaystyle \frac{-i{K}_3\eta {\tau }_3+K_3^{*}{\tau }_4}{-i\eta {\tau }_5}}$

The non trivial solutions (25)-(28) exist if its determinant is zero. Hence the frequency equation obtained as:

where, for all values $i=1,2,3,4$

Using specific values for isotropic medium, we can obtain different results from Eq. (29):

Case: 1 If we substitute ${\tau }_q\ne 0,{\tau }_v\ne 0$ and ${\tau }_T\ne 0$ in Eq. (29), then the frequency equation considered to be Rayleigh wave in Three Phase Lag (TPL) model.

Case: 2 If we substitute ${\tau }_q={\tau }_v={\tau }_T=0$ , $K_1^{*}\ne 0$ and $K_3^{*}\ne 0$ in Eq. (29), then the frequency equation reduced to Rayleigh wave in Green-Nagdhi (GN-III) model with dissipation of energy.

Case: 3 If we substitute $K_1^{*}=0$ , $K_3^{*}=0$ and ${\tau }_q={\tau }_v=0$ , ${\tau }_T=T_0>0$ in Eq. (29) and neglecting the term ${\tau }_q^2$ , then the frequency equation reduced to Rayleigh wave in Lord-Shulman (L-S) model.

Since $c^{-1}=B^{-1}+i{\chi }^{-1}F$ whereas $\eta =E+iF$ (here $E=\frac{\chi }{B}$ ). In Eq. (13), the exponential term is replaced by $iE(x-Bt)-Fx$ . Specific loss represents internal resistance of thermoelastic medium that is represented by $W=4\pi |\frac{F}{E}|$ . Penetration depth represents the depth by which a wave can penetrates the thermoelastic material and interpret the depreciation of waves inside the material. It is represented by $Z=\frac{1}{F}$

Case: 4 If we assume that $c_{11}=c_{33}=2\mu +\lambda$ and $c_{13}=\lambda ,c_{44}=\mu ,{\tau }_q={\tau }_T=0,K_3^{*}=K_1^{*}=0,{\beta }_1={\beta }_3=\beta ,K_3=K_1=K$ in the Eq. (29), we obtain the equation as:

where ${\zeta }_1^2=1-\frac{m_1^2}{k^2},{\zeta }_2^2=1-\frac{m_2^2}{k^2},{\zeta }_3^2=1-\frac{{\Delta }^2}{k^2}$ , ${\Delta }^2=\frac{k^2{c}^2}{c_2^2}$ , $c_1^2=\frac{\lambda +2\mu }{\rho }$ and $c_2^2=\frac{\mu }{\rho }$ and $m_1^2$ and $m_2^2$ are the roots of biquadrate equation

As a result, Eq. (30) represents the Rayleigh wave's frequency equation in thermally insulated isotropic half space.

Case: 5 Ignoring the coupling between temperature (${\beta }_1=0$ ) and strain field in Eq. (29), Rayleigh wave propagating in orthotropic half space have the frequency equation as follows (agrees with the conclusion of Abd-Alla et al. [10]:

By considering the value of $c_{11}=2\mu +\lambda =c_{33},c_{44}=\mu ,c_{13}=\lambda$ in Eq. (31), the above equation transformed into:

As a result, Eq. (32) represents Rayleigh wave's frequency equation in isotropic half space. Eq. (32) can be rewritten as:

The solution c = 0 of this equation reflects the static condition of Rayleigh wave. At particular condition of Poisson relation ($\lambda =\mu$ ), then $c_1=\sqrt{3}{c}_2$

Eq. (33) becomes:

Eq. (34) has real roots ${\left({\displaystyle \frac{c}{c_2}}\right)}^2=4,2+(2/\sqrt{3}),2-(2/\sqrt{3})$ , The first two roots do not fulfil the criterion for the reduction of amplitude by depth, i.e. condition $c<c_2$ . The last root yields the velocity of surface wave as:

$c=0.919c_2$ satisfy the frequency Eq. (32) of Rayleigh wave in isotropic half space. Hence, Eq. (35) depicts that there is existence of Rayleigh surface waves at stress free surface with phase velocity $c<c_2<c_1$ and polarised in the vertical plane passing through to the direction of propagation.

Relevant cobalt material parameter Biswas and Mukhopadahyay [14] Table 1 has been considered to demonstrate the variation of various parameter like non-dimensional speed, attenuation coefficient, penetration depth and specific loss in Rayleigh wave propagation.