Recovering the Water-Wave Profile from Pressure Measurements

A new method is proposed to recover the water-wave surface elevation from pressure data obtained at the bottom of the fluid. The new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. The solvability properties of the nonlocal equation are rigorously analyzed using the implicit function theorem.

[1]  Philippe Guyenne,et al.  Hamiltonian long–wave expansions for water waves over a rough bottom , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  M. Ablowitz,et al.  SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS , 2008 .

[3]  L. E. Scales,et al.  Non-linear least squares , 1985 .

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  W. Strauss,et al.  Pressure beneath a Stokes wave , 2009 .

[6]  Adrian Constantin,et al.  The trajectories of particles in Stokes waves , 2006 .

[7]  A. Constantin.  On the particle paths in solitary water waves , 2009 .

[8]  J. F. Toland,et al.  On solitary water-waves of finite amplitude , 1981 .

[9]  Y. Chiu,et al.  Transfer function between wave height and wave pressure for progressive waves , 1994 .

[10]  M. Donelan,et al.  Measuring waves with pressure transducers , 1987 .

[11]  Cheng‐Han Tsai,et al.  Wave measurements by pressure transducers using artificial neural networks , 2009 .

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Hsiang Wang,et al.  MEASUREMENT OF SURFACE WAVES FROM SUBSURFACE GAGE , 1984 .

[14]  B. Deconinck,et al.  The instability of periodic surface gravity waves , 2011, Journal of Fluid Mechanics.

[15]  Peter Teunissen,et al.  Nonlinear least squares , 1990 .

[16]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[17]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[18]  Athanassios S. Fokas,et al.  On a new non-local formulation of water waves , 2006, Journal of Fluid Mechanics.

[19]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[20]  J. Escher,et al.  Pressure Beneath a Solitary Water Wave: Mathematical Theory and Experiments , 2011 .

[21]  E. Spielvogel.  A Variational principle for waves of infinite depth , 1970 .

[22]  R. Schnabel,et al.  10. Nonlinear Least Squares , 1996 .

[23]  E. Varvaruca,et al.  Singularities of Bernoulli Free Boundaries , 2006 .

[24]  N. Wiener,et al.  Fourier Transforms in the Complex Domain , 1934 .

[25]  Hsien-Wen Li,et al.  On the recovery of surface wave by pressure transfer function , 2005 .

[26]  Robert J. Weaver,et al.  Rapidly installed temporary gauging for hurricane waves and surge, and application to Hurricane Gustav , 2010 .

[27]  T. Schlurmann,et al.  On the Recovery of the Free Surface from the Pressure within Periodic Traveling Water Waves , 2008 .

[28]  A. Tørum,et al.  WAVE MEASUREMENTS BY A PRESSURE TYPE WAVE GAUGE , 1968 .

[29]  E. Cokelet,et al.  Steep gravity waves in water of arbitrary uniform depth , 1977, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[30]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[31]  David R. Basco,et al.  Water Wave Mechanics for Engineers and Scientists , 1985 .

[32]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[33]  Joachim Escher,et al.  Particle trajectories in solitary water waves , 2007 .