Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 301816, 12 pages
http://dx.doi.org/10.5402/2011/301816
Research Article

Propagation of Plane Waves in a Thermally Conducting Mixture

Department of Mathematics, Post Graduate Government College, Sector 11, Chandigarh 160 011, India

Received 10 March 2011; Accepted 4 May 2011

Academic Editors: Y. Dimakopoulos, S. Li, and Y. Liu

Copyright © 2011 Baljeet Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The governing equations for generalized thermoelasticity of a mixture of an elastic solid and a Newtonian fluid are formulated in the context of Lord-Shulman and Green-Lindsay theories of generalized thermoelasticity. These equations are solved to show the existence of three coupled longitudinal waves and two coupled transverse waves, which are dispersive in nature. Reflection from a thermally insulated stress-free surface is considered for incidence of coupled longitudinal wave. The speeds and reflection coefficients of plane waves are computed numerically for a particular model.

1. Introduction

The soil consists of an assemblage of particles with different sizes and shapes which form a skeleton whose voids are filled by water and air or gas. The word “soil,” therefore, implies a mixture of assorted mineral grains with various fluids. The first continuum theory of mixtures was proposed by Truesdell [1] and Truesdell and Toupin [2] in terms of the kinematic and thermodynamic variables associated with each constituent of the mixture. Another theory presented by Green and Naghdi [3] uses the kinematic and thermodynamic variables associated with the mixture as a whole. The classical theory of mixtures was discussed by Atkin and Craine [4] and Bowen [5]. Bedford and Drumheller [6] formulated the theories of immiscible and structured mixtures. Various other theories of mixtures were also developed by Muller [7], Dunwoody [8], Krishnaswamy and Batra [9], Iesan [1012], Rajagopal and Tao [13], and Pompei and Scalia [14].

Many engineering materials, as well as soils, rocks, granular materials, sand, and underground water mixtures may be modeled more realistically by means of micropolar continua. Eringen [15] developed a theory of micropolar mixture of porous media (nonreacting mixture of micropolar elastic solid and a micropolar viscous fluid at a single temperature) to include the rotational degrees of freedom. Eringen [15] also obtained the field equations of a mixture of an elastic solid and a Newtonian fluid as a special case by dropping the micropolar effects. In the present paper, the governing equations for generalized thermoelasticity of mixture of an elastic solid and a Newtonian fluid are formulated in context of Lord-Shulman [16] and Green-Lindsay [17] theories of generalized thermoelasticity. These equations are solved to show the existence of various coupled plane waves. The reflection of these plane waves is studied at thermally insulated stress-free surface and the reflection coefficients are computed and shown graphically with the angle of incidence. Wave propagation in such a media may find its applications in consolidation and oil exploration problems.

2. Equations of Motion

We consider a binary mixture of an elastic solid and a Newtonian fluid at the same temperature where no chemical reaction takes place between the two species of the mixture. According to the last section given in Eringen [15], the equations of motion in a linear isotropic thermoelastic mixture of an elastic solid and Newtonian fluid in absence of external loads are written as𝜆𝑆+2𝜇𝑆𝐮𝑆𝜇𝑆××𝐮𝑆̇𝐮𝜉Ṡ𝐮𝐹𝑠𝑇0+𝛽0𝑇+𝑡1̇𝑇=𝜌𝑆̈𝐮𝑆,𝜆(2.1)𝐹+2𝜇𝐹̇𝐮𝐹𝜇𝐹̇𝐮××𝐹̇𝐮+𝜉𝑆̇𝐮𝐹+𝑠𝑇0+𝛽0𝑇+𝑡1̇𝑇=𝜌𝐹̈𝐮𝐹,𝑠̇𝐮(2.2)Ṡ𝐮𝐹+𝐾𝑇0𝛽𝑇𝜌0̇𝐮𝑆+𝑡0Ω̈𝐮𝑆+𝜌𝐹̇𝐮𝛿𝐹+𝑡0Ω̈𝐮𝐹=𝜌𝐶0̇𝑇+𝑡0̈𝑇,(2.3) where 𝐮𝑆,𝐮𝐹 are displacement vectors in solid and fluid phase, respectively, and 𝑇 is the temperature variable. The coefficients 𝜆𝑆,𝜇𝑆,𝛽0 are the thermoelastic constants for isotropic elastic solids, 𝜆𝐹,𝜇𝐹 are fluid viscosities, 𝐾 is classical Fourier constant, 𝑠 is the heat generation due to velocity difference, 𝜉 is the momentum generation coefficient due to the velocity difference, 𝐶0 is specific heat at constant deformation, 𝑇0 is an ambient temperature, 𝛿=(𝜕𝜓/𝜕𝜌𝐹)0, 𝜓 is Helmholtz’s free energy function, 𝜌,𝜌𝑆,𝜌𝐹are the densities of mixture, solid, and fluid in the natural state, superposed dot indicates the temporal derivative, and other symbols have their usual meanings.

