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Behaviour of Transverse Wave at an Imperfectly Corrugated Interface of a Functionally Graded Structure

  • METHODS OF WAVE SCATTERING IN INHOMOGENEOUS MEDIA
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Abstract

An investigation of the transverse wave has been carried out in a structure of functionally graded material. The structure consists of layer and a semi-infinite medium. The layer is composed of a functionally graded magneto-electro-elastic material with a quadratic gradedness parameter. The semi-infinite medium is composed of functionally graded piezoelectric with a quadratic gradedness parameter. The interface between the layer and half-space is taken as loosely bonded and corrugated, whereas the upper boundary is assumed to be stress-free and corrugated. Moreover, the solutions for layer and half-space are obtained using the basic variable-separable method to reduce the partial differential equation to the ordinary differential equation. And ordinary differential equation solutions are obtained by using the WKB approximation technique. The solutions with boundary conditions lead to dispersion relations in the determinant form of two cases: electrically open and short. To know the impact of the parameter involved, a particular model consists of BaTiO3–CoFe2O4 magneto-electro-elastic material layer and BaTiO3 piezoelectric semi-infinite medium have been taken. Graphs of phase velocity versus wave number are plotted, which interpret the numerical outcomes physically.

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ACKNOWLEDGMENTS

One of the authors gratefully acknowledges to SRMIST, Kattankulathur, India for facilitating with best research facility and providing research fellowship to carryout our research. The authors are also thankful to the Editors and reviewers for their valuable comments and suggestions to improve the quality of the paper.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. Department of Mathematics, Faculty of Engineering and Technology, College of Engineering and Technology, SRM Institute of Science and Technology, 603203, Kattankulathur, India

    A. Akshaya, S. Kumar & K. Hemalatha

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Appendices

APPENDIX A

$${{s}_{1}} = 1 + {{\delta }_{1}}{{\xi }_{2}},~\,\,\,{{s}_{2}} = 1 + {{\delta }_{1}}\left( {h + {{\xi }_{1}}} \right),\,\,\,~{{s}_{3}} = 1 + {{\delta }_{2}}{{\xi }_{1}},$$
$${{v}_{1}} = k\sqrt {\frac{{\rho _{1}^{0}{{c}^{2}}}}{{{{M}^{0}}}} - 1} + \frac{{\delta _{1}^{2}}}{{2ks_{1}^{2}}}\sqrt {\frac{{{{M}^{0}}}}{{\rho _{1}^{0}{{c}^{2}} - {{M}^{0}}}}} ,$$
$${{v}_{2}} = k - \frac{{\delta _{1}^{2}}}{{2ks_{1}^{2}}},$$
$${{v}_{3}} = k\sqrt {\frac{{\rho _{1}^{0}{{c}^{2}}}}{{{{M}^{0}}}} - 1} + \frac{{\delta _{1}^{2}}}{{2ks_{2}^{2}}}\sqrt {\frac{{{{M}^{0}}}}{{\rho _{1}^{0}{{c}^{2}} - {{M}^{0}}}}} ,$$
$${{v}_{4}} = k - \frac{{\delta _{1}^{2}}}{{2ks_{2}^{2}}},$$
$${{v}_{5}} = k\sqrt {\frac{{\rho _{2}^{1}{{c}^{2}}}}{{{{M}^{1}}}} - 1} + \frac{{\delta _{2}^{2}}}{{2ks_{3}^{2}}}\sqrt {\frac{{{{M}^{1}}}}{{\rho _{2}^{1}{{c}^{2}} - {{M}^{1}}}}} ,$$
$${{v}_{6}} = k - \frac{{\delta _{2}^{2}}}{{2ks_{3}^{2}}},$$
$${{p}_{1}} = - k\left( {h + {{\xi }_{2}}} \right)\sqrt {\frac{{\rho _{1}^{0}{{c}^{2}}}}{{{{M}^{0}}}} - 1} - \frac{{{{\delta }_{1}}}}{{2k{{s}_{1}}}}\sqrt {\frac{{{{M}^{0}}}}{{\rho _{1}^{0}{{c}^{2}} - {{M}^{0}}}}} ,$$
$${{p}_{2}} = - k\left( {h + {{\xi }_{2}}} \right) + \frac{{{{\delta }_{1}}}}{{2k{{s}_{1}}}},$$
$${{p}_{3}} = k\left( { - h + {{\xi }_{2}}} \right),$$
$${{p}_{4}} = k{{\xi }_{1}}\sqrt {\frac{{\rho _{1}^{0}{{c}^{2}}}}{{{{M}^{0}}}} - 1} - \frac{{{{\delta }_{1}}}}{{2k{{s}_{2}}}}\sqrt {\frac{{{{M}^{0}}}}{{\rho _{1}^{0}{{c}^{2}} - {{M}^{0}}}}} ,$$
$${{p}_{5}} = k{{\xi }_{1}} + \frac{{{{\delta }_{1}}}}{{2k{{s}_{2}}}},$$
$${{p}_{6}} = k{{\xi }_{1}}\sqrt {\frac{{\rho _{2}^{1}{{c}^{2}}}}{{{{M}^{1}}}} - 1} - \frac{{{{\delta }_{2}}}}{{2k{{s}_{3}}}}\sqrt {\frac{{{{M}^{1}}}}{{\rho _{2}^{1}{{c}^{2}} - {{M}^{1}}}}} ,$$
$${{p}_{7}} = k{{\xi }_{1}} + \frac{{{{\delta }_{2}}}}{{2k{{s}_{3}}}},$$
$${{A}_{0}} = e_{{15}}^{0} - \kappa _{{11}}^{0}{{m}_{2}} - \beta _{{11}}^{0}{{m}_{3}},$$
$${{A}_{1}} = h_{{15}}^{0} - \beta _{{11}}^{0}{{m}_{2}} - \mu _{{11}}^{0}{{m}_{3}}.$$

