Study of reflection and transmission of plane waves at thermoelastic-diffusive solid/liquid interface

The paper concentrates on the study of reflection and transmission characteristics of waves at the interface of a thermoelasticdiffusive solid half-space underlying an inviscid liquid. The analytic expressions for amplitude ratios, reflection and transmission coefficients, in terms of incident angles and material parameters have been obtained for quasi-longitudinal ( qP ) and quasi-transverse ( qSV ) wave incidence. The normal and grazing incidence cases have also been derived and discussed. The total reflection phenomenon has also been discussed. The energy law has been shown to be obeyed by the incidence, reflected and transmitted waves by deriving energy equation and its simulation. The numerical computations of reflection and transmission coefficients have been carried out for copper material half-space in contact with water at the interface by using MATLAB software. The computer simulated results have been presented graphically in order to bring out clear comparison of various situations.


INTRODUCTION
Diffusion is defined as a movement of the particles from a region of high concentration to the low concentration. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. Today, the study of thermal diffusion phenomenon got a great deal of interest due to its wide ranging applications in geophysics and industrial applications. The concentration obeys the famous Fick's law, which does not take into consideration the mutual interaction between the solvent and the solute or the effect of temperature on this interaction. However, there is a certain degree of coupling with temperature and temperature gradients as temperature speeds up the diffusion process. The thermo-diffusion in elastic solids is due to coupling of temperature, mass diffusion and strain fields addition to heat and mass exchange with the environment. Nowacki ( , 1976 developed the theory of coupled thermoelastic diffusion. The recent development of generalized theory of thermoelastic diffusion by Sherief et al. (2004) allows the finite speed of propagation of thermo-diffusive waves. Sharma (2007) discussed the propagation of plane harmonic waves in generalized thermo-elasto-diffusive solid. Kumar et al. (2014) studied the propagation of Rayleigh waves in a homogeneous isotropic micro-stretch generalized thermoelastic diffusion solid half-space.
The phenomenon of wave reflection and refraction is a fundamental topic in many fields such as seismology, geophysics, earthquake engineering, non-destructive evaluation, etc. Jeffreys (1930) and Gutenberg (1944) considered the reflection of elastic plane waves at a solid half space. Knott (1899) derived the general equations for reflection and refraction of waves at plane boundaries. Singh (2005) discussed the reflection of P and SV waves from the free surface of an elastic solid with generalized thermo-diffusion. Singh (2006) studied the reflection of SV waves from the free surface of an elastic solid in generalized thermo-elastic diffusion. Sharma and Sharma (2010) investigated the reflection characteristics of acousto-diffusive waves from the surface of a semiconductor half-space which is subjected to stress free, isoconcentrated and stress free, impermeable conditions. Kumar and Kansal (2012) considered the reflection and refraction of plane waves at the interface of an elastic solid half-space and a thermoelastic diffusive solid halfspace. Bijarnia and Singh (2012) investigated the propagation of plane waves in a transversely isotropic generalized thermoelastic solid half-space with diffusion and they also studied the reflection of these plane waves from a thermally insulated free surface. Kumar et al. (2013) studied the reflection and refraction phenomenon due to plane wave's incident obliquely at a plane interface between uniform elastic solid half-space and microstretch thermoelastic diffusion solid half-space. Wu, et al. (1990) investigated the reflection and transmission of elastic waves from the boundary of a fluid-saturated porous solid. Lin, et al. (2005) studied the reflection of plane waves in a poroelastic half-space saturated with inviscid fluid. Deresiewicz (1960) studied the reflection of plane waves from a plane stress-free boundary in the coupled theory of thermoelasticity. Sharma, et al. (2003) studied the reflection of generalized thermoelastic from the boundary of a half space. Sinha and Sinha (1974) and Elsibai (1996, 1997) investigated the reflection of thermoelastic waves from the free surface of a solid half-space and at the interface of two semiinfinite media in welded contact, in the context of generalized thermoelasticity. Das, et al. (2008) explored the reflection of generalized thermoelastic waves from isothermal and insulated boundaries of a half space.  studied the reflection of piezo-thermoelastic waves from the charge free and stress free boundary of transversely isotropic half space. Abd-Alla and Al-Dawy (2000) discussed the reflection of thermoelastic plane wave at a generalized thermoelastic half space with one and two relaxation times. Singh (2010) applied the LS and GL theories of generalized thermoelasticity to study the reflection from a thermally insulated stressfree thermoelastic solid half-space of monoclinic type. Sharma et al. (2012) studied the reflection and transmission characteristics of acoustic waves at an interface of a semiconductor half-space underlying an inviscid liquid. Sharma and Bhargava (2014) investigated the reflection and transmission of thermoelastic plane waves at an imperfect interface between a thermal conducting viscous-liquid and generalized thermoelastic solid half-space.  studied the generalized Rayleigh waves in thermoelastic solids under viscous fluid loading.
Keeping in view the above state facts and applications of reflection/transmission phenomenon in thermoelastic-diffusive solid under the interaction of fluid, the present paper is devoted to discuss the reflection and transmission of plane waves at the interface between such continua. The effects of incident angles, material parameters and fluid loading on reflection and transmission coefficients of various possible waves due to the incident qP and qSV waves have been considered. The analytical results so obtained have been verified numerically and are illustrated graphically.

