Recovering the Water-Wave Profile from Pressure Measurements

Abstract

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.

On the recovery of traveling water waves with vorticity from the pressure at the bed

“…We include the detail of the algebra in Appendix A for completeness. The first-order approximation of the result in [OVDH12] agrees with (3.28); see (A.9). The second-order approximation of the result in [OVDH12] may be written, abusing notation, as…”

“…, up to functions of y. We may continue to assume that (see (3.13)) In the case of zero vorticity, in [OVDH12], for instance, likewise, one determines the velocity potential up to a constant, but the reconstruction formula does not require knowledge of the velocity potential itself; see Appendix A for the detail. Another shortcoming is that the wave speed agrees with the bifurcation speed up to the order of ǫ 2 in the regime of small amplitude waves, even in the case of zero vorticity.…”

“…whence (3.15) and (3.16) become We compare (3.28) and (3.29) to asymptotic approximations of the exact formula in [OVDH12], for instance, but in the physical variables. We include the detail of the algebra in Appendix A for completeness.…”

“…We compare the results to asymptotic approximations of the exact formula in [OVDH12], for instance. The results in [OVDH12, Section 4.2] may be written, in the dimensionless variables, abusing notation, as…”

“…From such scant data, can one recover the wave height? Incidentally traveling waves may be used as a means to understand more general wave motions; see [OVDH12,DOV12], for instance.…”