Propagation and reflection of thermoelastic wave in a rotating nonlocal fractional order porous medium under Hall current influence

This investigation relates to the research on Hall current on propagation and reflection of elastic waves through non-local fractional-order thermoelastic rotating medium with voids. The system is split up into longitudinal and transverse components using the Helmholtz vector rule. It is observed that, through the frequency dispersion relation four coupled quasi-waves exist in the medium. The rotating solid modifies the nature of purely longitudinal and transverse waves toward the quasi-type waves. All the propagating waves are dispersive as they depend upon angular frequency. The quasi-longitudinal wave qP and quasi-transverse wave qSV faces cut-off frequencies. The nonlocal parameter affect all the waves except the quasi void wave. Analytically, the reflection coefficients of the wave are computed using suitable boundary conditions. MATLAB software is used to perform numerical computations for a chosen solid material. The amplitude ratios and the speed of propagation of the wave are plotted graphically for rotational frequency, nonlocal, fractional order, and Hall current parameter. The significant effect of the physical parameters on the computed results has been observed. The cut-off frequency of the waves is also presented graphically. The energy conservation law is proved in the form of energy ratios. The earlier findings in the literature are obtained as special case in the absence of rotation, Hall current parameter and porous voids.

Many fields, including seismology, geophysics, earthquake, engineering etc. have shown interest in the study of reflection and refraction of plane waves.Research on these phenomena is largely extremely important for provision of crucial details about the internal make-up of the earth's structure and are of huge importance in fields such as acoustic and mining considering the actual uses and theoretical research.Surface reflection and energy partioning have been studied for elastic media 1-4 .Biot's theory has been the foundation for numerous investigations of transmission of waves in a poroelastic half-space that is saturated.
Yang and Sato 5,6 used Biot's theory to check how an earthquake affect a place with saturated soil.Eringen 7 studied the nonlocal linear elasticity theory and plane waves dispersion.Thermoelasticity theory was first put forth by Biot 8 which resolves the inconsistency in the uncouple theory, which describe that the temperature is unaffected by elastic charges.The idea of nonlocal theory was given by Eringen 9,10 and Eringen and Edelen.The core idea of Eringen's nonlocal theory is that every material in the continuum gets energies that are dispersed due to its extensive correspondence with every other particle in continuum domain.Generalised thermoelastic heat model was added by Sarkar and Tomar 11 to study the impact of voids and nonlocal parameters on plane wave propagation.
In order to examine a few physical issues involving real materials as polymer, rocks, etc., the concept of fractional calculus is important.Fractional order derivatives can be used to better discuss the chemical characteristics of such a substance.Numerous real-world fields, such as quantum field theory, nuclear physics, control engineering, electromagnetic, chemistry, signal processing, quantum mechanics, astronomy, etc., can benefit from

Basic relation
In this section, some basic constitutive relations are expressed as follows the non-local stress-strain relation is expressed as follows; 49 where the strain tensor be denoted by where e ij is known as strain, u ij is called displacement component.∇ 2 is called the Laplacian operator, α is fractional order parameter, δ ij is Kronecker delta function,T = T ′ − T 0 , T is used to indicate the body's natural temperature such that | T T 0 | ≤ 1 , the Lame's constants are and µ , isothermal compressibility is denoted by K T and T ′ represents the material's amplitude temperature, and e 1 = ǫ 0 a cl where ǫ 0 is a material constant and rep- resents the length of the internal properties.The thermoelastic material's energy expression for linear theory is denoted by following relation where q i,i = −(ρC e T + ( αT 0 K T )e ij ) is the expression for motion without body forces and C e represents the specific heat.When there are no body forces, the equation of motion for a nonlocal isotropic thermoelastic solid with porous voids can be expressed as In the thermoelastic material's nonlocal heat transfer is where τ 0 is the relaxation time and K is the thermal conductivity, ρ is the density.The fractional order parameter is defined as follows 14 ; (1) (2) e ij = 1 2 (u i,j + u j,i ), where Ŵ is constant and represents the gamma function so that 0 ≤α≤1 .Using the universal Ohm's law and taking into account the impact of Hall current on a medium with fixed conductivity σ 0 24 the current density is expressed as follows; Equation of motion in rotating frame of reference is obtained by putting Eqs.(1-3) into Eqs.( 4) and (5)   Similarly, equation of voids is taken from 39