The use of symbol Ω, in (2.3), makes these fundamental equations possible for the two different theories of the generalized thermoelasticity. For the L-S (Lord-Shulman) theory 𝑡1=0,Ω=1 and for G-L (Green-Lindsay) theory 𝑡1>0 and Ω=0. The thermal relaxations 𝑡0 and 𝑡1 satisfy the inequality 𝑡1𝑡00 for the G-L theory only.

By introducing the scalar potentials 𝑞𝑆,𝑞𝐹 and vector potentials 𝐔𝑆,𝐔𝐹 through Helmholtz representation of vector field, we can write𝐮𝑆=𝑞𝑆+×𝐔𝑆,𝐔𝑆𝐮=0,𝐹=𝑞𝐹+×𝐔𝐹,𝐔𝐹=0.(2.4)

Using (2.4) into (2.1) to (2.3), we obtain the following five equations, that is, three coupled equations in 𝑞𝑆,𝑞𝐹,𝑇 and two coupled equations in 𝐔𝑆,𝐔𝐹;𝑐21𝑆2𝑞𝑆𝜉𝑆̇𝑞𝑆̇𝑞𝐹𝛽1𝑆𝑇+𝑡1̇𝑇=̈𝑞𝑆,𝑐21𝐹2̇𝑞𝐹+𝜉𝐹̇𝑞𝑆̇𝑞𝐹+𝛽1𝐹𝑇+𝑡1̇𝑇=̈𝑞𝐹,𝐾2𝑇+𝑠2̇𝑞𝑆̇𝑞𝐹𝛽01+𝑡0Ω𝜕𝜕𝑡2̇𝑞𝑆𝛿1+𝑡0Ω𝜕𝜕𝑡2̇𝑞𝐹=1+𝑡0𝜕̇𝑐𝜕𝑡𝑇,(2.5)22𝑆2𝐔𝑆𝜉𝑆̇𝐔𝑆̇𝐔𝐹=̈𝐔𝑆,𝑐22𝐹2̇𝐔𝐹+𝜉𝐹̇𝐔𝑆̇𝐔𝐹=̈𝐔𝐹,(2.6) where, 𝑐21𝑆=(𝜆𝑆+2𝜇𝑆)/𝜌𝑆,𝑐22𝑆=𝜇𝑆/𝜌𝑆,𝜉𝑆=𝜉/𝜌𝑆,𝛽1𝑆=𝛽1/𝜌𝑆,𝑐21𝐹=(𝜆𝐹+2𝜇𝐹)/𝜌𝐹,𝑐22𝐹=𝜇𝐹/𝜌𝐹,𝜉𝐹=𝜉/𝜌𝐹,𝛽1𝐹=𝛽1/𝜌𝐹,𝛽1=𝑠/𝑇0+𝛽0,𝛽0=𝛽0/𝐶0,𝛿=𝜌𝛿/𝐶0,𝐾=𝐾/𝜌𝐶0𝑇0,𝑠=𝑠/𝜌𝐶0.

3. Wave Propagation

In this section, the plane wave propagation in an infinite thermally conducting mixture of elastic solid and Newtonian fluid is studied. In order to solve (2.5), we consider the following form of plane waves propagating in the positive direction of a unit vector 𝐧𝑞𝑆,𝑞𝐹=,𝑇𝑞𝑆,𝑞𝐹,𝑇[]exp𝜄𝑘(𝐧𝐫𝑉𝑡),(3.1) where 𝑞𝑆,𝑞𝐹,𝑇 are the constant complex scalar wave amplitudes, 𝜄=1, 𝐫 is the position vector, 𝑉 is the phase velocity in the direction of 𝐧, k is the wave number, and 𝜔(=𝑘𝑉) is angular frequency. Using the values of 𝑞𝑆,𝑞𝐹, and 𝑇 from (3.1) into (2.5), we obtain a set of three homogeneous equations in three unknown amplitudes 𝑞𝑆,𝑞𝐹,𝑇. After eliminating these three unknowns, we get the following cubic equationΓ3+𝐴Γ2+𝐵Γ+𝐶=0,(3.2) where Γ=𝑉2 and the expressions for 𝐴, 𝐵, 𝐶 are given in Appendix A. The three roots 𝑉21,𝑉22,𝑉23 of (3.2) correspond to complex speeds of three dispersive coupled longitudinal waves in a thermally conducting mixture of elastic solid and Newtonian fluid for both L-S and G-L theories.

