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Nonlinear surface SH waves in a half-space covered by an irregular layer

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Abstract

Research on the shear horizontal (SH) waves propagating in a nonlinear elastic half-space coated with a nonlinear elastic layer having slow variation in the boundary surfaces is presented. It is assumed that both the free surface and interface change as a function of the distance in the direction of propagation of the waves. By employing a perturbation method, nonlinear self-modulation of the surface SH waves for a general geometry is characterized by a generalized nonlinear Schrödinger (GNLS) equation with variable coefficients depending on functions representing the irregularities of boundary surfaces in addition to linear and nonlinear material parameters of the medium. The solitary wavelike solutions of this GNLS equation are obtained for certain special cases of the irregular boundary surfaces. It has been demonstrated by graphs that the propagation characteristics of SH waves are significantly affected by slowly varying layer and nonlinear properties of the media.

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Author information

Authors and Affiliations

  1. Department of Naval Architecture and Marine Engineering, Faculty of Engineering, Piri Reis University, 34940, Tuzla, Istanbul, Turkey

    Ekin Deliktas-Ozdemir

  2. Department of Mathematics, Faculty of Science and Letters, İstanbul Technical University, 34469, Maslak, Istanbul, Turkey

    Mevlüt Teymür

Authors
  1. Ekin Deliktas-Ozdemir
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  2. Mevlüt Teymür
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Correspondence to Ekin Deliktas-Ozdemir.

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Appendices

Appendices

A.1

$$\begin{aligned} {\mathcal {D}}_{1}= & {} 2iR_{1} {\mathcal {M}}_{2}^{(1)}+2 \Lambda _{1}\frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(1)}+R_{1}{\mathcal {N}}^{(1)}+2iR_{1}c_1^2 k \frac{\partial \textit{A}_1}{\partial \xi } -{\mathcal {P}}_{1}+2i\mathcal {\zeta }_{1}^{(1)}{} \textit{A}_1, \\ {\mathcal {D}}_{2}= & {} (-2iR_{1}/pkc_{1}^{2})\left( -\omega + \frac{kc_{1}^{2}}{V_g}\right) \frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(1)}+2c_1^2\frac{k}{p} \frac{\partial k}{\partial \xi }{} \textit{A}_1 R_{1}, \\ {\mathcal {D}}_{3}= & {} 2iR_{2} {\mathcal {M}}_{2}^{(1)}+2 \Lambda _{2}\frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(1)}+R_{2}{\mathcal {N}}^{(1)}+2iR_{2}c_1^2 k \frac{\partial \textit{A}_1}{\partial \xi }-{\mathcal {P}}_{2}+2i\mathcal {\zeta }_{2}^{(1)}{} \textit{A}_1,\\ {\mathcal {D}}_{4}= & {} (2iR_{2}/pkc_{1}^{2})\left( -\omega +\frac{kc_{1}^{2}}{V_g}\right) \frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(1)}-2c_1^2\frac{k}{p} \frac{\partial k}{\partial \xi }{} \textit{A}_1 R_2, \\ {\mathcal {D}}_{5}= & {} {\mathcal {Q}}_{1},\quad {\mathcal {D}}_{6}={\mathcal {Q}}_{2} \\ {\mathcal {D}}_{7}= & {} 2iR_{3} {\mathcal {M}}_{2}^{(2)}+2 \Lambda _{3}\frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(2)}+R_{3}{\mathcal {N}}^{(2)}+2iR_{3}c_2^2 k \frac{\partial \textit{A}_1}{\partial \xi }+2i\mathcal {\zeta }_{3}^{(2)}{} \textit{A}_1, \\ {\mathcal {D}}_{8}= & {} (2R_{3}/vkc_{2}^{2})\left( -\omega +\frac{kc_{2}^{2}}{V_g}\right) \frac{\partial }{\partial \tau }{\mathcal {M}}_{1}^{(2)}+2ic_2^2\frac{k}{v} \frac{\partial k}{\partial \xi }{} \textit{A}_1 R_3,\\ {\mathcal {D}}_{9}= & {} n_{2}k^{4}(-3+2v^{2}+9v^{4})|R_{3}|^{2}R_{3}|\textit{A}_{1}|^{2}{} \textit{A}_{1} , \end{aligned}$$

with

$$\begin{aligned} {\mathcal {M}}_{\beta }^{(\alpha )}=\left( -\omega +\frac{kc_{\alpha }^{2}}{V_g}\right) \frac{\partial \textit{A}_{\beta } }{\partial \tau }, \quad {\mathcal {N}}^{(\alpha )}=\left( \frac{c_{\alpha }^{2}}{V_g^2}-1\right) \frac{\partial ^{2}{} \textit{A}_{1} }{\partial \tau ^{2}}, \quad \Lambda _{\alpha }=\left( \frac{1}{V_g}\frac{\partial R_{\alpha }}{\partial k}+\frac{\partial R_{\alpha }}{\partial \omega }\right) \end{aligned}$$
$$\begin{aligned} {\mathcal {P}}_{1}= & {} n_{1}k^{4}\left( |R_{1}|^{2}R_{1}(9+2p^{2}+9p^{4})\right) |\textit{A}_{1}|^{2}{} \textit{A}_{1}, \quad {\mathcal {P}}_{2}=n_{1}k^{4}\left( |R_{2}|^{2}R_{2}(9+2p^{2}+9p^{4})\right) |\textit{A}_{1}|^{2}{} \textit{A}_{1}, \\ \mathcal {\zeta }_{\beta }^{(\alpha )}= & {} k c_{\alpha }^{2} \frac{\partial R_{\beta }}{\partial \xi }+\frac{R_{\beta }}{2}c_{\alpha }^{2} \frac{\partial k}{\partial \xi }, \\ {\mathcal {Q}}_{1}= & {} n_{1}k^{4}|\textit{A}_{1}|^{2}{} \textit{A}_{1}R_{1}^{3}(-3-2p^{2}+9p^{4}),\quad {\mathcal {Q}}_{2}=n_{1}k^{4}|\textit{A}_{1}|^{2}{} \textit{A}_{1}R_{2}^{3}(-3-2p^{2}+9p^{4}), \end{aligned}$$