Special case
When α → 0 , relation (5) reducing to the classical theory of coupled thermoelasticity and when integer order α → 1 , the Lord and Shulman thermoelasticity hypothesis can be obtained from Eq. ( 5).

Mathematical modeling of the problem
We consider the semiconducting nanostructure medium with finite conductivity and fractional-order three phase lag model (TPL) thermoelastic model with constant temperature T 0 as initial thermo-elasticity, a magnetic field that is initially applied, so that H0 = (0, H0 , 0) , ū0 = ( ū1 , 0, ū2 ) along the y direction.Furthermore, the plane of propagation for the waves is chozen as the xz-plane.The displacement vector ū has components , it is also sup- posed that Ē = 0 .The current densities J 1 and J 3 using Eq. ( 8) are computed as: Equation (9-10) are transformed to following component form.
Since Eqs. ( 14) and ( 15) are coupled equations therefore, we apply following decomposition rule of Helmholtz is employed to uncouple the system where scalar potential φ(x, z, t) yields an irrotational vector field and vector potential ψ(x, z, t) produces a solenoid vector field.The model can be made non-dimensional with the aid of the following set of quantities: www.nature.com/scientificreports/Making use of relation (18), the governing dimensionless system (14-17) in terms of potential functions becomes (for simplicity omitting the bar sign) as follows: It can be observed that set of Eqs.(20-23) are coupled in four functions φ, ψ, T, ϕ .In the next section, the solu- tion of the coupled system will be calculated.

Propagation of waves
The following harmonic wave solution can be used to satisfy the wave equations: Where φ, ψ, T and φ are constant amplitudes of propagating waves, ω is the angular frequency, θ is angle of propa- gation vector, ξ represents the wave number and the vector constant, where r = (xi + yj + zk) is the position vec- tor.By substituting relation (24), Eqs.(20-23) are transformed to following algebraic system of linear equation.www.nature.com/scientificreports/For unknown φ, ψ, T and φ the non-trivial solution to homogenous linear system (25-28) exist if the coefficient matrix's determinant vanishes, where For the propagation of plane waves in nonlocal thermoelastic solid media, Equation (30) provides the dispersion relation, which corresponds to speed of propagations.The relation (30) gives eight roots of ξ such that ±ξ 1 , ±ξ 2 , ±ξ 3 , ±ξ 4 .In the medium, there are four waves that corresponds to four positive values of +ξ , i = 1, 2, 3, 4 .Let these waves are quasi longitudinal wave (qP-wave), thermal wave (qT-wave), quasi transverse wave (qSV-wave) and void wave (qV-wave).The expression Q 1 , Q 2 , Q 3 , Q 4 , Q 5 are complex valued, therefore, speed of the wave is a complex quantity such that ξ = ξ r + iξ i , where ξ = ξ r is real part and ξ = ξ i is imaginary part.Moreover, its real part ( ξ = ξ r ) gives speed of propagation and the imaginary part ( ξ = ξ i ) gives attenu- ation coefficient.The rotation in the solid disturbs the isotropy and makes it like anisotropic.The anisotropic behavior of the solid converts the purely longitudinal and transverse type of the waves into quasi-longitudinal and quasi-transverse waves.

