Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-27T07:57:21.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On the recovery of solitary wave profiles from pressure measurements

Published online by Cambridge University Press:  05 April 2012

A. Constantin
A. Constantin*
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria
*
†Email address for correspondence: adrian.constantin@kcl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive an explicit formula that permits the recovery of the profile of an irrotational solitary water wave from pressure data measured at the flat bed of the fluid domain. The formula is valid for the governing equations and applies to waves of small and large amplitude.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 699 , 25 May 2012 , pp. 376 - 384
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.Google Scholar
2. Amick, C. J. 1987 Bounds for water waves. Arch. Rat. Mech. Anal. 99, 91–114.CrossRefGoogle Scholar
3. Amick, C. J., Fraenkel, L. E. & Toland, J. F. 1982 On the Stokes conjecture for the wave of extreme form. Acta Mathematica 148, 193–214.CrossRefGoogle Scholar
4. Amick, C. J. & Toland, J. F. 1981 On solitary water waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9–95.CrossRefGoogle Scholar
5. Baquerizo, A. & Losada, M. A. 1995 Transfer function between wave height and wave pressure for progressive waves. Coast. Engng 24, 351–353.CrossRefGoogle Scholar
6. Beale, J. T. 1981 The existence of solitary water waves. Commun. Pure Appl. Maths 30, 373–389.CrossRefGoogle Scholar
7. Bishop, C. T. & Donelan, M. A. 1987 Measuring waves with pressure transducers. Coast. Engng 11, 309–328.Google Scholar
8. Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535.Google Scholar
9. Constantin, A. 2011 Nonlinear water waves with applications to wave-current interactions and tsunamis. In CBMS-NSF Regional Conference Series on Applied Mathematics, vol. 81. SIAM.Google Scholar
10. Constantin, A. & Escher, J. 2007 Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431.CrossRefGoogle Scholar
11. Constantin, A. & Escher, J. 2011 Analyticity of periodic travelling free surface water waves with vorticity. Ann. of Math. 173, 559–568.Google Scholar
12. Constantin, A., Escher, J. & Hsu, H.-C. 2011 Pressure beneath a solitary water wave: mathematical theory and experiments. Arch. Rat. Mech. Anal. 201, 251–269.Google Scholar
13. Constantin, A. & Strauss, W. 2010 Pressure beneath a Stokes wave. Commun. Pure Appl. Maths 63, 533–557.Google Scholar
14. Craig, W. & Sternberg, P. 1988 Symmetry of solitary waves. Commun. Part. Diff. Equ. 13, 603–633.Google Scholar
15. Deconinck, B., Henderson, D., Oliveras, K. L. & Vasan, V. 2011 Recovering the water-wave surface from pressure measurements. Proceedings of 10th International Conference WAVES, Vancouver July 25–29, 2011.Google Scholar
16. Dym, H. & McKean, H. P. 1972 Fourier Series and Integrals. Academic Press.Google Scholar
17. Escher, J. & Schlurmann, T. 2008 On the recovery of the free surface from the pressure within periodic travelling water waves. J. Nonlinear Math. Phys. 15, 50–57.Google Scholar
18. Friedlander, F. G. 1998 Introduction to the Theory of Distributions. Cambridge University Press.Google Scholar
19. Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Commun. Pure Appl. Maths 7, 517–550.CrossRefGoogle Scholar
20. Hur, V. M. 2011 Analyticity of rotational flows beneath solitary water waves. IMRN, doi:10.1093/imrn/rnr123.Google Scholar
21. Kuo, Y.-Y. & Chiu, J.-F. 1994 Transfer function between the wave height and wave pressure for progressive waves. Coast. Engng 23, 81–93.Google Scholar
22. McLeod, J. B. 1983 The rate of decay of solitary waves of finite amplitude. Appl. Anal. 17 (1983), 37–50.CrossRefGoogle Scholar
23. Oliveras, K. L., Vasan, V., Deconinck, B. & Henderson, D. 2012 Recovering surface elevation from pressure data. SIAM J. Appl. Maths (in press).Google Scholar
24. Reed, M. & Simon, B. 1975 Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press.Google Scholar
25. Spielvogel, E. R. 1970 A variational principle for waves of infinite depth. Arch. Rat. Mech. Anal. 39, 189–205.CrossRefGoogle Scholar
26. Stoker, J. J. 1957 Water Waves. Interscience.Google Scholar
27. Strichartz, R. S. 1994 A Guide to Distribution Theory and Fourier Transforms. CRC Press.Google Scholar
28. Tsai, C.-H., Huang, M.-C., Young, F.-J., Lin, Y.-C. & Li, H. W. 2005 On the recovery of surface wave by pressure transfer function. Ocean Engng 32, 1247–1259.CrossRefGoogle Scholar
29. Varvaruca, E. 2006 Singularities of Bernoulli free boundaries. Commun. Part. Diff. Equ. 31, 1451–1477.CrossRefGoogle Scholar