RECOVERING SURFACE PROFILES OF SOLITARY WAVES ON A UNIFORM STREAM FROM PRESSURE MEASUREMENTS

. 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.


Introduction.
There are several different ways to measure surface wave elevation. A convenient way is through the use of underwater pressure transducers. In fact, the pressure plays an important role in qualitative studies of travelling water waves in irrotational flow and is essential in the description of the particle trajectories beneath the waves (see [6]- [8], [11]). The pressure is also important in quantitative studies because the elevation of a surface water waves is often determined in field investigation from pressure data obtained at the sea bed, especially used in the tsunami detection made by the Pacific Tsunami Warning Center (see [2], [17], [19]).
The standard approach to reconstruct the free surface from pressure measurements consists of assuming that the hydrostatic assumption is sufficiently accurate (see [19]). However, one can not expect that this hydrostatic assumption can capture the nonlinear effects and the discussion in Bishop and Donelan [3] shows that even for waves of moderate amplitude predictions errors are significant. Within the linear regime of water waves of small amplitude in finite water depth, one can derive a better approximation (see [14]), but it is limited for waves of moderate amplitude compared with the experimental data by Tsai et al. [21].
These considerations motivate the quest for a reconstruction formula that accounts for nonlinear effects and that is thus applicable to waves of moderate and large amplitude. Recently, nonlinear nonlocal equations relating the dynamic pressure on the bed and the wave profiles were obtained without approximation from the governing equations (see [9], [13], [19]). Clamond and Constantin [4] and Clamond [5] obtained exact tractable relations and straightforward numerical procedures can be used to derive the free surface from the pressure at the bed.
The above discussions are focused on irrotational travelling water waves. It is well known that wave-current interaction is important for sediment transport and 3036 HUNG-CHU HSU pollution dispersion in the nearshore zone. It is therefore highly desirable to extend the theoretical investigations to the irrotational solitary waves with uniform current. In this paper, we follow the method of Constantin [9] to derive an explicit formula providing a parametric representation of the wave profile in the irrotational solitary waves with uniform current in terms of the pressure on the flat bed. It is noted that the case of non-uniform currents is quite open [16].
Here the velocity filed is represented by (u = u(X − ct),v = v(X − ct)), where c > 0 is the wave speed, P = P (X − ct, Y ) denotes the pressure and g is the gravitational constant of acceleration. The absence of vorticity is expressed by the irrotational condition throughout the flow. The boundary conditions are where P atm is the constant atmospheric pressure. The flow is uniform for X → ±∞, with the free surface approaching the height d > 0 above the flat bed, and must hold for non-trivial solutions (see [1]). Moreover, all solitary waves are waves of elevation above their asymptotic flat limit, symmetric about a single crest and with a strictly monotonic wave profile on either side of this crest (see [12]). Hence, for solitary waves we additionally impose the far-field conditions Throughout the paper we are only concerned with smooth solitary waves. For these waves it is known that the wave speed exceeds the horizontal velocity component: In this paper we consider that the ambient flow is a uniform current U , so that we can write the solution as follows Substituting (12) into (1)- (7), we can obtain the problem as Let us introduce in the moving frame the velocity potential φ by Using (13), we can also define the stream function ψ by that is for a suitable constant m ∈ R. It follows from the normalization in (23) and from (17)-(18) that ψ vanishes on the free surface y = η(x) and that ψ = m on the flat bed y = −d. In particular, we have 2 ) + gy +P =c throughout the flow (that is Bernoulli's law). Summarizing, the governing equations for irrotational two-dimensional solitary waves on a uniform current above a flat bed are equivalent to the study of the following elliptic free boundary problem coupled with the asymptotic limits, The Cauchy-Riemann equations and (27)-(29) imply that the complex functions is holomorphic, so that the smooth functions φ and ψ are harmonic conjugated.
H is an analytical bijection from Ω on to R × [−d, 0], whered =cd/c (see [10,16]). It is convenient to denote the coordinates in H(Ω) = R 2 × [−d, 0] by (q, p), that is, Introducing the height function and Thus (16) is equivalent to the following nonlinear boundary value problems: with the asymptotic limits 4. Main result. The above considerations will be developed by means of Fourier analysis. In using the Fourier transform the growth at infinity is crucial. Following Constantin [9], we can provide an approach that just relies upon classical distribution theory. Indeed, from (37), we have so that Bernoulli's law (26) becomes Evaluating the above relation on p = −d, we infer from (38) that with f (q) =P (q, −d) − P atm − gd. Taking into account (11) and (37), this yields The function h is bounded so that it represents a tempered distribution, cf. Friedlander [15]. We perform Fourier transforms in this distributional sense-throughout this paper all the references to (tempered) distributions are with respect to the q-variable, and the p-variable is treated as a parameter. Taking the Fourier transformĥ in the q-variable of the partial differential equation in (38), we obtain the differential equationĥ whereĥ(k, p) = F {h} (k) = R h(k, p)e −ikq dq, k ∈ R, stands for the Fourier transform of the function q → h(q, p).
Equation (44) has the general solution The left-hand side is a tempered distribution (being the Fourier transform of a tempered distribution), while on the right-hand side A and B are merely distributions so that their multiplication with smooth functions is well-defined Sinceĥ(k, −d) = 0 by (38), we have B(k) = −A(k)e −2kd so that for some distribution A(k). Now both sides of (43) are bounded functions and therefore represent tempered distributions. Taking the Fourier transform of (43) we obtain thatĥ in the sense of tempered distributions. On the other hand, differentiating (46) with respect to p and subsequently setting p = −d, we get in the sense of tempered distributions. Comparing this with (47) enables us to infer that as distributions. Thus, in view of (46), we obtain in the sense of distributions. Notice that in view of equation (35), the inverse of H(Ω) : R × [−p, 0] →Ω is given by for some smooth function θ, so that (x, η(x)) = (θ(q, 0), h(q, 0) −d).
Since h p = θ q by (37), setting we obtain if we recall (50). On the other hand, (50) also yields where δ is the Dirac mass, since F {1} (k) = 2πδ(k). Note that Therefore (54) and (55) can be written as and respectively, holding in the sense of distributions. Moreover, from (43) we get The decay properties ofũ(x, −d) as |x| → ∞, established in Craig and Sternberg [12] and in Mcleod [18], together with (36) ensure that f and its derivative are squareintegrable. Consequently lim |q|→∞ f (q) = 0, cf. Strichartz [20]. The relation guarantees that the expression on the left-hand side is a square-integrable function. This will be also the case for its Fourier transform, cf. Strichartz [17]. Thus (57)-(58) are not just equalities in the sense of distributions but are equalities for functions. Finally, since (37) ensures θ 0 (q) → 1 for q → ∞, a glance at (52), (57) and (58) yields (61) with x = θ 0 and η = h 0 , Finally, the exact recovery formula for the wave profile of a solitary wave on a uniform current is obtained. This parametric representation of the surface wave profile is valid for smooth solitary wave with solutions of the governing equations, up to the wave of greatest height. The usefulness (61) holds without the restriction to small-amplitude waves and is explicit. The usefulness solution of (61) lies in showing that the surface elevation of a solitary water wave on a uniform stream can always be recovered from pressure data at the flat bed of the fluid domain. This is important since a direct measurement of the elevation is difficult, and often such data are inferred from pressure measurements. It is interesting to pursue a numerical or experimental study to investigate the practical use of (61). This is work in progress.