Skip to main content
SpringerLink
Log in

Magnetoelastic shear wave propagation in pre-stressed anisotropic media under gravity

  • Research Article
  • Published:
Acta Geophysica Aims and scope Submit manuscript

Abstract

The present study investigates the propagation of shear wave (horizontally polarized) in two initially stressed heterogeneous anisotropic (magnetoelastic transversely isotropic) layers in the crust overlying a transversely isotropic gravitating semi-infinite medium. Heterogeneities in both the anisotropic layers are caused due to exponential variation (case-I) and linear variation (case-II) in the elastic constants with respect to the space variable pointing positively downwards. The dispersion relations have been established in closed form using Whittaker’s asymptotic expansion and were found to be in the well-agreement to the classical Love wave equations. The substantial effects of magnetoelastic coupling parameters, heterogeneity parameters, horizontal compressive initial stresses, Biot’s gravity parameter, and wave number on the phase velocity of shear waves have been computed and depicted by means of a graph. As a special case, dispersion equations have been deduced when the two layers and half-space are isotropic and homogeneous. The comparative study for both cases of heterogeneity of the layers has been performed and also depicted by means of graphical illustrations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Abd-Alla AM, Ahmad SM (1999) Propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium. Appl Math Comput 106(2–3):265–275. doi:10.1016/S0096-3003(98)10128-5

    Google Scholar 

  • Acharya DP, Roy I, Sengupta S (2009) Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media. Acta Mech 202(1–4):35–45. doi:10.1007/s00707-008-0027-5

    Article  Google Scholar 

  • Anderson DL (1989) Theory of the Earth. Blackwell Scientific Publ, Boston

    Google Scholar 

  • Backus GE (1962) Long-wave elastic anisotropy produced by horizontal layering. J Geophys Res 67(11):4427–4440. doi:10.1029/JZ067i011p04427

    Article  Google Scholar 

  • Bhattacharya J (1962) On the dispersion curve for Love wave due to irregularity in the thickness of the transversely isotropic crustal layer. Gerl Beitr Geophys 6:324–334

    Google Scholar 

  • Bieńkowski A, Szewczyk R, Salach J (2010) Industrial application of magnetoelastic force and torque sensors. Acta Phys Pol A 118(5):1008–1009

    Article  Google Scholar 

  • Biot MA (1965) Mechanics of incremental deformations. Wiley, New York

    Google Scholar 

  • Chattopadhyay A (1975) On the propagation of Love type waves in an intermediate non- homogeneous layers lying between two semi-infinite homogeneous elastic media. Gerl Beitr Geophys 84(3/4):327–334

    Google Scholar 

  • Chattopadhyay A, Choudhury S (1990) Propagation, reflection and transmission of magnetoelastic shear waves in a self-reinforced medium. Int J Eng Sci 28(6):485–495. doi:10.1016/0020-7225(90)90051-J

    Article  Google Scholar 

  • Chattopadhyay A, Kar BK (1977a) On the propagation of Love-type waves in a non-homogeneous internal stratum of finite thickness lying between two semi-infinite isotropic media. Gerl Beitr Geophys 86(5):407–412

    Google Scholar 

  • Chattopadhyay A, Kar BK (1977b) On the dispersion curves of Love type waves in an internal stratum of finite thickness under initial stress lying between two semi-infinite isotropic elastic media. Gerl Beitr Geophys 86:493–497

    Google Scholar 

  • Chattopadhyay A, Singh AK (2014) Propagation of a crack due to magneto-elastic shear waves in a self-reinforced medium. J Vib Cont 20(3):406–420. doi:10.1177/1077546312458134

    Article  Google Scholar 

  • Chattopadhyay A, Gupta S, Sharma VK, Kumari P (2009) Propagation of shear waves in viscoelastic medium at irregular boundaries. Acta Geophys 58(2):195–214. doi:10.2478/s11600-009-0060-3

    Google Scholar 

  • Chattopadhyay A, Gupta S, Sharma VK, Kumari P (2010) Effect of point source and heterogeneity on the propagation of SH waves. Int J Appl Math Mech 6(9):76–89

    Google Scholar 

  • Dey S, Addy SK (1978) Love waves under initial stresses. Acta Geophys Pol 26(1):47–54

    Google Scholar 

  • Ding G, Dravinski M (1996) Scattering of in multi-layered media with irregular interfaces. Earthq Eng Struct Dyn 25(12):1391–1404. doi:10.1002/(SICI)1096-9845(199612)25:12<1391:AID-EQE617>3.0.CO;2-W