Equation (2.6) can be solved by considering the following form of vector potentials:𝐔𝑆,𝐔𝐹=𝐀𝑆,𝐀F[]exp𝜄𝑘(𝐧𝐫𝑉𝑡),(3.3) where 𝐀𝑆,𝐀𝐹 are constant complex vector wave amplitudes. With the help of (3.3), (2.6) leads to homogeneous vector equations in 𝐀𝑆 and 𝐀𝐹. After elimination of 𝐀𝑆 and 𝐀𝐹, the following quadratic equation in 𝑉2 is obtained:𝑉22+𝜄𝜔𝑐22𝐹+𝜄𝜔𝜉𝑆𝑘2+𝜄𝜔𝜉𝐹𝑘2𝑐22𝑆𝑉2𝜄𝜔𝑐22𝑆𝑐22𝐹+𝜄𝜔𝑐22𝑆𝜉𝐹𝑘2+𝜔𝑐22𝐹𝜉𝑆𝑘2=0.(3.4) Equation (3.4) gives two roots 𝑉24,𝑉25, which correspond to complex speeds of two dispersive coupled transverse waves in a thermally conducting mixture of elastic solid and Newtonian fluid. Hence, the analytical solution indicates the existence of five dispersive plane waves in two-dimensional model of a thermally conducting mixture of an elastic solid and a Newtonian fluid for both L-S and G-L theories.

4. Particular Cases

(i)In absence of fluid, (3.2) reduces to the following quadratic equation: 𝜏𝑚𝑉22𝜏𝑚𝑐21𝑆+𝐾𝜏𝑉2+𝑐21𝑆𝐾𝜏=0,(4.1) which gives complex speeds of two coupled longitudinal waves for both L-S and G-L theories whereas (3.4) reduces to a linear equation which gives the speed of transverse wave. (ii)In absence of thermal effects, the cubic equation (3.2) reduces to the following quadratic equation: 𝑉22+𝜄𝜔𝑐21𝐹𝑐21𝑆+𝜄𝜔𝜉𝑆+𝜉𝐹𝑘2𝑉2𝜔2𝑐21𝐹𝜉𝑆𝑘2+𝜄𝜔𝑐21𝑆𝑐21𝐹+𝜄𝜔𝑐21𝑆𝜉𝐹𝑘2=0.(4.2) The roots of (4.2) correspond to the complex phase speeds of two coupled longitudinal waves in a mixture of an elastic solid and a Newtonian fluid.

5. Reflection from Traction-Free Thermally Insulated Surface

We consider the half-space of the medium with x-axis along the thermally insulated stress-free surface and z-axis into the medium. For incident coupled longitudinal waves or coupled transverse waves, there will be five reflected waves into the medium as shown in Figure 1. The required boundary conditions at stress-free thermally insulated surface are𝑡𝑆𝑧𝑧=0,𝑡𝑆𝑧𝑥=0,𝑡𝐹𝑧𝑧=0,𝑡𝐹𝑧𝑥=0,𝜕𝑇𝜕𝑧=0.(5.1) Here, 𝑡𝑆𝑧𝑧=𝜆𝑆𝑢𝑆1,1+𝜆𝑆+2𝜇𝑆𝑢𝑆3,3𝛽0𝑇+𝑡1̇𝑇,𝑡𝑆𝑧𝑥=𝜇𝑆𝑢𝑆1,3+𝑢𝑆3,1,𝑡𝐹𝑧𝑧=𝜆𝐹̇𝑢𝐹1,1+𝜆𝐹+2𝜇𝐹̇𝑢𝐹3,3,𝑡𝐹𝑧𝑥=𝜇𝐹̇𝑢𝐹1,3+̇𝑢𝐹3,1.(5.2)

301816.fig.001
Figure 1: Geometry of the problem showing incident and reflected waves.