Special case of the model
In the absence of the rotation, Hall current and voids i.e. � = m = α ′ = υ = τ ′ = γ = χ = 0 then Eqs. ( 9) and (10) are transformed as follows: The non-trivial solution of equations yields the same quadratic polynomial as obtained by Subani and Aangeeta 14 .Furthermore, it can be seen that Eq. ( 21) becomes un-coupled in potential ψ corresponding to SV-wave.Its speed of propagation is computed as follows; Equation ( 33) is same as Eq. ( 32) obtained by Subani and Aangeeta 14 .The SV wave's speed is only dependent upon the non-local parameter and angular frequency.The speed becomes non-dispersive in the local medium as it does not depend upon the wave frequency.

Reflection of qP wave
Let quasi-longitudinal wave (qP-wave) be the incident wave at the free boundary resulting in generation of four reflected waves named as qP, qT, qSV, and qV-wave as shown in Fig.
(1).The incident wave makes an angle of incidence (θ) with respect to the normal.The appropriate potentials of reflected and incident waves are taken into consideration as Where coupling parameters are defined as follows; Now we employ the following boundary conditions which are necessary to solve the suggested problem.Because there is no stress at the boundary surface at z = 0 and thermally insulated, we get, www.nature.com/scientificreports/ The boundary conditions (39) are transformed to the following system of algebraic equation with the help of Eqs.(34-37), the following system of algebraic equation is obtained

Energy conservation
In this section, we discuss the energy conservation of the system.The computed results can be validated in the context of energy conservation.Following 50 , the energy transmission rate per unit area is given by Defining the quantities E 1 , E 2 , E 3 and E 4 as energy ratio corresponding to reflected qP, qT, qSV and qV wave to the incident wave.These energy ratios are computed as follows: Where A 0 = G i and using the values of c 1i and c 2i the expression of energy becomes

Graphical discussion of the analytical results
Now the computed results are studied graphically under certain physical parameters.The following values of copper like material's elastic constants are taken from 14 for this purpose.In the following paragraphs, the impact of different physical parameters including non-local parameter, rotational frequency, fractional order, and Hall current parameter on the speed of wave propagation and their corresponding amplitude ratios is studied.

Effect of nonlocal parameter on propagation speed
Figure 2a,d show the variation of speed of the quasi longitudinal wave (qP), quasi thermal wave (qT), quasi transverse wave (qSV) and quasi void wave (qV) versus angular frequency ω for different values of the nonlocal parameter.The chosen nonlocal parameters are 0, 0.5, and 0.9.Keeping in view its physical significance that it is in fact a characteristic length of the solid, the value of e 1 = 0 corresponds to local medium (classical medium), and e 1 = 0 corresponds to the nonlocal nature of the solid.Figure 2a shows the effect of e 1 on |v 1 | by taking α = 0.4, m = 0.06 .And furthermore, a solid is rotating with = 0.007 .Here, we discuss the speed of the qP wave.For e 1 = 0.0 , the curve is increasing slowly with increasing values of angular frequency, for e 1 → 0.5 curve (39) change in fractional volume of the void is zero, i.e. ∂ϕ ∂z = 0 at z = 0.
Vol.:(0123456789) www.nature.com/scientificreports/ is firstly increasing slowly and then decreasing sy with respect to ω (when 0 < ω < 2.0 ).When ω → 2.0 , |v 1 | is zero.For ω > 2.0 curve shows increasing behavior.And shows the cut-off frequency at ω = 2.0 .For the nonlocal parameter e 1 = 0.9 , the curve is increasing and then decreasing, and it has been observed that when ω → 1.1 , |v 1 | is zero.Afterwards, for ω > 1.1 the curve is exponentially increasing.And shows the cut-off frequency at ω = 1.1 .So the speed of the qP wave is decreasing with increasing values of the nonlocal parameter.Figure 2b graphically displays the speed of quasi thermal wave (qT) propagation.It can be seen that speed is exponentially increasing with increasing values of angular frequency.The speed of the quasi thermal wave (qT) is decreasing with increasing values of non-local parameter.The speed is maximum for lower values of angular frequency, and it has maximum values at ω = 3 .Figure 2c shows the effect of e 1 on |v 3 | by taking α = 0.03, m = 0.6 .And furthermore, a solid is rotating with = 0.04 .For e 1 = 0.0 , the curve is increasing with increasing values of angular frequency and then shows a constant line with respect to ω ( ω > 0.8 ).It means that when ω > 0.8 , the speed of the qSV wave is independent of angular frequency.For e 1 = 0.5 , the curve is firstly increasing and then decreasing with respect to ω (when 0 < ω < 2.0 ).When ω → 2.0 , |v 3 | is zero.For ω > 2.0 , the curve shows increasing behavior.And shows cut-off frequency when ω → 2.0 .For the nonlocal parameter e 1 = 0.9 , the curve is increasing, and it has been observed that when ω → 1.0 , |v 3 | is zero.Afterwards, for ω > 1.0 , the curve is exponentially increasing.And shows the cut-off frequency when ω → 2.0 .So the speed of the qSV wave is decreasing with increasing values of the nonlocal parameter.www.nature.com/scientificreports/ The effect of e 1 on |v 4 | is studied by taking α = 0.04, m = 0.9 in Fig. 2d.And furthermore, a solid is rotating with = 1 .The speed of the quasi void wave (qV-wave) under different values of the nonlocal parameter is increasing when 0 < ω < 1.4 and decreasing when ω > 1.4 .A spike has been observed at ω = 1.4 .Physically, spikes mean that the magnitude is reaching infinity.The amplitude of a wave is too high means approaching to infinity.The speed of the quasi void wave is independent of the non-local parameter.