APPENDIX B

$${{r}_{{11}}} = {{M}^{0}}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} \left( {{{v}_{1}} - k\xi _{2}^{'}} \right)} \right){{e}^{{i{{p}_{1}}}}},$$
$${{r}_{{12}}} = - {{M}^{0}}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} \left( {{{v}_{1}} + k\xi _{2}^{'}} \right)} \right){{e}^{{ - i{{p}_{1}}}}},$$
$${{r}_{{13}}} = e_{{15}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} \left( {{{v}_{2}} - k\xi _{2}^{'}} \right)} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{14}}} = - e_{{15}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} \left( {{{v}_{2}} + k\xi _{2}^{'}} \right)} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{15}}} = h_{{15}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} \left( {{{v}_{2}} - k\xi _{2}^{'}} \right)} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{16}}} = ~ - h_{{15}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} \left( {{{v}_{2}} + k\xi _{2}^{'}} \right)} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{17}}} = {{r}_{{18}}} = {{r}_{{19}}} = {{r}_{{110}}} = {{r}_{{111}}} = 0;$$
$${{r}_{{21}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{1}}} }}{{e}^{{i{{p}_{1}}}}},~\,\,\,\,~{{r}_{{22}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{1}}} }}{{e}^{{ - i{{p}_{1}}}}},$$
$${{r}_{{23}}} = \frac{1}{{\sqrt {{{s}_{1}}} }}{{e}^{{{{p}_{2}}}}},\,\,\,~~{{r}_{{24}}} = \frac{1}{{\sqrt {{{s}_{1}}} }}{{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{25}}} = {{r}_{{26}}} = {{r}_{{27}}} = {{r}_{{28}}} = {{r}_{{29}}} = 0,$$
$${{r}_{{210}}} = ~ - {{e}^{{{{p}_{3}}}}},\,\,\,\,~~{{r}_{{211}}} = 0;$$
$${{r}_{{31}}} = {{A}_{0}}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} {{v}_{1}}} \right){{e}^{{i{{p}_{1}}}}},$$
$${{r}_{{32}}} = - {{A}_{0}}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} {{v}_{1}}} \right){{e}^{{ - i{{p}_{1}}}}},$$
$${{r}_{{33}}} = - \kappa _{{11}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{34}}} = \kappa _{{11}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{35}}} = - \beta _{{11}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{36}}} = ~\,\,\beta _{{11}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{37}}} = {{r}_{{38}}} = {{r}_{{39}}} = 0,~\,\,\,\,~{{r}_{{310}}} = ~\,\,\,{{\kappa }_{0}}k{{s}_{1}}{{e}^{{{{p}_{3}}}}},\,\,\,~~{{r}_{{311}}} = 0;$$
$$~{{r}_{{41}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{1}}} }}{{e}^{{i{{p}_{1}}}}},~\,\,\,\,~{{r}_{{42}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{1}}} }}{{e}^{{ - i{{p}_{1}}}}},$$
$${{r}_{{43}}} = {{r}_{{44}}} = 0,$$
$${{r}_{{45}}} = \frac{1}{{\sqrt {{{s}_{1}}} }}{{e}^{{{{p}_{2}}}}},\,\,\,\,~~{{r}_{{46}}} = \frac{1}{{\sqrt {{{s}_{1}}} }}{{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{47}}} = {{r}_{{48}}} = {{r}_{{49}}} = {{r}_{{410}}} = \,\,~0,~\,\,\,\,~{{r}_{{411}}} = - {{e}^{{{{p}_{3}}}}};$$
$${{r}_{{51}}} = {{A}_{1}}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} v} \right){{e}^{{i{{p}_{1}}}}},$$
$${{r}_{{52}}} = - {{A}_{1}}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + i\sqrt {{{s}_{1}}} {{v}_{1}}} \right){{e}^{{ - i{{p}_{1}}}}},$$
$${{r}_{{53}}} = \beta _{{11}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{54}}} = - \beta _{{11}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{55}}} = - \mu _{{11}}^{0}s_{1}^{2}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{{{p}_{2}}}}},$$
$${{r}_{{56}}} = ~\,\,\mu _{{11}}^{0}s_{1}^{2}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{1}}} }} + \sqrt {{{s}_{1}}} {{v}_{2}}} \right){{e}^{{ - {{p}_{2}}}}},$$
$${{r}_{{57}}} = {{r}_{{58}}} = {{r}_{{59}}} = 0,\,\,\,~~{{r}_{{510}}} = 0,\,\,\,\,~~{{r}_{{511}}} = {{\mu }_{0}}k{{s}_{1}}{{e}^{{{{p}_{3}}}}};$$
$${{r}_{{61}}} = {{M}^{0}}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} \left( {{{v}_{3}} - k\xi _{1}^{'}} \right)} \right){{e}^{{i{{p}_{4}}}}},$$
$${{r}_{{62}}} = - {{M}^{0}}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} \left( {{{v}_{3}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{63}}} = e_{{15}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} \left( {{{v}_{4}} - k\xi _{1}^{'}} \right)} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{64}}} = - e_{{15}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} \left( {{{v}_{4}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{65}}} = h_{{15}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} \left( {{{v}_{4}} - k\xi _{1}^{'}} \right)} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{66}}} = \,\,~ - h_{{15}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} \left( {{{v}_{4}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{67}}} = - {{A}_{2}}{{s}_{2}}s_{3}^{2}\left( {\frac{{{{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + i\sqrt {{{s}_{3}}} \left( {{{v}_{5}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - i{{p}_{6}}}}},$$