The appropriate potentials required to satisfy the above boundary conditions are 𝑞𝑆=𝐴0exp𝜄𝑘1sin𝜃0𝑥cos𝜃0𝑧𝜄𝜔1𝑡𝐴+Σ𝑖exp𝜄𝑘𝑖sin𝜃𝑖𝑥+cos𝜃𝑖𝑧𝜄𝜔𝑖𝑡𝑞,(𝑖=1,2,3),𝐹=𝜉1𝐴0exp𝜄𝑘1sin𝜃0𝑥cos𝜃0𝑧𝜄𝜔1𝑡𝜉+Σ𝑖𝐴𝑖exp𝜄𝑘𝑖sin𝜃𝑖𝑥+cos𝜃𝑖𝑧𝜄𝜔𝑖𝑡,(𝑖=1,2,3),𝑇=𝜂1𝐴0exp𝜄𝑘1sin𝜃0𝑥cos𝜃0𝑧𝜄𝜔1𝑡𝜂+Σ𝑖𝐴𝑖exp𝜄𝑘𝑖sin𝜃𝑖𝑥+cos𝜃𝑖𝑧𝜄𝜔𝑖𝑡𝑈,(𝑖=1,2,3),𝑆=𝐵0exp𝜄𝑘4sin𝜃0𝑥cos𝜃0𝑧𝜄𝜔4𝑡𝐴+Σ𝑗exp𝜄𝑘𝑗sin𝜃𝑗𝑥+cos𝜃𝑗𝑧𝜄𝜔𝑗𝑡𝑈,(𝑗=4,5),𝐹=𝜂4𝐵0exp𝜄𝑘4sin𝜃0𝑥cos𝜃0𝑧𝜄𝜔4𝑡𝜂+Σ𝑗𝐴𝑗exp𝜄𝑘𝑗sin𝜃𝑗𝑥+cos𝜃𝑗𝑧𝜄𝜔𝑗𝑡,(𝑗=4,5).(5.3) Here 𝐴0,𝐴1,𝐴2,𝐴3,𝐵0,𝐴4 and 𝐴5 are the amplitude of incident coupled longitudinal I (CL I), reflected coupled longitudinal I (CL I), reflected coupled longitudinal II (CL II), reflected coupled longitudinal III (CL III), incident coupled transverse I (CT I), reflected coupled transverse I (CT I) and reflected coupled transverse II (CT II) waves, respectively, where 𝐵0=0 for incident coupled longitudinal I (CL I) wave, 𝐴0=0 for incident coupled transverse I (CT I) wave, and𝜉𝑖=𝜄𝜔𝛽1𝑆𝜉𝑆/𝑘2𝛽1𝐹𝑉𝑖2𝑐21𝑆+𝜄𝜔𝜉𝑆/𝑘2𝜄𝜔𝛽1𝐹𝜉𝑆/𝑘2𝛽1𝑆𝑉𝑖2+𝜄𝜔𝑐21𝐹+𝜄𝜔𝜉𝑆/𝑘2𝜂,(𝑖=1,2,3),𝑖=𝜄𝜔𝜉𝐹𝑠/1𝜔𝜏0Ω+𝛿𝑘2𝑠/1𝜔𝜏0Ω𝛽0𝑉𝑖2+𝜄𝜔𝑐21𝐹+𝜄𝜔𝜉𝐹/𝑘2𝜏𝑚𝐾/𝜏𝑛𝑉𝑖2𝑉𝑖2+𝜄𝜔𝑐21𝐹+𝜄𝜔𝜉𝐹/𝑘2+𝛽1𝐹𝜏𝑚𝑠/1𝜔𝜏0Ω+𝛿,𝜂(𝑖=1,2,3),𝑗=𝜄𝜔𝜉𝐹/𝑘2𝑉𝑗2+𝜄𝜔𝑐22𝐹+𝜄𝜔𝜉𝐹/𝑘2,(𝑗=4,5),(5.4) and the relations between angles of incidence and reflection are given assin𝜃0𝑐1or𝑐4=sin𝜃1𝑐1=sin𝜃2𝑐2=sin𝜃3𝑐3=sin𝜃4𝑐4=sin𝜃5𝑐5,(5.5) and, 𝑉𝑗1=𝑐𝑗1𝜄𝜔1𝑞𝑗, where 𝑐𝑗 and 𝑞𝑗, (𝑗 = 1,2,…,5) are real phase speeds and attenuations of coupled longitudinal and transverse waves.