Effect of rotational frequency on propagation speed
Figure 3a,d, show the variation of speed for rotating and non-rotating medium versus angular frequency.Where = 0 corresponds to non-rotating medium and = 0 corresponds to rotating medium.The chosen values of are 0, 0.5, 1.0.Figure 3a illustrates the quasi-longitudinal wave's (qP) speed propagation and indicates the effect of on |v 1 | is studied by taking the α = 0.05, m = 0.05, e 1 = 0.5 .The speed of the qP wave is decreasing or moving down with increasing values of rotational frequency when 0 < ω < 2.0 .For ω > 2.0 , speed is increasing with increasing values of rotational frequency.Curves are first increasing with respect to angular frequency and then decreasing.It has been observed that when ω → 2.0 , |v 1 | is zero.After that, for ω > 2.0 , curves are increasing exponentially.It means that the cut-off frequency in the range of 0 to 2.0 due to rotation speed decreases, and reverse effect detected beyond the cut-off frequency.Figure 3b graphically represents the speed of quasi thermal wave (qT) propagation.It can be seen that the speed is exponentially increasing with increasing values of angular frequency.The speed of the quasi thermal wave (qT) is decreasing with increasing values of rotational frequency.The speed is maximum for lower values of rotational frequency.Figure 3c illustrates the (qSV) wave speed propagation and shows the effect of on |v 3 | by taking the α = 0.09, m = 0.05, e 1 = 0.5 .The speed of the qSV wave is decreasing under different values of when the angular frequency is between 0 and almost 2.0, and then increasing under different values of when the angular frequency exceeds 2.0.It has been observed that at ω = 2.0 , |v 3 | is closed to 0. It means that the cut-off frequency in the range of 0 to 2.0 due to rotation speed decreases, but beyond the cut-off frequency, rotation increases the speed of propagation.Graphical representation of the quasi void wave (qV-wave) propagation is shown in Fig. 3d.It can be seen that speed is increasing with respect to angular frequency in rotating and non-rotating medium.The speed of the quasi void wave (V-wave) is decreasing with increasing values of rotational frequency.The speed is maximum for lower values of rotational frequency.Since the speed of each wave depends on the angular frequency, we concluded that all waves are dispersive.

