Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Nonlinear Internal Waves

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_363-3

Definition of the Subject

The objective of the present work is to study the generation and propagation of nonlinear internal waves in the frame of the shallow water theory. These waves are generated inside a stratified fluid occupying a semi-infinite channel of finite and constant depth by a wave maker situated in motion at the finite extremity of the channel. A distortion process is carried out to the variables and the nonlinear equations of the problem using a certain small parameter characterizing the motion of the wave maker and double series representations for the unknown functions are introduced. This procedure leads to a solution of the problem including a secular term, vanishing at the position of the wave maker. This inconvenient result is remedied using a multiple-scale transformation of variables, and it is shown that the free-surface and the interface elevations satisfy the well-known KdV equation. The initial conditions necessary for the solution of the KdV equations are...

Keywords

Internal Wave Fluid Layer Boussinesq Equation Secular Term Stratify Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Bibliography

Primary Literature

  1. Abou-Dina MS, Hassan FM (2006) Generation and propagation of nonlinear tsunamis in shallow water by a moving topography. Appl Math Comput 177:785–806CrossRefMATHMathSciNetGoogle Scholar
  2. Abou-Dina MS, Helal MA (1990) The influence of submerged obstacle on an incident wave in stratified shallow water. Eur J Mech B Fluids 9(6):545–564MATHGoogle Scholar
  3. Abou-Dina MS, Helal MA (1992) The effect of a fixed barrier on an incident progressive wave in shallow water. Il Nuovo Cimento 107B(3):331–344ADSCrossRefGoogle Scholar
  4. Abou-Dina MS, Helal MA (1995) The effect of a fixed submerged obstacle on an incident wave in stratified shallow water (Mathematical aspects). Il Nuovo Cimento B 110(8):927–942ADSCrossRefGoogle Scholar
  5. Barthelemy E, Kabbaj A, Germain JP (2000) Long surface wave scattered by a step in a two-layer fluid. Fluid Dyn Res 26:235–255ADSCrossRefMATHMathSciNetGoogle Scholar
  6. Benjamin TB (1966) Internal waves of finite amplitude and permanent form. J Fluid Mech 25:241–270ADSCrossRefMATHMathSciNetGoogle Scholar
  7. Benjamin TB (1967) Internal waves of permanent form of great depth. J Fluid Mech 29:559–592ADSCrossRefMATHGoogle Scholar
  8. Benney DJ, LIN CC (1960) On the secondary motion induced by oscillations in a shear flow. Phys Fluids 3:656–657ADSCrossRefGoogle Scholar
  9. Boussinesq MJ (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Acad Sci Paris CR Acad Sci 72:755–759MATHGoogle Scholar
  10. Chen Y, Liu PL-F (1995) Modified Boussinesq equations and associated parabolic models for water wave propagation. J Fluid Mech 288:351–381ADSCrossRefMATHMathSciNetGoogle Scholar
  11. Choi W, Camassa R (1996) Weakly nonlinear internal waves in a two-fluid system. J Fluid Mech 313:83–103ADSCrossRefMATHMathSciNetGoogle Scholar
  12. Choi W, Camassa R (1999) Fully nonlinear internal waves in a two-fluid system. J Fluid Mech 396:1–36ADSCrossRefMATHMathSciNetGoogle Scholar
  13. Craig W, Guyenne P, Kalisch H (2005) Hamiltonian long-wave expansions for free surfaces and interfaces. Commun Pure Appl Math 18:1587–1641CrossRefMathSciNetGoogle Scholar
  14. Davis RE, Acrivos A (1967) Solitary internal waves in deep water. J Fluid Mech 29:593–607ADSCrossRefMATHGoogle Scholar
  15. Garett C, Munk W (1979) Internal waves in the ocean. Annu Rev Fluid Mech 11:339–369ADSCrossRefGoogle Scholar