Using the appropriate potentials given by (5.3) with the required boundary conditions (5.1), the following nonhomogeneous system of five equations is obtainedΣ𝑎𝑖𝑗𝑍𝑗=𝑏𝑖,(𝑖,𝑗=1,2,,5),(5.6) where 𝑎𝑖𝑗,𝑏𝑖, and 𝑍𝑗 are given in Appendix B.

6. Numerical Results and Discussion

In this section, the complex absolutes of phase speeds and reflection coefficients are computed with the following physical constants at 𝑇0=293K,𝜆𝑆=7.59×109Nm2, 𝜇𝑆=1.89×109Nm2,𝜆𝐹=2.14×109Nm2, 𝜇𝐹=0.45×109Nm2,𝜌𝑆=2192Kgm3,𝜌𝐹=1010Kgm3, 𝐶0=96.3JKg1K1, 𝐾=2.51Wm1K1, 𝑠=0.021J, 𝜉=750Kgm3s1, 𝑡0=0.005𝑠,𝑡1=0.006𝑠,𝛽0=0.0005,𝛿=0.0001,𝑛=0.15.

The densities of solid, fluid, and mixture are related as 𝜌=(1𝑛)𝜌𝑆+𝑛𝜌𝐹, where 𝑛 is porosity of the mixture.

Numerical computations of phase speeds and reflection coefficients are restricted to the particular case of Lord-Shulman theory only. The phase speeds of the coupled waves are shown graphically in Figure 2 for the range 5Hz𝜔50Hz of frequency. It is observed that the phase speeds of all coupled longitudinal waves increase with the increase in frequency. In absence of thermal effects, the phase speed of CL I wave decreases at each value of frequency, whereas the phase speed of CL II increases. The wave CL III will disappear in absence of thermal parameters. The phase speeds of coupled transverse waves (CT I and CT II) remain unaffected by thermal effects.

fig2
Figure 2: Thermal effects on speeds of coupled waves against frequency.

Reflection coefficients are computed for incident CL I wave only. The reflection coefficients of coupled longitudinal waves are shown graphically in Figure 3 against the angle of incidence. The reflection coefficients of CL I and CL II waves change with the increase in angle of incidence. The comparison of solid and dotted curves shows the effects of thermal parameters on these coefficients. The reflected CL III wave will disappear in absence of thermal effects. The reflection coefficients of CT I and CT II waves also change with the increase in angle of incidence. These waves are also affected by thermal parameters as shown in Figure 4.

fig3
Figure 3: Thermal effects on coefficients of reflected coupled longitudinal waves for incident CL I wave.
fig4
Figure 4: Thermal effects on coefficients of reflected coupled transverse waves for incident CL I wave.

7. Conclusions

The generalized thermoelasticity of a mixture of an elastic solid and a Newtonian fluid is developed in accordance with Lord-Shulman and Green-Lindsay theories. The solutions of governing equations suggest that there will exist three coupled longitudinal waves and two coupled transverse waves in the present model. From numerical results, it is observed that the phase speeds of coupled waves increase with the increase in frequency. The presence of thermal parameters affects the phase speeds of coupled longitudinal waves only. Reflection coefficients of all reflected coupled waves are affected significantly by thermal disturbances in the material.

