American Institute of Mathematical Sciences

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August  2014, 34(8): 3035-3043. doi: 10.3934/dcds.2014.34.3035

Recovering surface profiles of solitary waves on a uniform stream from pressure measurements

Hung-Chu Hsu 1,

1. 

Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom

Received  April 2013 Revised  September 2013 Published  January 2014

In this paper, we derive an explicit formula that permits to recover the free surface wave profile of an irrotational solitary wave with a uniform underlying current from pressure data measured at the flat bed of the fluid. The formula is valid for the governing equations and applies to waves of small and large amplitude.
Keywords: solitary wave, Pressure, current..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Hung-Chu Hsu. Recovering surface profiles of solitary waves on a uniform stream from pressure measurements. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3035-3043. doi: 10.3934/dcds.2014.34.3035
References:
[1]

C. J. Amick and J. F. Toland, On solitary water waves of finite amplitude, Arch. Rat. Mech. Anal., 76 (1981), 9-95. doi: 10.1007/BF00250799.  Google Scholar

[2]

A. Baquerizo and M. A. Losada, Transfer function between wave height and wave pressure for progressive waves, by Y.-Y. Kuo and J.-F. Chiu: Comments, Coast. Engng., 24 (1995), 351-353. doi: 10.1016/0378-3839(94)00038-Y.  Google Scholar

[3]

C. T. Bishop and M. A. Donelan, Measuring waves with pressure transducers, Coast. Engng., 11 (1987), 309-328. doi: 10.1016/0378-3839(87)90031-7.  Google Scholar

[4]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-575. doi: 10.1017/jfm.2012.490.  Google Scholar

[5]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom pressure measurements, J. Fluid Mech., 726 (2013), 547-558. doi: 10.1017/jfm.2013.253.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

A. Constantin, On the particle paths in solitary water waves, Quart. Appl. Math., 68 (2010), 81-90.  Google Scholar

[9]

A. Constantin, On the recovery of solitary wave profiles from pressure measurements, J. Fluid Mech., 699 (2012), 376-384. doi: 10.1017/jfm.2012.114.  Google Scholar

[10]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: Mathematical theory and experiments, Arch. Rat. Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.  Google Scholar

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[12]

W. Craig and P. Sternberg, Symmetry of solitary waves, Commun. Part. Diff. Equ., 13 (1988), 603-633. doi: 10.1080/03605308808820554.  Google Scholar

[13]

B. Deconinck, D. Henderson, K. L. Oliveras and V. Vasan, Recovering the water-wave surface from pressure measurements, in 10th International Conference on Mathematical and Numerical Aspects of Waves, WAVES 2011, Vancouver, 2011. Google Scholar

[14]

J. Escher and T. Schlurmann, On the recovery of the free surface from the pressure within periodic traveling water waves, J. Nonlinear Math. Phys., 15 (2008), 50-57. doi: 10.2991/jnmp.2008.15.s2.4.  Google Scholar

[15]

F. G. Friedlander, Introduction to the Theory of Distributions, Second edition, Cambridge University Press, Cambridge, 1998.  Google Scholar

[16]

D. Henry, On the pressure transfer function for solitary water waves with vorticity, Math. Ann., 357 (2013), 23-30. doi: 10.1007/s00208-013-0899-0.  Google Scholar

[17]

Y.-Y. Kuo and Y.-F. Chiu, Transfer function between the wave height and wave pressure for progressive waves, Coast. Engng., 23 (1994), 81-93. doi: 10.1016/0378-3839(94)90016-7.  Google Scholar

[18]

J. B. Mcleod, The rate of decay of solitary waves of finite amplitude, Appl. Anal., 17 (1983), 37-50. doi: 10.1080/00036818308839482.  Google Scholar

[19]

K. L. Oliveras, V. Vasan, B. Deconinck and D. Henderson, Recovering the water-wave profile from pressure measurements, SIAM J. Appl. Math., 72 (2012), 897-918. doi: 10.1137/110853285.  Google Scholar