$${{r}_{{68}}} = - e_{{15}}^{1}{{s}_{2}}s_{3}^{2}\left( {\frac{{{{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + \sqrt {{{s}_{3}}} \left( {{{v}_{6}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - {{p}_{7}}}}},$$
$${{r}_{{69}}} = {{r}_{{610}}} = {{r}_{{611}}} = 0;$$
$${{r}_{{71}}} = \frac{{ - K}}{{\sqrt {{{s}_{2}}} }}{{e}^{{i{{p}_{4}}}}},~\,\,\,~{{r}_{{72}}} = - \frac{K}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{73}}} = {{r}_{{74}}} = {{r}_{{75}}} = {{r}_{{76}}} = 0,$$
$${{r}_{{77}}} = {{A}_{2}}s_{3}^{2}\left( {\frac{{{{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + i\sqrt {{{s}_{3}}} \left( {{{v}_{5}} + k\xi _{1}^{'}} \right) + K} \right){{e}^{{ - i{{p}_{6}}}}},$$
$${{r}_{{78}}} = - e_{{15}}^{1}s_{3}^{2}\left( {\frac{{ - {{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + \sqrt {{{s}_{3}}} \left( {{{v}_{6}} + k\xi _{1}^{'}} \right)} \right){{e}^{{ - {{p}_{7}}}}},$$
$${{r}_{{79}}} = {{r}_{{710}}} = \,\,~{{r}_{{711}}} = 0;$$
$${{r}_{{81}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{i{{p}_{4}}}}},\,\,\,\,~~{{r}_{{82}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{83}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{{{p}_{5}}}}},~\,\,\,\,~{{r}_{{84}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{85}}} = {{r}_{{86}}} = 0,$$
$${{r}_{{87}}} = - \frac{{e_{{15}}^{1}}}{{\kappa _{{11}}^{1}}}\frac{1}{{\sqrt {{{s}_{3}}} }}{{e}^{{ - i{{p}_{6}}}}},$$
$${{r}_{{88}}} = - \frac{1}{{\sqrt {{{s}_{3}}} }}{{e}^{{ - {{p}_{7}}}}},$$
$${{r}_{{89}}} = {{r}_{{810}}} = \,\,~{{r}_{{811}}} = 0;$$
$${{r}_{{91}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{i{{p}_{4}}}}},~\,\,\,\,~{{r}_{{92}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{93}}} = {{r}_{{94}}} = 0,\,\,\,~~{{r}_{{95}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{{{p}_{5}}}}},$$
$${{r}_{{96}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - {{p}_{5}}}}},~\,\,\,\,~{{r}_{{97}}} = {{r}_{{98}}} = 0,$$
$${{r}_{{99}}} = - \frac{1}{{\sqrt {{{s}_{3}}} }}{{e}^{{ - {{p}_{7}}}}},~~{{r}_{{910}}} = ~{{r}_{{911}}} = 0;$$
$${{r}_{{1001}}} = {{A}_{0}}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} {{v}_{3}}} \right){{e}^{{i{{p}_{4}}}}},~$$
$${{r}_{{1002}}} = - {{A}_{0}}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} {{v}_{3}}} \right){{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{1003}}} = - \kappa _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{1004}}} = \kappa _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{1005}}} = - \beta _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{1006}}} = \,\,~\beta _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{1007}}} = 0,$$
$${{r}_{{1008}}} = - \kappa _{{11}}^{1}{{s}_{2}}s_{3}^{2}\left( {\frac{{{{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + \sqrt {{{s}_{3}}} {{v}_{6}}} \right){{e}^{{ - {{p}_{7}}}}},$$
$${{r}_{{1009}}} = {{r}_{{1010}}} = {{r}_{{1011}}} = 0;$$
$${{r}_{{1101}}} = {{A}_{1}}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} {{v}_{3}}} \right){{e}^{{i{{p}_{4}}}}},$$
$${{r}_{{1102}}} = - {{A}_{1}}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + i\sqrt {{{s}_{2}}} {{v}_{3}}} \right){{e}^{{ - i{{p}_{4}}}}},$$
$${{r}_{{1103}}} = \beta _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{1104}}} = - \beta _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{1105}}} = - \mu _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{ - {{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{{{p}_{5}}}}},$$
$${{r}_{{1106}}} = \,\,~\mu _{{11}}^{0}s_{2}^{2}{{s}_{3}}\left( {\frac{{{{\delta }_{1}}}}{{2\sqrt {{{s}_{2}}} }} + \sqrt {{{s}_{2}}} {{v}_{4}}} \right){{e}^{{ - {{p}_{5}}}}},$$
$${{r}_{{1107}}} = {{r}_{{1108}}} = 0,$$
$${{r}_{{1109}}} = - \mu _{{11}}^{1}{{s}_{2}}s_{3}^{2}\left( {\frac{{{{\delta }_{2}}}}{{2\sqrt {{{s}_{3}}} }} + \sqrt {{{s}_{3}}} {{v}_{6}}} \right){{e}^{{ - {{p}_{7}}}}},$$
$${{r}_{{110}}} = {{r}_{{1111}}} = 0.$$