FORMULATION OF THE PROBLEM
We consider a homogeneous isotropic, thermoelastic-diffusive solid in the undeformed state initially at uniform temperature 0 T , underlying an inviscid liquid half-space. We take origin of the rectangular Cartesian co-ordinate system Oxyz at any point on the plane surface (interface) with z -axis directed normally into the solid half-space, which is thus represented by 0  z as shown in Figure 1. We choose the x-axis along the direction of propagation of waves in such a way that all the particles on the line parallel to the y-axis are equally displaced. Therefore, all the field quantities are independent of y -co-ordinate. Further, the disturbances are assumed to be The basic governing field equations of motion, heat conduction equation, mass equation and constitutive relations for a solid medium, in the absence of heat sources and body forces, are given by Sherief et al. (2004)  2  j for Green and Lindsay (GL) theory (1972). The superposed dot notation is used for time differentiation. The basic governing equations for inviscid fluid (liquid) medium are given by  is the bulk modulus; L  and   are the density and coefficient of volume thermal expansion; is the velocity vector and L T is the temperature deviation in the liquid temperature from its ambient temperature  0 T .
In order to facilitate the analysis, we define the following dimensionless quantities , , , Upon introducing expressions (17) in equations (9)-(15), and noting that The equation (18) corresponds to decoupled shear motion which remains independent of temperature and mass concentration changes.

BOUNDARY CONDITIONS
The boundary conditions to be satisfied at the solid-liquid interface ( are the non-dimensional phase velocity, frequency and wave number of waves respectively. The primes have been suppressed for convenience. Upon using solution (27) in equations (18)-(23), one obtains a system of algebraic equations in unknowns A, B, C, D and E. The condition for the existence of non-trivial solution of this system of equations on simplification, provides us In case elastic and thermal fields are uncoupled The equation (30) and (31) will be used for reductions in the following analysis.

REFLECTION AND TRANSMISSION AT SOLID-LIQUID INTERFACE
In this section we shall discuss reflection and transmission of waves at the interface of thermoelasto-diffusive solid and inviscid fluid for qP -wave and qSV -wave incidence cases.

Quasi longitudinal
Suppose that a qP -wave is incident at the interface from the solid half-space. Then the total wave fields after reflection and transmission of waves from the interface are given by In the absence of mass diffusion( In the absence of mass diffusion and thermal variations The equation (41) with the help of equation (28) implies that This is the modified form of the Snell's law, which in the absence of thermal, mass diffusion, viscosity and fluid fields The system of equations (A.1)-(A.5) with the help of equation (41) can be expressed as where A , p Z and B are defined in the Appendix.
Solving the system of equations (44), the amplitude ratios Thus for the grazing incidence, the reflected qP wave annihilates the incident qP wave and there is no reflection or transmission of other waves through the interface. In the absence of mass diffusion Here the quantities In the absence of mass diffusion and thermal fields respectively. The relations (48) are in complete agreement with the corresponding equations of Achenbach (1973). Clearly qP wave is reflected as well as transmitted in case of normal incidence, however, for grazing incidence the reflected qP -wave annihilates the incident one.

) (qP -wave incidence at free surface
In case the liquid media is absent Thus, the incident qP -wave is reflected as qP -wave without change in phase in case of normal incidence and the reflected qP -wave annihilates the incident qP -wave for grazing incidence one.
It may be noted from the above analytical expressions for the reflection/transmission coefficients that the characteristics of reflected and transmitted waves depends on material parameters and incidence angle in addition to thermal variation and fluid loading effects.

Quasi transverse   qSV -wave incidence
Now consider the reflection and transmission of a plane qSV -wave for similar conditions on the boundary as in Section 5.1 above. The total displacement field in this case is given by Thus the shear ( qSV ) wave is reflected as qSV -wave in case of normal incidence and the reflected ( qSV ) wave annihilates the incident wave for grazing incidence.
In the absence of mass diffusion Thus only shear ( qSV ) wave is reflected as qSV -wave in case of normal incidence and the reflected ( qSV ) wave annihilates the incident wave for grazing incidence case. The other waves do not reflect or transmit in either case. where L  and   L are defined in equations (51) and (52) respectively.