[20]

R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994.  Google Scholar

[21]

C.-H. Tsai, M.-C. Huang, F.-J. Young, Y.-C. Lin and H.-W. Li, On the recovery of surface wave by pressure transfer function, Ocean Engng., 32 (2005), 1247-1259. doi: 10.1016/j.oceaneng.2004.10.020.  Google Scholar

show all references

References:
[1]

C. J. Amick and J. F. Toland, On solitary water waves of finite amplitude, Arch. Rat. Mech. Anal., 76 (1981), 9-95. doi: 10.1007/BF00250799.  Google Scholar

[2]

A. Baquerizo and M. A. Losada, Transfer function between wave height and wave pressure for progressive waves, by Y.-Y. Kuo and J.-F. Chiu: Comments, Coast. Engng., 24 (1995), 351-353. doi: 10.1016/0378-3839(94)00038-Y.  Google Scholar

[3]

C. T. Bishop and M. A. Donelan, Measuring waves with pressure transducers, Coast. Engng., 11 (1987), 309-328. doi: 10.1016/0378-3839(87)90031-7.  Google Scholar

[4]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-575. doi: 10.1017/jfm.2012.490.  Google Scholar

[5]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom pressure measurements, J. Fluid Mech., 726 (2013), 547-558. doi: 10.1017/jfm.2013.253.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

A. Constantin, On the particle paths in solitary water waves, Quart. Appl. Math., 68 (2010), 81-90.  Google Scholar

[9]

A. Constantin, On the recovery of solitary wave profiles from pressure measurements, J. Fluid Mech., 699 (2012), 376-384. doi: 10.1017/jfm.2012.114.  Google Scholar

[10]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: Mathematical theory and experiments, Arch. Rat. Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.  Google Scholar

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[12]

W. Craig and P. Sternberg, Symmetry of solitary waves, Commun. Part. Diff. Equ., 13 (1988), 603-633. doi: 10.1080/03605308808820554.  Google Scholar

[13]

B. Deconinck, D. Henderson, K. L. Oliveras and V. Vasan, Recovering the water-wave surface from pressure measurements, in 10th International Conference on Mathematical and Numerical Aspects of Waves, WAVES 2011, Vancouver, 2011. Google Scholar

[14]

J. Escher and T. Schlurmann, On the recovery of the free surface from the pressure within periodic traveling water waves, J. Nonlinear Math. Phys., 15 (2008), 50-57. doi: 10.2991/jnmp.2008.15.s2.4.  Google Scholar

[15]

F. G. Friedlander, Introduction to the Theory of Distributions, Second edition, Cambridge University Press, Cambridge, 1998.  Google Scholar

[16]

D. Henry, On the pressure transfer function for solitary water waves with vorticity, Math. Ann., 357 (2013), 23-30. doi: 10.1007/s00208-013-0899-0.  Google Scholar

[17]

Y.-Y. Kuo and Y.-F. Chiu, Transfer function between the wave height and wave pressure for progressive waves, Coast. Engng., 23 (1994), 81-93. doi: 10.1016/0378-3839(94)90016-7.  Google Scholar

[18]

J. B. Mcleod, The rate of decay of solitary waves of finite amplitude, Appl. Anal., 17 (1983), 37-50. doi: 10.1080/00036818308839482.  Google Scholar

[19]

K. L. Oliveras, V. Vasan, B. Deconinck and D. Henderson, Recovering the water-wave profile from pressure measurements, SIAM J. Appl. Math., 72 (2012), 897-918. doi: 10.1137/110853285.  Google Scholar

[20]

R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994.  Google Scholar

[21]

C.-H. Tsai, M.-C. Huang, F.-J. Young, Y.-C. Lin and H.-W. Li, On the recovery of surface wave by pressure transfer function, Ocean Engng., 32 (2005), 1247-1259. doi: 10.1016/j.oceaneng.2004.10.020.  Google Scholar

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