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Ocean Dynamics

, Volume 69, Issue 1, pp 21–27 | Cite as

Modulational instability of two obliquely interacting waves in two-layer fluid domain

  • Anushri PurkaitEmail author
  • Suma Debsarma
Article

Abstract

This research report is concerned with the derivation of two evolution equations of two obliquely interacting wave packets in a two-layer fluid domain in which the lower fluid is of infinite depth. These two evolution equations are then employed to perform stability analysis of two obliquely interacting uniform wave trains. Results of the stability analysis are shown graphically. It is observed that the growth rate of instability increases as the thickness of the upper lighter fluid increases. If the angle of interaction between the two waves is acute then the growth rate of instability decreases with the increase in the angle; but the result is reversed if the angle of interaction is obtuse. Also, the growth rate of instability of one wave train increases with the increase in the amplitude of the other wave train.

Keywords

Evolution equation Gravity waves Nonlinear interaction Stability Two-layer fluid 

Notes

Acknowledgements

Authors are grateful to a reviewer for the helpful comments and suggestions.

References

  1. Choi W, Camassa R (1999) Fully nonlinear internal waves in a two-fluid system. J Fluid Mech 396:1–36.  https://doi.org/10.1017/S0022112099005820 CrossRefGoogle Scholar
  2. Debsarma S, Das KP, Kirby JT (2010) Fully nonlinear higher-order model equations for long internal waves in a two-fluid system. J Fluid Mech 654:281–303.  https://doi.org/10.1017/50022112010000601 CrossRefGoogle Scholar
  3. Dysthe KB (1979) Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc R Soc Lond A369:105–114CrossRefGoogle Scholar
  4. Gramstad O, Trulsen K (2011) Fourth-order coupled nonlinear Schrödinger equations for gravity waves on deep water. Phys Fluids 2(1-9):062101Google Scholar
  5. Kundu S, Debsarma S, Das KP (2013) Modulational instability in crossing sea states over finite depth water. Phys Fluids 25(1-13):066605.  https://doi.org/10.1016/j.ocemod.2015.07.017 CrossRefGoogle Scholar
  6. Kurkina OE, Kurkin AA, Soomere T, Pelinovsky EN, Rouvinskaya EA (2011) Higher-order(2 + 4) Korteweg-de Vries-like equation for interfacial waves in a symmetric three layer fluid. vol 23, pp 116602(1-13).  https://doi.org/10.1063/1.3657816
  7. Onorato M, Osborne AR, Serio M (2006) Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys Rev Lett 96(1-4):014503.  https://doi.org/10.1103/PhysRevLett.96.014503 CrossRefGoogle Scholar
  8. Senapati S, Debsarma S, Das KP (2010) Fourth-order evolution equation for a surface gravity wave packet in a two layer fluid. Int J Appl Mech Engg 15(4):1215–1225Google Scholar
  9. Shukla PK, Kourakis I, Eliasson B, Marklund M, Stenflo L (2006) Instability and evolution of nonlinearly interacting water waves. Phys Rev Lett 97(1-4):094501.  https://doi.org/10.1103/PhysRevLett.97.094501 CrossRefGoogle Scholar
  10. Yih C (1974) Progressive waves of permanent form in continuously stratified fluids. Phys Fluids 17(1974):1489.  https://doi.org/10.1063/1.1694923 CrossRefGoogle Scholar
  11. Yuan Y, Li J, Cheng Y (2007) Validity ranges of interfacial wave theories in a two-layer fluid system. Acta Mech Sin 23:597–607.  https://doi.org/10.1007/s10409-007-0101-6 CrossRefGoogle Scholar
  12. Yuen HC, Lake BM (1982) Nonlinear dynamics of deep-water gravity waves. Adv Appl Mech 22:67–229.  https://doi.org/10.1016/S0065-2156(08)70066.8 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mozilpur Shyamsundar Balika Vidyalaya (High)JaynagarIndia
  2. 2.Applied MathematicsUniversity of CalcuttaKolkataIndia

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