Singularities in Surface Waves

  • G. Ruíz ChavarríaEmail author
  • T. Rodriguez Luna
Conference paper
Part of the Environmental Science and Engineering book series (ESE)


In this paper we investigate the evolution of surface waves produced by a parabolic wave maker. This system exhibits, among other, spatial focusing, wave breaking, the presence of caustics and points of full destructive interference (dislocations). The first approximation to describe this system is the ray theory (also known as geometrical optics). According to it, the wave amplitude becomes infinite along a caustic. However this does not happen because geometrical optics is only an approximation which does not take into account the wave behavior of the system. Otherwise, in ray theory the wave breaking does not hold and interference occurs only in regions delimited by caustics. A second step is the use of a diffraction integral. For linear waves this task has been made by Pearcey (1946) (Pearcey, Philos Mag 37 (1946) 311–317) for electromagnetic waves. However the system under study is non linear and some features have not counterpart in the linear theory. In the paper our attention is focused on three types of singularities: caustics, wave breaking and dislocations. The study we made is both experimental and numerical. The experiments were conducted with two different methods, namely, Schlieren synthetic for small amplitudes and Fourier Transform Profilometry. With respect the numerical simulations, the Navier-Stokes and continuity equations were solved in polar coordinates in the shallow water approximation.


Surface Wave Wave Amplitude Wave Field Wave Breaking Fringe Pattern 
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.



G. Ruíz Chavarría acknowledges DGAPA-UNAM by support for a sabbatical period at IRPHE between September 2010 and August 2011. Additionally the authors acknowledge support by DGAPA-UNAM under project 116312, Vorticidad y ondas no lineales en fluidos. Authors acknowledge also Eric Falcon from University Paris Diderot for his assistance in the implementation of the Fourier Transform Profilometry.


  1. Babanin A (2011) Breaking and dissipation of the ocean surface waves. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  2. Cobelli PJ, Maurel A, Pagneux V, Petitjeans P (2009) Global measurement of water waves by Fourier transform profilometry. Exp Fluids 46:1037–1047CrossRefGoogle Scholar
  3. Elmore WC, Heald MA (1969) Phys waves. McGraw Hill, New YorkGoogle Scholar
  4. Maurel A, Cobelli PJ, Pagneux V, Petitjeans P (2009) Experimental and theoretical inspection of the phase to height relation in Fourier transform profilometry. Appl Opt 48:380–392CrossRefGoogle Scholar
  5. Meunier P, Leweke T (2003) Analysis and minimization of errors due to high gradients in particle image velocimetry. Exp Fluids 35:408–421.
  6. Moisy F, Rabaud M, Salsac K (2009) A synthetic Schlieren method for the measurement of the topography of a liquid surface. Exp Fluids 46:1021–1036CrossRefGoogle Scholar
  7. Paris RB, Kaminsky D (2001) Asymptotic Mellin-Barnes integrals. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. Pearcey I (1946) The structure of an electromagnetic field in the neighborhood of a cusp of a caustic. Philos Mag 37:311–317CrossRefGoogle Scholar
  9. Ruiz-Chavarria G, Le Bars M, Le Gal P (2014) Focusing of surface waves in experimental and computational fluid mechanics. 315–325. ISSN 1431-2492Google Scholar
  10. Zemenzer S (2011) Etude experimentale du deferlement des vagues en eau profonde par focalisation spatiale; Rapport de stage, Universite de la MediterraneeGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoMexicoMexico

Personalised recommendations