    Article  Google Scholar 

  • Dutta S (1963) Love wave in a non-homogeneous internal stratum lying between two semi-infinite isotropic media. Geophysics 28(2):156–160. doi:10.1190/1.1439162

    Article  Google Scholar 

  • Gogna ML (1976) On Love waves in heterogeneous layered media. Geophys J Roy Astr Soc 45(2):357–370. doi:10.1111/j.1365-246X.1976.tb00331.x

    Article  Google Scholar 

  • Guz AN (2002) Elastic waves in bodies with initial (residual) stresses. Int Appl Mech 38(1):23–59. doi:10.1023/A:1015379824503

    Article  Google Scholar 

  • Kumari N, Sahu SA, Chattopadhyay A, Singh AK (2015) Influence of heterogeneity on the propagation behavior of Love-type waves in a layered isotropic structure. Int J Geomech 16(2):04015062. doi:10.1061/(ASCE)GM.1943-5622.0000541

    Article  Google Scholar 

  • Sahu SA, Saroj PK, Paswan B (2014) Shear waves in a heterogeneous fiber-reinforced layer over a half-space under gravity. Int J Geomech 15(2):04014048. doi:10.1061/(ASCE)GM.1943-5622.0000404

    Article  Google Scholar 

  • Sur SP (1963) Note on stresses produced by a shearing force moving over the boundary of a semi-infinite transversely isotropic solid. Pure Appl Geophys 55(1):72–76. doi:10.1007/BF02011217

    Article  Google Scholar 

  • Whittaker ET, Watson GN (1991) A Course of Modern Analysis. Universal Book Stall, New Delhi

    Google Scholar 

Download references

Acknowledgements

The authors convey their sincere thanks to Indian School of Mines, Dhanbad for providing JRF to Ms. Nirmala Kumari and also facilitating us with its best facility. Authors extend their sincere thanks to Mr. Santan Kumar for his valuable suggestion in development of the manuscript.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand, India

    Nirmala Kumari, Amares Chattopadhyay, Abhishek K. Singh & Sanjeev A. Sahu

Authors
  1. Nirmala Kumari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amares Chattopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abhishek K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sanjeev A. Sahu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nirmala Kumari.

Appendices

Appendix 1

$$\alpha_{2} = ikS_{2} + \eta ,\,\,\,\;T_{2} = k\sqrt {\frac{{S_{2}^{2} }}{{4R_{2}^{2} }} + \frac{{\left( {{{c^{2} } \mathord{\left/ {\vphantom {{c^{2} } {\beta_{2}^{2} }}} \right. \kern-0pt} {\beta_{2}^{2} }}\, - Q_{2} } \right)}}{{R_{2} }}\;} ,$$
$$Q = \left( {{{L_{3} } \mathord{\left/ {\vphantom {{L_{3} } {L_{2}^{\left( 0 \right)} }}} \right. \kern-0pt} {L_{2}^{\left( 0 \right)} }}} \right)\left\{ {\frac{g}{{4a_{1} }} + k\left( {\frac{Gs}{2} - 1 + \frac{{G^{2} \left( {s + 0.5} \right)^{2} }}{{2\left[ {4 + G\left( {s + 0.5} \right)^{2} } \right]}}} \right)} \right\}.$$