APPENDIX C

$${{r}_{{ij}}} = {{t}_{{mn}}},\,\,\,\,\forall i = 1,~6,~7,~8,~9,~10,~11;~\,\,\,~j = 1,~2,..,9;$$
$$m = 1,~4,~5,~6,~7,~8,~9;\,\,\,~~n = 1,~2,..,9$$
$${{t}_{{21}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{i{{p}_{4}}}}},\,\,\,~~{{t}_{{22}}} = \frac{{{{m}_{2}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - i{{p}_{4}}}}},$$
$${{t}_{{23}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{{{p}_{5}}}}},\,\,\,\,~{{t}_{{24}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - {{p}_{5}}}}},$$
$${{t}_{{25}}} = {{t}_{{26}}} = {{t}_{{27}}} = {{t}_{{28}}} = {{t}_{{29}}} = 0;$$
$${{t}_{{31}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{i{{p}_{4}}}}},~\,\,\,~{{t}_{{32}}} = \frac{{{{m}_{3}}}}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - i{{p}_{4}}}}},$$
$${{t}_{{33}}} = {{t}_{{34}}} = 0,~\,\,\,~{{t}_{{35}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{{{p}_{5}}}}},$$
$${{t}_{{36}}} = \frac{1}{{\sqrt {{{s}_{2}}} }}{{e}^{{ - {{p}_{5}}}}},~\,\,\,\,~~{{t}_{{37}}} = {{t}_{{38}}} = {{t}_{{39}}} = 0.$$

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Akshaya, A., Kumar, S. & Hemalatha, K. Behaviour of Transverse Wave at an Imperfectly Corrugated Interface of a Functionally Graded Structure. Phys. Wave Phen. 32, 117–134 (2024). https://doi.org/10.3103/S1541308X24700067

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