# American Institute of Mathematical Sciences

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

 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.
Citation: Hung-Chu Hsu. Recovering surface profiles of solitary waves on a uniform stream from pressure measurements. Discrete and 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. [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. [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. [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. [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. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [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. [8] A. Constantin, On the particle paths in solitary water waves, Quart. Appl. Math., 68 (2010), 81-90. [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. [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. [11] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [12] W. Craig and P. Sternberg, Symmetry of solitary waves, Commun. Part. Diff. Equ., 13 (1988), 603-633. doi: 10.1080/03605308808820554. [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. [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. [15] F. G. Friedlander, Introduction to the Theory of Distributions, Second edition, Cambridge University Press, Cambridge, 1998. [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. [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. [18] J. B. Mcleod, The rate of decay of solitary waves of finite amplitude, Appl. Anal., 17 (1983), 37-50. doi: 10.1080/00036818308839482. [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. [20] R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. [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.

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##### 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. [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. [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. [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. [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. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [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. [8] A. Constantin, On the particle paths in solitary water waves, Quart. Appl. Math., 68 (2010), 81-90. [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. [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. [11] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [12] W. Craig and P. Sternberg, Symmetry of solitary waves, Commun. Part. Diff. Equ., 13 (1988), 603-633. doi: 10.1080/03605308808820554. [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. [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. [15] F. G. Friedlander, Introduction to the Theory of Distributions, Second edition, Cambridge University Press, Cambridge, 1998. [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. [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. [18] J. B. Mcleod, The rate of decay of solitary waves of finite amplitude, Appl. Anal., 17 (1983), 37-50. doi: 10.1080/00036818308839482. [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. [20] R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. [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.
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