Appendix 2

$$h_{1} = \frac{{ik\,\,t_{H}^{\left( 1 \right)} \sin 2\varphi_{1} }}{{\left( {1 + t_{H}^{\left( 1 \right)} \sin^{2} \varphi_{1} } \right)}},\;\quad \xi_{1} = \frac{{k^{2} }}{{R_{1} }}\left[ {{{c^{2} } \mathord{\left/ {\vphantom {{c^{2} } {\beta_{1}^{2} }}} \right. \kern-0pt} {\beta_{1}^{2} }} - \left( {\frac{{N_{1}^{\left( 0 \right)} }}{{L_{1}^{\left( 0 \right)} }} + \frac{{P_{1}^{\left( 0 \right)} }}{{2L_{1}^{\left( 0 \right)} }} + t_{H}^{\left( 1 \right)} \cos^{2} \varphi } \right)} \right],$$
$$k_{1} = - \frac{{{{h_{1} } \mathord{\left/ {\vphantom {{h_{1} } 2}} \right. \kern-0pt} 2}}}{{R_{1} \sqrt {h_{1}^{2} - 4\xi_{1} } }},\quad \mu_{1} = \frac{1}{4} \times \sqrt {1 + {1 \mathord{\left/ {\vphantom {1 {R_{1}^{2} }}} \right. \kern-0pt} {R_{1}^{2} }} - {2 \mathord{\left/ {\vphantom {2 {R_{1} }}} \right. \kern-0pt} {R_{1} }}} ,$$
$$h_{2} = \frac{{ik\left\{ {t_{H}^{\left( 2 \right)} \sin 2\varphi } \right\}}}{{R_{2} }},\quad \chi_{2} \left( z \right) = \frac{1}{{R_{2} }}\left\{ {ik\,\left( {t_{H}^{\left( 2 \right)} \sin 2\varphi } \right) + \frac{b}{{\left( {1 + bz} \right)}}} \right\},$$
$$\xi_{2} = \frac{{k^{2} }}{{R_{2} }}\left[ {{{c^{2} } \mathord{\left/ {\vphantom {{c^{2} } {\beta_{2}^{2} }}} \right. \kern-0pt} {\beta_{2}^{2} }} - \left( {\frac{{N_{2}^{\left( 0 \right)} }}{{L_{2}^{\left( 0 \right)} }} + \frac{{P_{2}^{\left( 0 \right)} }}{{2L_{2}^{\left( 0 \right)} }} + t_{H}^{\left( 2 \right)} \cos^{2} \varphi } \right)} \right],$$
$$k_{2} = - \frac{{{{h_{2} } \mathord{\left/ {\vphantom {{h_{2} } 2}} \right. \kern-0pt} 2}}}{{R_{2} \sqrt {h_{2}^{2} - 4\xi_{2} } }},\quad \mu_{2} = \frac{1}{4} \times \sqrt {1 + {1 \mathord{\left/ {\vphantom {1 {R_{2}^{2} }}} \right. \kern-0pt} {R_{2}^{2} }} - {2 \mathord{\left/ {\vphantom {2 {R_{2} }}} \right. \kern-0pt} {R_{2} }}},$$
$$m_{1} = \left\{ {\mu_{1}^{2} - \left( {k_{1} - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)^{2} } \right\},\quad m_{2} = \left\{ {\mu_{1}^{2} - \left( {k_{1} + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)^{2} } \right\},$$
$$m_{3} = \left\{ {\mu_{2}^{2} - \left( {k_{2} - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)^{2} } \right\},\quad m_{4} = \left\{ {\mu_{2}^{2} - \left( {k_{2} + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)^{2} } \right\},$$
$$\begin{aligned} f_{1} = \frac{{\left( {1 - aH_{2} } \right)^{{k_{1} + 1 - {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}}} }}{{q_{1}^{{ - k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} + aq_{1} } \right)H_{2} - q_{1} } \right\}}} L_{1}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{1} }}{2} - \frac{{R_{1} h_{1} }}{2} - \frac{a}{{2\left( {1 - aH_{2} } \right)}}} \right)\left( {1 + \frac{{m_{1} }}{{\left( {1 - aH_{2} } \right)q_{1} }}} \right) + } \right. \\ \left. {\quad \quad aq_{1} R_{1} \left\{ {\left( {\frac{{k_{1} }}{{\left( {1 - aH_{2} } \right)q_{1} }} - \frac{1}{2}} \right)\left( {1 + \frac{{m_{1} }}{{\left( {1 - aH_{2} } \right)q_{1} }}} \right) - \frac{{m_{1} }}{{\left( {1 - aH_{2} } \right)^{2} q_{1}^{2} }}} \right\}} \right], \\ \end{aligned}$$
$$\begin{aligned} f_{2} &= \frac{{\left( {1 - aH_{2} } \right)^{{1 - k_{1} - {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}}} }}{{\left[ { - q_{1} } \right]^{{k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} - aq_{1} } \right)H_{2} + q_{1} } \right\}}} L_{1}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{1} }}{2} - \frac{{R_{1} h_{1} }}{2} - \frac{a}{{2\left( {1 - aH_{2} } \right)}}} \right)\left( {1 - \frac{{m_{2} }}{{\left( {1 - aH_{2} } \right)q_{1} }}} \right) + } \right. \\ \hfill & \quad \left. aq_{1} R_{1} \left\{ {\left( {\frac{{ - k_{1} }}{{\left( {1 - aH_{2} } \right)q_{1} }} + \frac{1}{2}} \right)\left( {1 - \frac{{m_{2} }}{{\left( {1 - aH_{2} } \right)q_{1} }}} \right) + \frac{{m_{2} }}{{\left( {1 - aH_{2} } \right)^{2} q_{1}^{2} }}} \right\} \right], \\ \end{aligned}$$