Effect of fractional order on propagation speed
In Fig. 4a,d, the propagation speed of the qP, qT, qSV, qV waves for different values of fractional order versus angular frequency is shown.The fractional order parameter's value must between 0 and 1.When we take α = 1 , fractional order study becomes integer order study.The values we choose here are 0.2, 0.4, and 0.6.Figure 4a illustrates the quasi-longitudinal wave's (qP) speed propagation and shows the effect of α on |v 1 | by taking the m = 0.08, e 1 = 0.4, = 0.008 .The speed of the qP wave is increasing or moving up with increasing values of fractional order.Curves are first increasing with respect to angular frequency and then decreasing.It has been observed that at ω = 2.5 , |v 1 | is zero.After that, for ω > 2.5 , curves are increasing exponentially, and the speed of the qP wave is increasing with respect to angular frequency.It means that the cut-off frequency in the range of 0 to 2.5 due to rotation speed increases, and same effect is beyond the cut-off frequency.Figure 4b graphically displays the speed of quasi thermal wave (qT) propagation.It can be seen that the speed is increasing with respect to the angular frequency.The speed of the qT wave is decreasing with increasing values of fractional order.The speed is maximum for lower values of fractional order.The speed of the qSV wave propagation is displayed in Fig. 4c.It can be seen that the speed of the qSV wave is increasing with respect to the angular frequency.Since the speed is frequency dependent.So it is called a dispersive wave.The speed of the qSV wave is decreasing when fractional order increasing from 0.2 to 0.4 and increasing when fractional order exceeds 0.4.
Figure 4d represents the quasi void wave's (qV-wave) speed propagation, and the effect of α on |v 4 | is studied by taking the m = 0.5, = 0.2, e 1 = 0.6 .It can be seen that for some frequencies, it shows increasing and decreasing behavior and spikes have been observed at some particular frequencies.Physically, spikes mean that magnitude is showing to infinity.The highest amplitude has been observed at α = 0.4 .For α = 0.2 , a spike has been observed at ω = 0.6 .For α = 0.4 , a spike has been observed at ω = 0.4 .For α = 0.6 , a spike has been observed at ω = 1.1 .This shows that spikes shift to the origin when fractional order increases from 0.2 to 0.4 and move away from the origin when fractional order exceeds 0.4.

Effect of Hall current on propagation speed
Figure 5a,d illustrate the propagation speed of the qP, qT, qSV and qV wave for different values of Hall current parameter versus angular frequency.The values that are selected for Hall current are m = 0.0, 0.3, 0.6 .Figure 5a illustrates the quasi longitudinal wave's (qP) speed propagation and shows the effect of "m ′′ on |v 1 | by taking the α = 0.06, � = 0.02, e 1 = 0.6 .The speed of the qP wave is decreasing with increasing values of the Hall current parameter.And observed cut-off frequency at ω = 1.6 .Curves are first increasing with respect to the angular fre- quency and then decreasing.It has been observed that when ω → 1.6 , |v 1 | is zero.After that, for ω > 1.6 , curves are increasing exponentially.It means that the cut-off frequency in the range of 0 to 1. 6 Hall current parameter decreases the speed of the qP wave and has same effect beyond this range.The speed of quasi thermal wave (qT) propagation is shown in Fig. 5b.It can be seen that the speed is increasing with respect to the angular frequency.The speed of the qT wave is decreasing with increasing values of the Hall current parameter.The speed reaches its maximum at m = 0.0 .Figure 5c indicates the quasi-transverse wave's (qSV) speed propagation and shows the effect of "m ′′ on |v 3 | by taking the α = 0.06, � = 0.2, e 1 = 0.04 .The speed of the qSV wave is decreasing or moving down with increasing values of the Hall current parameter (0 < ω < 1.3) .It shows the cut-off frequency at ω = 1.3 .For ω > 1.3 , speed is increasing with increasing values of the Hall current parameter.Curves are first increasing with respect to the angular frequency and then decreasing.After that, for ω > 1.3 , curves are increasing exponentially.It means that the cut-off frequency in the range of 0 to 1.3 due to the Hall current parameter speed decreases, and the opposite effect is beyond the cut-off frequency.The speed of propagation of the quasi void wave (qV-wave) is represented in Fig. 5d, which shows that the speed is increasing with respect to the angular frequency.So it is called the dispersive wave.The speed of the qV wave is decreasing with increasing values of the Hall current parameter.