A.2

$$\begin{aligned} F_{1}= & {} - \frac{1}{8}{\mathrm{{e}}^{3f_2kv}}\left( {f_1 + h} \right) {k^4}\left( {9 + 2{p^2} + 9{p^4}} \right) \beta _1 \sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^3}, \\ F_{2}= & {} \frac{1}{32pv}{\mathrm{{e}}^{3f_2kv}}{k^3}\{4p\left( {9 + 2{v^2} - 3{v^4}} \right) \beta _2\gamma - 4f_2 kp\left( {9 + 2{p^2} + 9{p^4}} \right) v\beta _1\sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^2}\\&-\, 4\left( { - 9 + 2{p^2} + 3{p^4}} \right) v\beta _1\tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \\&+\, \left( {27 + 2{p^2} - 33{p^4}} \right) v\beta _1\sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^2} \tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \},\\ F_{3}= & {} -\frac{{\mathrm{{e}}^{3f_2kv}}{k^2}\sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^3}}{32 p^2 v^2}\{ 3{p^2}\beta _2\left( { - 3 + 2{v^2} + 9{v^4}} \right) \cos \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \\&+\, \left( { - 3{v^2}\beta _1 + {p^2}\left( { - 3\beta _2 + 9{v^4}\beta _2 + {v^2}\left( { - 2\beta _1 + 9{p^2}\beta _1 + 2\beta _2} \right) } \right) } \right) \cos \left[ {3\left( {f_1 - f_2 + h} \right) kp} \right] \\&-\,4f_2kp {v^2}\beta _1\left( {9 + 2{p^2} + 9{p^4}} \right) \sin \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \}. \end{aligned}$$

A.3

$$\begin{aligned} G_{1}= & {} \frac{{}{\mathrm{{e}}^{f_2kv}}\left( {f_1 + h} \right) \sec \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] }{2p^2v}\{2{k^2}{p^2}v\left( {p\tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \left( {f{_1^\prime } - f{_2^\prime }} \right) + vf{_2^\prime }} \right) \\&+\, \left( {v + p\left( {2f_2kp + v\left( {p - 2\left( {f_1 - f_2 + h} \right) k\tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] } \right) } \right) } \right) k'\}, \\ G_{2}= & {} \frac{1}{{2k{p^3}{v^3}}}{}{\mathrm{{e}}^{f_2kv}}\{2{k^2}{p^2}{v^3}(- f_2k{p^2}\tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] f{_2^\prime } - v\tan \left[ {\left( {f1 - f_2 + h} \right) kp} \right] f{_2^\prime } \\&+\,2{k^2}{p^3}{v^3}\left( {\left( {f_2kv - \gamma } \right) f{_2^\prime } + \sec {{\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] }^2}\left( { - f{_1^\prime } + f{_2^\prime }} \right) } \right) ) \\&+\,p\left( {f_2k{v^3} + {p^2}\left( {\gamma + v\left( { - f{_2^2}{k^2}v\left( { - 2 + \gamma } \right) - v\gamma - f_2k\left( {{v^2}\left( { - 1 + \gamma } \right) + \gamma } \right) } \right) } \right) } \right) k' \\&+\, 2\left( {f_1 - f_2 + h} \right) kp{v^3}\sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^2}\\&+ {v^2}\left( { - v + {p^2}\left( { - v + f_2k\left( { - 2 + f_2kv} \right) } \right) } \right) \tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] k' \}, \\ G_{3}= & {} -\frac{{}{\mathrm{{e}}^{f_2kv}}f_2}{2kp^3v^3}\{ 2{k^2}{p^3}{v^3}f{_2^\prime } + p\left( { - f_2k{v^3} + {p^2}\left( { - 1 + f_2kv + {v^2}} \right) } \right) k' \\&+\,2kp{v^3}\sec {\left[ {\left( {f_1 - f_2 + h} \right) kp} \right] ^2}\left( {k{p^2}\left( {f{_1^\prime } - f{_2^\prime }} \right) - \left( {f_1 - f_2 + h} \right) k'} \right) \\&+\,{v^2}\tan \left[ {\left( {f_1 - f_2 + h} \right) kp} \right] \left( {2{k^2}{p^2}{v^2}f{_2^\prime } + \left( {v + {p^2}\left( {2f_2k + v} \right) } \right) k'} \right) \}. \end{aligned}$$

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Deliktas-Ozdemir, E., Teymür, M. Nonlinear surface SH waves in a half-space covered by an irregular layer. Z. Angew. Math. Phys. 73, 145 (2022). https://doi.org/10.1007/s00033-022-01783-z

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