$$f_{3} = \frac{{\left( {1 - aH_{1} } \right)^{{k_{1} - {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}}} }}{{q_{1}^{{ - k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} + aq_{1} } \right)H_{1} - q_{1} } \right\}}} \left( {1 + \frac{{m_{1} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right),$$
$$f_{4} = \frac{{\left( {1 - aH_{1} } \right)^{{ - \left( {k_{1} + {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}} \right)}} }}{{\left[ { - q_{1} } \right]^{{k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} - aq_{1} } \right)H_{1} + q_{1} } \right\}}} \left( {1 - \frac{{m_{2} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right),$$
$$f_{5} = \frac{{\left( {1 - bH_{1} } \right)^{{k_{2} - {1 \mathord{\left/ {\vphantom {1 {2R_{2} \,\,}}} \right. \kern-0pt} {2R_{2} \,\,}}}} }}{{\left[ {q_{2} } \right]^{{ - k_{2} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{2} + bq_{2} } \right)H_{1} - q_{2} } \right\}}} \left( {1 + \frac{{m_{3} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right),$$
$$f_{6} = \frac{{\left( {1 - bH_{1} } \right)^{{ - \left( {k_{2} + {1 \mathord{\left/ {\vphantom {1 {2R_{2} \,\,}}} \right. \kern-0pt} {2R_{2} \,\,}}} \right)}} }}{{\left[ { - q_{2} } \right]^{{k_{2} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{2} - bq_{2} } \right)H_{1} + q_{2} } \right\}}} \left( {1 - \frac{{m_{4} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right),$$
$$\begin{aligned} f_{7} &= \frac{{\left( {1 - aH_{1} } \right)^{{1 + k_{1} - {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}}} }}{{\left[ {q_{1} } \right]^{{ - k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} + aq_{1} } \right)H_{1} - q_{1} } \right\}}} L_{1}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{1} }}{2} - \frac{{R_{1} h_{1} }}{2} - \frac{a}{{2\left( {1 - aH_{1} } \right)}}} \right)\left( {1 + \frac{{m_{1} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right) + } \right. \\ & \quad \left. {aq_{1} R_{1} \left\{ {\left( {\frac{{k_{1} }}{{\left( {1 - aH_{1} } \right)q_{1} }} - \frac{1}{2}} \right)\left( {1 + \frac{{m_{1} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right) - \frac{{m_{1} }}{{\left( {1 - aH_{1} } \right)^{2} q_{1}^{2} }}} \right\}} \right], \\ \end{aligned}$$
$$\begin{aligned} \hfill f_{8} = \frac{{\left( {1 - aH_{1} } \right)^{{1 - \left( {k_{1} + {1 \mathord{\left/ {\vphantom {1 {2R_{1} \,\,}}} \right. \kern-0pt} {2R_{1} \,\,}}} \right)}} }}{{\left[ { - q_{1} } \right]^{{k_{1} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{1} - aq_{1} } \right)H_{1} + q_{1} } \right\}}} L_{1}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{1} }}{2} - \frac{{R_{1} h_{1} }}{2} - \frac{a}{{2\left( {1 - aH_{1} } \right)}}} \right)\left( {1 - \frac{{m_{2} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right) + } \right. \\ \hfill \left. {aq_{1} R_{1} \left\{ {\left( {\frac{{ - k_{1} }}{{\left( {1 - aH_{1} } \right)q_{1} }} + \frac{1}{2}} \right)\left( {1 - \frac{{m_{2} }}{{\left( {1 - aH_{1} } \right)q_{1} }}} \right) + \frac{{m_{2} }}{{\left( {1 - aH_{1} } \right)^{2} q_{1}^{2} }}} \right\}} \right], \\ \end{aligned}$$