Quasi transverse  
In the absence of mass diffusion, fluid and thermal fields Thus only shear wave is reflected as qSV wave without any change of phase in case of normal incidence and reflected qSV -wave annihilates the incident qSV wave in case of grazing incidence. It is noticed that the reflection/transmission characteristics of waves depends upon material parameters and incidence angle in this case too.

TOTAL REFLECTION
In this section, the case of total reflection beyond critical angle has been discussed. We consider the equation (56) are real quantities. Then j V , j Q respectively represent phase speed and attenuation coefficients of these waves.
Upon using relations (67) in equation (66) Thus the thermal part and elastic part of reflected P-wave propagates horizontally in Xdirection and these quantities decay exponentially with depth.
Moreover, we have also considered the relation    numerical results later on) and beyond this cutoff point maximum incident shear wave power is converted to reflected shear wave.

ENERGY EQUATIONS
From the principle of energy, the energy carried to the boundary by the incident wave must be equal to the energy carried away from the boundary by the reflected and transmitted waves.
For the incident qP waves the particle velocities are )} cos sin ( exp{ sin If we take the kinetic energy per unit volume as , then the energy flux for the waves mentioned above may be computed by multiplying the total energy per unit volume by the velocity of propagation and the area of the wave front involved. Here the cross-sectional areas of the incident, reflected and transmitted waves are proportional to the cosines of the angles made by the ray directions of the waves with the normal to the interface. Thus, we may write the equality between the incident, reflected and transmitted waves in case of qP waves for the unit area on the interface as In the absence of mass diffusion In the absence of mass diffusion and thermal fields In this case, the energy equation may be obtained by writing the equality between the incident qSV wave and the sum of the reflected qP wave, reflected qSV wave and transmitted qP wave energies for the unit area on the interface as   In the absence of mass diffusion In the absence of diffusion and thermal fields Here TED stands for thermoelastic-diffusion, TE for thermoelasticity and E for elastic one. It is shown with the help of numerical results in the following section that the energy equation is satisfied in each case.

NUMERICAL RESULTS AND DISCUSSIONS
In this section the reflection and transmission coefficients for qP and qSV wave incidences at an interface between thermoelastic-diffusive solid and inviscid fluid have been computed numerically. The material chosen for this purpose is Copper, the physical data for which is given as in Sherief et al. (2004) Table 1   .
The values of the reflection  Figures 2-11, and the satisfaction of energy equations at different angles of incidence is given in Table 2. Here, TED refers to thermoelastic diffusion case, ED to elastic-diffusive case and TE to thermo-elastic one. Figures 7-11, shows the energy distribution of reflection coefficients in case of qSV wave incidence at the interface and stress free surfaces. It is noticed that for shear wave incidence, there exists two critical angles in case of TED and TE: for transmitted longitudinal wave ( 0 9 . 21 0    ) and for reflected shear wave ( 6 9 . 29 0    ) beyond which it becomes evanescent in Z -direction and whole of the incident shear wave power is converted to the reflected shear wave.
It is also noticed that the reflected longitudinal wave increases to attain its maximum value at 0 30   due to high stresses generated in the material at this angle of incidence and after that it decreases sharply to become zero at is quite close to zero though not exactly zero, a phenomenon closely analogous to Brewster angle in optics [Kino, (1987)] has been observed. This may be used to convert a shear wave to longitudinal wave.  From Table 2, it is inferred that in case of qP wave incidence energy equation is approximately satisfied at all the incidence angles though it exactly holds near the normal and grazing incident angles. Therefore, there is a slight dissipation of energy at the other angles of incidence. In case of qSV wave incidence, the energy equation holds approximately everywhere except in the neighborhoods of critical angle and 0 80   where large amount of energy dissipation has been noticed. These conclusions are in agreement with Mott (1971).

CONCLUDING REMARKS
1. The analytic expressions for reflection and transmission coefficients for qP and qSV wave incident obliquely at the interface between thermoelastic-diffusive solid and inviscid fluid media have been derived.
2. For qP wave incidence at the surface, the significant amount of energy is carried out by transmitted longitudinal wave in the presence of liquid, however the maximum amount of energy is carried out by reflected longitudinal wave in the absence of liquid.
3. For qSV wave incidence at the surface, the maximum amount of energy is carried out by reflected longitudinal wave before the critical angle occurs and there after reflected shear wave becomes prominent. 4. It is observed that the distribution of energy through the reflected and transmitted waves obey energy equations in the respective cases of wave incidence with some exceptions in the neighbourhood of critical angle. 5. The reflection and transmission coefficients have been observed to depend on the material parameters and angle of incidence. 6. Significant effect of mass diffusion and thermal variations has been noticed on the reflection/transmision characteristics of waves. 7. The phenomenon of total reflection reveals that the thermal and elastic parts of reflected qP wave propagate horizontally but decay exponentially with depth.
8. The study may find applications in semiconductor, seismology and signal processing.