Effect of nonlocal parameter on amplitude ratios
Figure 6a,d show the variation of amplitude ratios of the (qP), (qT), (qSV), and qV waves versus angle of incidence 0 • ≤ θ ≤ 90 • for different values of nonlocal parameter.The non-local parameter's values are chosen to be 0.0, 0.1, and 0.2 which compares the local and non-local parameters as well.The e 1 = 0 corresponds to local theory (classical theory), and e 1 = 0 corresponds to nonlocal theory.In Fig. 6a the impact of non-local parameter on amplitude ratios is shown.To check the effect, some values of the parameter have been fixed.The amplitude ratio of the qP wave is decreasing with increasing nonlocal parameter's values.It means that the amplitude ratio is dependent on a nonlocal parameter.Amplitude ratio is increasing with respect to the θ (0 • ≤ θ ≤ 25 www.nature.com/scientificreports/decreasing with respect to the θ when (θ ≥ 25 • ) till grazing incidence.The amplitude ratio has a peak value at θ = 25 • .Finally, at θ = π 2 , the amplitude ratios decrease to zero. Figure 6b represents the variation of α on |z 2 | by taking fixed values of � = 0.9, m = 0.5, e 1 = 0.08, ω = 655 .It indicates that with increasing non-local parameter's values, the amplitude ratio is increasing.It means that the amplitude ratio of the qT wave is dependent on local and non-local theories.We also noticed that for θ = 45 • , |z 2 | = 0 .Therefore, θ = 45 • acts as a critical angle for the existence of the qT wave.The same effect of non-local theory on |z 2 | is for θ ≥ 45 • till θ = π 2 .Figure 6c demonstrates the effect of the non-local parameter on |z 3 | .To check the effect, some values of the parameter have been fixed.The amplitude ratio of the qSV wave is decreasing with increasing nonlocal parameter's values.It means that the amplitude ratio is dependent on nonlocal parameter.The amplitude ratio is increasing with respect to the θ (0 • ≤ θ ≤ 10 • ) and decreasing with respect to θ when (θ ≥ 10 • ) till grazing incidence.The amplitude ratio has its peak value at θ = 10 • .One can observe from Fig. 6d that with increasing values of the non-local parameter, the amplitude ratio of quasi void wave is increasing.The amplitude ratio of the qV Wave is decreasing with respect to the angle of incidence ( θ ).Finally, at grazing incidence angles, the amplitude ratio decreases to zero.The values of rotational frequency are chosen to be 0.0, 0.4, and 0.8.This in fact, gives a comparison between rotating and non-rotating medium.The = 0 cor- responds to a non-rotating medium, and = 0 corresponds to rotating medium.One can see from Fig. 7a the effect of on |z 1 | by taking fixed values of parameters.The amplitude ratio of the qP wave is increasing under increasing values of rotational frequency (0 • ≤ θ ≤ 20 • ) and decreasing when θ ≥ 20 • .Curves are decreasing with respect to the angle of incidence (0 • ≤ θ ≤ 20 • ) and increasing with respect to θ (20 • ≤ θ ≤ 60 • ) and again slightly decreasing when θ ≥ 60 • till grazing incidence.Figure 7b indicates the variation of on |z 2 | by taking fixed values of α = 0.05, m = 0.07, e 1 = 0.1, ω = 655 .It indicates that with increasing values of rotational fre- quency, the amplitude ratio is increasing.It means that the amplitude ratio of the qT is dependent on rotating and non-rotating mediums.We also noticed that for θ = π 4 , |z 2 | = 0 .Therefore, θ = 45 • acts as a critical angle for the existence of the qT wave.The same effect of the rotational frequency on |z 2 | is for θ ≥ 45 • till θ = π 2 .The effect of on |z 3 | is studied by taking fixed values of α = 0.05, m = 0.07, e 1 = 0.1, ω = 655 in Fig. 7c.It indicates that with increasing values of rotational frequency, the amplitude ratio is increasing.It means that the amplitude ratio is dependent on the rotating and non-rotating mediums.We also noticed that for θ = 45 of on the amplitude ratios of the quasi void wave.The amplitude ratio of the qV wave is increasing when the rotational frequency is increasing.We can see that the amplitude ratio is decreasing with increasing values of θ.

