The Balance of Material Momentum Applied to Water Waves

  • Manfred Braun
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


The balance of material momentum is applied to the motion of an ideal, incompressible fluid with special emphasis on water waves. To this end, the fluid flow is represented by its material or Lagrangian description. A variational approach using Hamilton’s principle is employed, with the incompressibility condition incorporated into the Lagrangian by means of a Lagrange multiplier. The balance of material momentum is obtained in its standard form known from nonlinear elasticity, however with the peculiarity that the dynamic Eshelby stress becomes hydrostatic and its divergence reduces to the (negative) gradient of an “Eshelby pressure”. The balance is applied to Gerstner’s nonlinear theory of water waves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Clamond D (2007) On the Lagrangian description of steady surface gravity waves. Journal of Fluid Mechanics 589:433–454Google Scholar
  2. Constantin A (2001) On the deep water wave motion. Journal of Physics A: Mathematical and General 34(7):1405–1417Google Scholar
  3. Constantin A, Monismith SG (2017) Gerstner waves in the presence of mean currents and rotation. Journal of Fluid Mechanics 820:511–528Google Scholar
  4. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Advances in Computational Mathematics 5(1):329–359Google Scholar
  5. Gerstner F (1804) Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile - Abh. Königl. Böhm. Ges. Wiss. Haase, PragGoogle Scholar
  6. Gołębiewska-Herrmann A (1981) On conservation laws of continuum mechanics. International Journal of Solids and Structures 17(1):1–9Google Scholar
  7. Gurtin ME (2000) Configurational Forces as Basic Concepts of Continuum Physics. Springer, New YorkGoogle Scholar
  8. Janssen P (2004) The Interaction of Ocean Waves and Wind. Cambridge University Press, CambridgeGoogle Scholar
  9. Kalisch H (2004) Periodic traveling water waves with isobaric streamlines. Journal of Nonlinear Mathematical Physics 11(4):461–471Google Scholar
  10. Kienzler R, Herrmann G (2000) Mechanics in Material Space - With Application to Defect and Fracture Mechanics. Springer, BerlinGoogle Scholar
  11. Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press, CambridgeGoogle Scholar
  12. Lazar M, Anastassiadis C (2007) Lie point symmetries, conservation and balance laws in linear gradient elastodynamics. Journal of Elasticity 88(1):5–25Google Scholar
  13. Le Méhauté B (1976) An Introduction to Hydrodynamics and Water Waves. Springer, New YorkGoogle Scholar
  14. Maugin GA (1993) Material Inhomogeneities in Elasticity. Chapman & Hall, LondonGoogle Scholar
  15. Maugin GA (2010) Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. Chapman & Hall/CRCGoogle Scholar
  16. Rahman M (1995) Water Waves. Clarendon Press, OxfordGoogle Scholar
  17. Rousseau M, Maugin GA, Berezovski M (2011) Elements of study on dynamic materials. Archive of Applied Mechanics 81(7):925–942Google Scholar
  18. Stokes GG (2009) On the Theory of Oscillatory Waves, Cambridge Library Collection - Mathematics, vol 1, Cambridge University Press, pp 197–229Google Scholar
  19. Stuhlmeier R (2015) Gerstner’s water wave and mass transport. Journal of Mathematical Fluid Mechanics 17(4):761–767Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Mechanics and RoboticsUniversity of Duisburg–EssenDuisburgGermany

Personalised recommendations