Appendices

A. Expressions for 𝐴, 𝐵, and 𝐶

𝐴=𝜄𝜔𝑐21𝐹𝜏𝑚𝑐21𝑆𝜏𝑚+𝜄𝜔𝜏𝑚𝜉𝑆+𝜉𝐹/𝑘2𝐾𝜏+𝑎2𝛽𝐹1𝜏𝑚+𝑎1+𝑎2𝛽𝑆1𝜏𝑚,𝜏𝑚𝐵=𝜄𝜔𝐾𝜏𝑐21𝐹+𝜉𝐹𝑘2𝑐21𝑆𝜄𝜔𝑐21𝐹𝜏𝑚+𝜄𝜔𝜉𝐹𝑘2𝜏𝑚𝐾𝜏+𝑎2𝛽𝐹1𝜏𝑚+𝜄𝜔𝜉𝑆𝑘2𝜄𝜔𝑐21𝐹𝜏𝑚𝐾𝜏𝑎1𝛽𝐹1𝜏𝑚+𝜄𝜔𝛽𝑆1𝜏𝑚𝑎2𝑐21𝐹+𝑎1𝑐21𝐹+𝑎1𝜉𝐹𝑘2/𝜏𝑚,𝐶=𝜄𝜔𝑐21𝑆𝐾𝜏𝑐21𝐹+𝜉𝐹/𝑘2+𝜔2𝐾𝜏𝑐21𝐹𝜉𝑆/𝑘2𝜏𝑚,(A.1) and, 𝑎1=𝑠𝛽0, 𝑎2=𝑠+𝛿, 𝑠=𝑠/(1𝜄𝜔𝑡0Ω), 𝐾𝜏=𝐾/(𝑡0Ω+𝜄/𝜔), 𝜏𝑚=1𝜄𝜔𝑡1, 𝜏𝑚=(1𝜄𝜔𝑡0)/(1𝜄𝜔𝑡0Ω).

B. Expressions for 𝑎𝑖𝑗,𝑏𝑖, and 𝑍𝑗

𝑎1𝑖=𝑘2𝑖𝜆𝑆+2𝜇𝑆cos2𝜃𝑖+𝛽01𝜄𝜔𝑖𝜏1𝜂𝑖𝑘2𝑖𝑎,(𝑖=1,2,3),1𝑗=2𝑘2𝑗𝜇𝑆sin𝜃𝑗cos𝜃𝑗𝑎(𝑗=4,5),2𝑖=𝑘2𝑖sin𝜃𝑖cos𝜃𝑖𝑎,(𝑖=1,2,3),2𝑗=𝑘2𝑗cos2𝜃𝑗,𝑎(𝑗=4,5),3𝑖=𝑘2𝑖𝜔𝑖𝜉𝑖𝜆𝐹+2𝜇𝐹cos2𝜃𝑖𝑎,(𝑖=1,2,3),3𝑗=2𝑘2𝑗𝜔𝑗𝜂𝑗𝜇𝐹sin𝜃𝑗cos𝜃𝑗𝑎,(𝑗=4,5),4𝑖=2𝑘2𝑖𝜔𝑖𝜉𝑖sin𝜃𝑖cos𝜃𝑖𝑎,(𝑖=1,2,3),4𝑗=𝑘2𝑗𝜔𝑗𝜂𝑗cos2𝜃𝑗𝑎,(𝑗=4,5),5𝑖=𝑘𝑖𝜂𝑖cos𝜃𝑖,𝑎(𝑖=1,2,3),5𝑗=0,(𝑗=4,5),(B.1) and (𝑎) for incident CL I wave (𝜃0=𝜃1),𝑏1=𝑘21𝜆𝑆+2𝜇𝑆cos2𝜃0+𝛽01𝜄𝜔1𝜏1𝜂1𝑘21,𝑏2=𝑘21sin𝜃0cos𝜃0,𝑏3=𝑘21𝜔1𝜉1𝜆𝐹+2𝜇𝐹cos2𝜃0,𝑏4=2𝑘21𝜔1𝜉1sin𝜃0cos𝜃0,𝑏5=𝑘1𝜂1cos𝜃0,𝑍1=𝐴1𝐴0,𝑍2=𝐴2𝐴0,𝑍3=𝐴3𝐴0,𝑍4=𝐴4𝐴0,𝑍5=𝐴5𝐴0(B.2)(𝑏) for incident CT I wave (𝜃0=𝜃4),𝑏1=2𝑘24𝜇𝑆sin𝜃0cos𝜃0,𝑏2=𝑘24cos2𝜃0,𝑏3=2𝑘24𝜔4𝜂4𝜇𝐹sin𝜃0cos𝜃0,𝑏4=𝑘24𝜔4𝜂4cos2𝜃0,𝑏5𝑍=0,1=𝐴1𝐵0,𝑍2=𝐴2𝐵0,𝑍3=𝐴3𝐵0,𝑍4=𝐴4𝐵0,𝑍5=𝐴5𝐵0.(B.3) Here, the amplitude ratios 𝑍1,𝑍2,𝑍3,𝑍4, and 𝑍5 correspond to reflected CL I wave, CL II wave, CL III wave, CT I wave, and CT II wave, respectively.

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