Effect of fractional order on amplitude ratios
In Fig. 8a,d, the variation of amplitude ratios of the (qP), (qT), (qSV) and qV waves verses angle of incidence 0 • ≤ θ ≤ 90 • under different values of fractional order parameter is shown.Different values of fractional order are selected to be 0.5, 0.6, and 0.7.Keeping in view that for α = 1 , the current study may produce the result of integer order wave theory.For fractional order investigation, α should be chosen as 0 < α < 1 .One can see from Fig. 8a the effect of α |z 1 | by taking fixed values of various parameters.The amplitude ratio of the qP wave is decreasing under increasing values of fractional order (0 • ≤ θ ≤ 15 • ) and increasing when θ ≥ 15 • .Curves are decreasing with respect to the angle of incidence (0 • ≤ θ ≤ 15 • ) and increasing with respect to θ (15 • ≤ θ ≤ 60 • ) and show a constant line when θ ≥ 60 • till grazing incidence.
The amplitude ratio of the qT wave is graphically represented in Fig. 8b, which shows that z 2 decreases for increasing values of fractional order (0 • ≤ θ ≤ 10 • ) afterwards, increasing with increasing values of α (θ ≥ 10 • ) till grazing angle of incidence.Amplitude ratio is increasing with respect to θ (0 • ≤ θ ≤ 10 • ) and decreasing with respect to θ when (θ ≥ 10 • ) till θ = π 2 .The amplitude ratio has its peak value at θ = 10 • .The effect of the fractional order on amplitude ratios is shown in Fig. 8c.To check the effect, some parameter's values have been fixed.The amplitude ratio of the qSV wave is decreasing with increasing values of the fractional order parameter.It means that the amplitude ratio is dependent on the fractional order parameter.Amplitude ratio is increasing with respect to θ (0 • ≤ θ ≤ 10 • ) and decreasing with respect to θ when (θ ≥ 10 • ) till θ = π 2 .The amplitude ratio has its peak value at θ = 10 • .One can observe from Fig. 8d the impact of α on the amplitude ratio of the quasi void wave.The amplitude ratio of the qV wave is increasing when the fractional order is increasing.We can see that amplitude ratio is decreasing with increasing values of θ .Finally, at grazing incidence angles, the amplitude ratio decreases to zero.again when θ ≥ 40 • till grazing incidence.One can observe from Fig. 9b the Hall current's effect on |z 2 | by taking α = 0.003, e 1 = 0.02, ω = 655 .Further, it is supposed that the solid is rotating with = 0.007 .For different values of the Hall current parameter, the curve is first increasing with respect to θ (0 • ≤ θ ≤ 45 • ) and afterwards, it is decreasing for 45 • ≤ θ ≤ 80 • beyond the θ = 80 • smooth curve has been observed.We also observed that the curves are moving down with increasing values of the Hall current parameter.It is significant to note that the amplitude ratio of the qT wave is decreasing with increasing Hall current parameter's values.It indicates that the amplitude ratio is dependent on the Hall current parameter.We can see the variation of "m ′′ on |z 3 | as shown in Fig. 9c.It can be seen that when the Hall current parameter's values increases the amplitude ratio of the qSV wave decreases (0 • ≤ θ ≤ 10 • ) afterwards, increasing with increasing values of "m ′′ (θ ≥ 10 • ) till the grazing angle of incidence.Amplitude ratio is increasing with respect to θ (0 • ≤ θ ≤ 10 • ) and decreasing with respect to θ when (θ ≥ 10 • ) till the grazing angle of incidence.Amplitude ratio has its peak value at θ = 10 • .Finally, at θ = π 2 , the amplitude ratio decreases to zero.The variation of "m ′′ on amplitude ratio of the quasi void wave is shown in Fig. 9d.The amplitude ratio of the qV wave is decreasing when the Hall current parameter is increasing.

Figure 1 .
Figure 1.Reflection view of the waves at the free boundary of the solid.

Figure 2 .
Figure 2. Speed of propagation versus ω for nonlocal parameter.

Figure 3 .
Figure 3. Speed of propagation versus ω for rotational frequency.

Figure 4 .
Figure 4. Speed of propagation versus ω for fractional order parameter.

Figure 5 .
Figure7a,dshow the variation of amplitude ratios of the (qP), (qT), (qSV) and qV waves versus angle of incidence 0 • ≤ θ ≤ 90 • under different values of rotational frequency.The values of rotational frequency are chosen to be 0.0, 0.4, and 0.8.This in fact, gives a comparison between rotating and non-rotating medium.The = 0 cor- responds to a non-rotating medium, and = 0 corresponds to rotating medium.One can see from Fig.7athe effect of on |z 1 | by taking fixed values of parameters.The amplitude ratio of the qP wave is increasing under increasing values of rotational frequency (0 • ≤ θ ≤ 20 • ) and decreasing when θ ≥ 20 • .Curves are decreasing with respect to the angle of incidence (0 • ≤ θ ≤ 20 • ) and increasing with respect to θ (20 • ≤ θ ≤ 60 • ) and again slightly decreasing when θ ≥ 60 • till grazing incidence.Figure7bindicates the variation of on |z 2 | by taking fixed values of α = 0.05, m = 0.07, e 1 = 0.1, ω = 655 .It indicates that with increasing values of rotational fre- quency, the amplitude ratio is increasing.It means that the amplitude ratio of the qT is dependent on rotating and non-rotating mediums.We also noticed that for θ = π 4 , |z 2 | = 0 .Therefore, θ = 45 • acts as a critical angle for the existence of the qT wave.The same effect of the rotational frequency on|z 2 | is for θ ≥ 45 • till θ = π2 .The effect of on |z 3 | is studied by taking fixed values of α = 0.05, m = 0.07, e 1 = 0.1, ω = 655 in Fig.7c.It indicates that with increasing values of rotational frequency, the amplitude ratio is increasing.It means that the amplitude ratio is dependent on the rotating and non-rotating mediums.We also noticed that for θ = 45 • , |z 3 | = 0 .Therefore, θ = 45 • act as a critical angle for the existence of the qSV wave.The same effect of the rota- tional frequency on |z 3 | is for θ ≥ 45 • till grazing angle of incidence.One can observe from Fig.7dthe impact

Figure 6 .
Figure 6.Amplitude ratios versus angle of incidence θ under non-local parameter.

Figure
Figure9a,d show the effect of amplitude ratios of (qP), (qT), (qSV) and qV-waves verses angle of incidence 0 • ≤ θ ≤ 90 • under different values of parameter(Hall current).Different values of Hall current are chosen to be m = 0.3, 0.6, 0.9 .One can see from Fig.9athe effect of "m ′′ on |z 1 | by taking the fixed values of various param- eters.The amplitude ratio of the qP wave is increasing under increasing values of the Hall current parameter (0 • ≤ θ ≤ 10 • ) , and reverse effect detected when θ ≥ 10 • .Curves are decreasing with respect to the angle of incidence (0 • ≤ θ ≤ 10 • ) and increasing with respect to θ (10 • ≤ θ ≤ 40 • ) and afterwards slightly decreasing

Figure 7 .
Figure 7. Amplitude ratios versus angle of incidence θ in rotating and non-rotating medium.

Figure 8 .
Figure 8. Amplitude ratios versus angle of incidence θ for fractional order.