$$\begin{aligned} f_{9} & = \frac{{\left( {1 - bH_{1} } \right)^{{1 + k_{2} - {1 \mathord{\left/ {\vphantom {1 {2R_{2} \,\,}}} \right. \kern-0pt} {2R_{2} \,\,}}}} }}{{\left[ {q_{2} } \right]^{{ - k_{2} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{2} + bq_{2} } \right)H_{1} - q_{2} } \right\}}} L_{2}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{2} }}{2} - \frac{{R_{2} h_{2} }}{2} - \frac{b}{{2\left( {1 - bH_{1} } \right)}}} \right)\left( {1 + \frac{{m_{3} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right) + } \right. \\ & \quad\left. {bq_{2} R_{2} \left\{ {\left( {\frac{{k_{2} }}{{\left( {1 - bH_{1} } \right)q_{2} }} - \frac{1}{2}} \right)\left( {1 + \frac{{m_{3} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right) - \frac{{m_{3} }}{{\left( {1 - bH_{1} } \right)^{2} q_{2}^{2} }}} \right\}} \right], \\ \end{aligned}$$
$$\begin{aligned} f_{10} &= \frac{{\left( {1 - bH_{1} } \right)^{{1 - \left( {k_{2} + {1 \mathord{\left/ {\vphantom {1 {2R_{2} \,\,}}} \right. \kern-0pt} {2R_{2}}}} \right)}} }}{{\left[ { - q_{2} } \right]^{{k_{2} }} }}e^{{\frac{1}{2}\left\{ {\left( {h_{2} - bq_{2} } \right)H_{1} + q_{2} } \right\}}} L_{2}^{\left( 0 \right)} \times \left[ {\left( {\frac{{ikS_{2} }}{2} - \frac{{R_{2} h_{2} }}{2} - \frac{b}{{2\left( {1 - bH_{1} } \right)}}} \right)\left( {1 - \frac{{m_{4} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right) + } \right. \\ & \quad \left. {bq_{2} R_{2} \left\{ {\left( {\frac{{ - k_{2} }}{{\left( {1 - bH_{1} } \right)q_{2} }} + \frac{1}{2}} \right)\left( {1 - \frac{{m_{4} }}{{\left( {1 - bH_{1} } \right)q_{2} }}} \right) + \frac{{m_{4} }}{{\left( {1 - bH_{1} } \right)^{2} q_{2}^{2} }}} \right\}} \right], \\ \end{aligned}$$
$$f_{11} = \frac{{e^{{{{ - q_{2} } \mathord{\left/ {\vphantom {{ - q_{2} } 2}} \right. \kern-0pt} 2}}} }}{{\left( {q_{2} } \right)^{{ - k_{2} }} }}\left( {1 + \frac{{m_{3} }}{{q_{2} }}} \right),\quad f_{12} = \frac{{e^{{{{q_{2} } \mathord{\left/ {\vphantom {{q_{2} } 2}} \right. \kern-0pt} 2}}} }}{{\left( { - q_{2} } \right)^{{k_{2} }} }}\left( {1 - \frac{{m_{4} }}{{q_{2} }}} \right),\quad f_{13} = \frac{{e^{{\frac{2}{G}}} }}{{\sqrt {a_{1} } }}\left[ { - \frac{4}{G}} \right]^{ - s} \left( {1 + \frac{{G\left( {s + 0.5} \right)^{2} }}{4}} \right),$$
$$f_{14} = L_{2}^{\left( 0 \right)} e^{{ - \frac{{q_{2} }}{2}}} \left( {q_{2} } \right)^{{k_{2} }} \left[ {\left( {\frac{{ikS_{2} }}{2} - \frac{{R_{2} h_{2} }}{2} - \frac{b}{2}} \right)\left( {1 + \frac{{m_{3} }}{{q_{2} }}} \right) + bq_{2} R_{2} \left\{ {\left( {\frac{{k_{2} }}{{q_{2} }} - \frac{1}{2}} \right)\left( {1 + \frac{{m_{3} }}{{q_{2} }}} \right) - \frac{{m_{3} }}{{q_{2}^{2} }}} \right\}} \right],$$
$$f_{15} = L_{2}^{\left( 0 \right)} e^{{\frac{{q_{2} }}{2}}} \left( { - q_{2} } \right)^{{ - k_{2} }} \left[ {\left( {\frac{{ikS_{2} }}{2} - \frac{{R_{2} h_{2} }}{2} - \frac{b}{2}} \right)\left( {1 - \frac{{m_{4} }}{{q_{2} }}} \right) + bq_{2} R_{2} \left\{ {\left( {\frac{{ - k_{2} }}{{q_{2} }} + \frac{1}{2}} \right)\left( {1 - \frac{{m_{4} }}{{q_{2} }}} \right) + \frac{{m_{4} }}{{q_{2}^{2} }}} \right\}} \right],$$
$$f_{16} = L_{3} \sqrt {a_{1} } \,e^{{\frac{2}{G}}} \left[ { - \frac{4}{G}} \right]^{ - s} \left[ {\frac{g}{{4a_{1} }} - k\left( {1 - \frac{sG}{2}} \right)\left( {1 + \frac{{G\left( {s + 0.5} \right)^{2} }}{4}} \right) + \frac{{kG^{2} \left( {s + 0.5} \right)^{2} }}{8}} \right].$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumari, N., Chattopadhyay, A., Singh, A.K. et al. Magnetoelastic shear wave propagation in pre-stressed anisotropic media under gravity. Acta Geophys. 65, 189–205 (2017). https://doi.org/10.1007/s11600-017-0016-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11600-017-0016-y

Keywords

Search

Navigation