  16. Germain JP (1971) Sur le caractère limite de la théorie des mouvements des liquides parfaits en eau peu profonde. CR Acad Sci Paris Série A 273:1171–1174MathSciNetGoogle Scholar
  17. Germain JP (1972) Théorie générale d’un fluide parfait pesant en eau peu profonde de profondeur constante. CR Acad Sci Paris Série A 274:997–1000MATHMathSciNetGoogle Scholar
  18. Gobbi MF, Kirby JT, Wei G (2000) A fully nonlinear Boussinesq model for surface waves-part 2. Extension to O(kh)4. J Fluid Mech 405:181–210ADSCrossRefMATHMathSciNetGoogle Scholar
  19. Helal MA, Moline JM (1981) Nonlinear internal waves in shallow water: a theoretical and experimental study. Tellus 33:488–504ADSCrossRefMathSciNetGoogle Scholar
  20. Kabbaj A (1985) Contribution a l’etude du passage des ondes des gravite sur le talus continental et a la generation des ondes internes. These de doctorat d’etat, IMG Université de GrenobleGoogle Scholar
  21. Keulegan GH (1953) Hydrodynamical effects of gales on Lake Erie. J Res Natl Bur Std 50:99–109CrossRefGoogle Scholar
  22. Kubota T, Ko DRS, Dobbs LD (1978) Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth. AIAA J Hydrodyn 12:157–165Google Scholar
  23. LeBlond PH, Mysak LA (1978) Waves in ocean. Elsevier, AmsterdamGoogle Scholar
  24. Liu C-M, Lin M-C, Kong C-H (2008) Essential properties of Boussinesq equations for internal and surface waves in a two-fluid system. Ocean Eng 35:230–246CrossRefGoogle Scholar
  25. Long RR (1956) Solitary waves in one- and two-fluid systems. Tellus 8:460–471ADSCrossRefGoogle Scholar
  26. Lynett PJ, Liu PL-F (2002) A two-dimensional depth-integrated model for internal wave propagation over variable bathymetry. Wave Motion 36:221–240CrossRefMATHMathSciNetGoogle Scholar
  27. Madsen PA, Schaffer HA (1998) Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Philos Trans R Soc Lond A 356:3123–3184ADSCrossRefMATHMathSciNetGoogle Scholar
  28. Matsuno Y (1993) A unified theory of nonlinear wave propagation in two-layer fluid systems. J Phys Soc Jpn 62:1902–1916ADSCrossRefGoogle Scholar
  29. Nwogu O (1993) Alternative form of Boussinesq equations for nearshore wave propagation. J Waterways Port Coast Ocean Eng ASCE 119:618–638CrossRefGoogle Scholar
  30. Ono H (1975) Algebraic solitary waves in stratified fluids. J Phys Soc Jpn 39:1082–1091ADSCrossRefGoogle Scholar
  31. Peters AS, Stoker JJ (1960) Solitary waves in liquid having non-constant density. Comm Pure Appl Math 13:115–164CrossRefMATHMathSciNetGoogle Scholar
  32. Robinson RM (1969) The effect of a vertical barrier on internal waves. Deep-Sea Res 16:421–429Google Scholar
  33. Stoker JJ (1957) Water waves. Interscience, New YorkMATHGoogle Scholar
  34. Temperville A (1985) Contribution à la théorie des ondes de gravité en eau peu profonde. Thèse de doctorat d’état, IMG Université de GrenobleGoogle Scholar
  35. Wehausen JV, Laitone EV (1960) Surface waves. In: Handbuch der Physik, vol 9. Springer, BerlinGoogle Scholar
  36. Wei G, Kirby JT, Grilli ST, Subramanya R (1995) A fully nonlinear Boussinesq model for surface waves, part 1. Highly nonlinear, unsteady waves. J Fluid Mech 294:71–92ADSCrossRefMATHMathSciNetGoogle Scholar

Books and Reviews

  1. Germain JP, Guli L (1977) Passage d’une onde sur une barrier mince immergée en eau peu profonde. Ann Hydrog 5(746):7–11Google Scholar
  2. Pinettes M-J, Renouard D, Germain J-P (1995) Analytical and experimental study of the oblique passing of a solitary wave over a shelf in a two-layer fluid. Fluid Dyn Res 16:217–235ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt