2. Preliminaries

In the frame of reference moving with a travelling wave of permanent shape, the flow beneath the wave reduces to a steady motion with respect to the moving coordinate system. Thus, the wave phase speed $c$ is constant in any Galilean frame of reference. Let $(x,y)$ be a Cartesian coordinate system moving with the wave, $x$ being the horizontal coordinate and $y$ the upward vertical coordinate. We define accordingly the fluid domain as $\varOmega = \{(x,y):x\in {\mathbb R},-d \leqslant y \leqslant \eta (x)\}$, where $y=-d$ and $y=\eta (x)$ correspond, respectively, to the solid bottom and the free surface (both impermeable). In addition, $(u(x,y),v(x,y))$ denotes the velocity field in the moving frame. We assume the wave is $L=(2{\rm \pi} /k$) periodic (with $k\to 0$ for solitary waves) in the $x$-direction, and we denote by $y=0$ the mean water level. The latter equation expresses the fact that $\langle \eta \rangle =0$, where $\langle {\cdot } \rangle$ is the Eulerian average operator over one period, that is,

(2.1)\begin{equation} \langle \eta \rangle \stackrel{{\rm def}}{=} \frac{k}{2{\rm \pi}} \int_{-{\rm \pi}/k}^{{\rm \pi}/k} \eta(x) \, \mathrm{d}\kern0.7pt x = 0. \end{equation}

The flow is governed by the balance between the restoring gravity force and the inertia of the system. With constant density $\rho >0$, the equation of mass conservation and Euler equations (defined in $\varOmega$) are, respectively,

(2.2a–c)\begin{equation} \partial_x u + \partial_y v = 0, \quad u \partial_x u + v \partial_y u ={-}\partial_x P/\rho, \quad u \partial_x v + v \partial_y v ={-}\partial_y P/\rho - g,\end{equation}

where $P(x,y)$ denotes the pressure. As a general notation, subscripts ‘b’ denote all quantities written at the bed $y=-d$, whereas subscripts ‘s’ denote all quantities written at the free surface $y = \eta (x)$. The effect of surface tension being neglected, on the free surface we must have the dynamic boundary condition

(2.3)\begin{equation} {P}_{s} = P_{atm} , \end{equation}

where $P_{atm}$ is the (constant) atmospheric pressure. The free surface and the rigid bed are impermeable interfaces, giving the kinematic boundary conditions (with $\eta _x \stackrel{\rm def}{=}\mathrm {d}\eta (x)/\mathrm {d}\kern0.7pt x$)

(2.4a,b)\begin{equation} {v}_{s} = {u}_{s} \eta_x, \quad {v}_{b} = 0,\end{equation}

respectively, while the rotational character of the flow is ensured by requiring

(2.5)\begin{equation} \partial_x v - \partial_y u \stackrel{\rm def}{=} \omega \quad (= \text{const.}). \end{equation}

Equations (2.2a–c)–(2.5) are the governing equations for rotational (of constant vorticity $\omega$) travelling water waves in a frame of reference moving with the wave.

For incompressible flows where (2.2a) holds, we can define a streamfunction $\psi$ such that $u=\partial _y\psi$ and $v=-\partial _x\psi$. As the flow is steady and the free surface is impermeable, it follows that the free surface is a streamline, that is, the streamfunction is constant $\psi ={\psi }_{s}$ at the free surface (similarly, the streamfunction is constant $\psi ={\psi }_{b}$ at the bed).

Equations (2.2a–c) can be integrated to

(2.6)\begin{equation} 2 p + 2 g y + u^2 + v^2 = {B}_{s} - 2\omega (\psi - {\psi}_{s}), \end{equation}

for some constant ${B}_{s}$, where $p(x,y)\stackrel{\rm def}{=}[P(x,y)-P_{atm}]/\rho$ is a normalised relative pressure.

Equation (2.6) is a Bernoulli equation, and we note that the Bernoulli integral $B(\psi ) \stackrel{\rm def}{=} {B}_{s} - 2\omega (\psi - {\psi }_{s})$ is constant for an irrotational motion (i.e. when $\omega =0$).

3. Definition of the parameters

From the definition (2.1) of the mean water level and by averaging expression (2.6) written at the free surface, we obtain a definition for the constant ${B}_{s}$ in the form

(3.1)\begin{equation} {B}_{s} = \langle {u}_{s}^{2} + {v}_{s}^{2} \rangle . \end{equation}

As the frame of reference moving with the wave is Galilean, there is no mean acceleration. For steady waves with constant vorticity, the zero-mean horizontal acceleration condition is perforce satisfied, but the condition for zero-mean vertical acceleration yields

(3.2)\begin{align} 0 &= \left\langle \int_{{-}d}^\eta \left[ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right] \mathrm{d} y \right\rangle = \left\langle \int_{{-}d}^\eta \frac{\partial}{\partial y} \left[ \frac{u^2+v^2}{2} + \omega \psi \right] \mathrm{d} y \right\rangle \nonumber\\ &= \frac12 \langle {u}_{s}^{2} + {v}_{s}^{2} \rangle - \frac12 \langle {u}_{b}^{2} \rangle + \omega ({\psi}_{s} - {\psi}_{b}) . \end{align}

This furnishes at once an alternative relation for the Bernoulli constant,

(3.3)\begin{equation} {B}_{s} = \langle {u}_{b}^{2} \rangle + 2\omega ({\psi}_{b} - {\psi}_{s}) , \end{equation}

and, since ${p}_{s}=0$, relation (3.2) implies that the average pressure at the bottom is

(3.4)\begin{equation} \langle {p}_{b} \rangle ={-} \left\langle \int_{{-}d}^\eta \frac{\partial p} {\partial y} \,\mathrm{d} y \right\rangle = \left\langle \int_{{-}d}^\eta \left[ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + g\right] \mathrm{d} y \right\rangle = g d.\end{equation}

Relation (3.4) provides a mechanism for determining the mean water depth $d$ from bottom pressure measurements. For later convenience, define the alternative Bernoulli constant as

(3.5)\begin{equation} {B}_{b} \stackrel{\rm def}{=} {B}_{s} - 2\omega ({\psi}_{b} - {\psi}_{s}) = \langle {u}_{b}^{2} \rangle , \end{equation}

where, as expected, both Bernoulli constants ${B}_{b}$ and ${B}_{s}$ coincide for irrotational flows. The vorticity $\omega$ being constant, exploiting the free-surface impermeability gives

(3.6)\begin{align} \omega d &= \left\langle \int_{{-}d}^\eta \left[ \frac{\partial v} {\partial x} - \frac{\partial u}{\partial y} \right] \mathrm{d} y \right\rangle ={-} \langle \eta_x {v}_{s} \rangle - \langle {u}_{s} \rangle + \langle {u}_{b} \rangle \nonumber\\ &= \langle {u}_{b} \rangle - \langle (1 + \eta_x^{2} ) {u}_{s} \rangle . \end{align}

Expression (3.6) provides a means of determining the vorticity $\omega$ in terms of ${u}_{b}$, ${u}_{s}$, $d$ and $\eta$. In the same vein, a relation not involving velocity evaluation along the flat bed is given by

(3.7)\begin{align} \frac{\omega}{2} \langle h^2 \rangle &= \left\langle \int_{{-}d}^\eta \left[ \frac{\partial v} {\partial x} - \frac{\partial u}{\partial y} \right] (y+d) \, \mathrm{d} y \right\rangle ={-} \langle h \eta_x {v}_{s} \rangle - \langle h {u}_{s} \rangle + {\psi}_{s}-{\psi}_{b} \nonumber\\ &= {\psi}_{s} - {\psi}_{b} - \langle (1+\eta_x^{2}) h {u}_{s} \rangle , \end{align}

where $h(x)\stackrel{\rm def}{=}\eta (x)+d$ is the total water depth. Together with (3.5), relation (3.7) can be expressed as

(3.8)\begin{equation} {B}_{s} = {B}_{b} - \omega^2 \langle h^2 \rangle - 2 \omega \langle (1+\eta_x^{2}) h {u}_{s} \rangle . \end{equation}

As for irrotational motions, Stokes’ first and second definitions of the phase speed (Kishida & Sobey Reference Kishida and Sobey1988; Clamond Reference Clamond2017) can be applied, resulting in the expressions

(3.9)\begin{gather} c_1 \stackrel{\rm def}{=} - \langle {u}_{b} \rangle ={-}\omega d - \langle (1+\eta_x^{2}) {u}_{s} \rangle , \end{gather}

(3.10)\begin{gather}c_2 \stackrel{\rm def}{=} - \left\langle \frac{1}{d} \int_{{-}d}^\eta u \,\mathrm{d} y \right\rangle= \frac{{\psi}_{b} - {\psi}_{s}}{d} ={-} \frac{\omega d}{2} - \frac{\omega\langle \eta^2 \rangle }{2d} - \frac{\langle (1+\eta_x^{2}) h {u}_{s} \rangle }{d}. \end{gather}

Here $c_1$ and $c_2$ are the wave speeds observed in frames of reference without mean horizontal velocity at the bed, and without mean flow, respectively. In the irrotational case ($\omega =0$), it can be shown (Clamond & Dutykh Reference Clamond and Dutykh2018) that $c_2\rightarrow _1$ and ${B}_{b}={B}_{s}\rightarrow _1^2$ as $d\to \infty$ or as $k\to 0$, but $c_2\approx c_1$ in the linear wave regime. For constant vorticity $\omega \neq 0$, matters are more complex, even at the linear level: $c_2 \approx c_1-\omega d/2 \not \approx c_1$, while $c_1\approx c^\pm _0$, where the linear phase speed (Brink-Kjær Reference Brink-Kjær1976; Kishida & Sobey Reference Kishida and Sobey1988),

(3.11)\begin{equation} c^\pm_0={-}\omega d+\tfrac12 k^{{-}1}\omega\tanh(k d)\pm\tfrac12 k^{{-}1}\sqrt{\omega^2 \tanh(k d)^2+4gk\tanh(k d)}, \end{equation}

solves a linear dispersion relation with symmetry property $c_0^{\pm }(-\omega ) =-c_0^{\mp }(\omega )$. Hence, without loss of generality in subsequent considerations, we assume that $c_1>0$ (that is, the wave propagates towards the increasing $x$-direction in the frame of reference without mean velocity at the seabed) and allow the vorticity to take either sign.

4. Holomorphic functions

When $\omega =0$, the flow is irrotational and one can use the powerful theory of holomorphic functions (Clamond Reference Clamond2013; Clamond & Constantin Reference Clamond and Constantin2013; Clamond & Henry Reference Clamond and Henry2020). In the case where $\omega \neq 0$, one can still use this technique following Helmholtz representation (Aris Reference Aris1962). Thus, introduce $U$, $V$ and $\varPsi$ (functions of both $x$ and $y$) as

(4.1a–c)\begin{equation} U \stackrel{\rm def}{=} u + \omega (y+d), \quad V \stackrel{\rm def}{=} v, \quad \varPsi \stackrel{\rm def}{=} \psi + \tfrac12 \omega(y+d)^2,\end{equation}

from which, using (2.5), straightforward calculations show that $U = \partial _y \varPsi$, $V = -\partial _x \varPsi$ and $\partial _x V = \partial _y U$, implying that the velocity field $(U,V)$ is curl-free. At the bed $y=-d$, we have

(4.2a–c)\begin{equation} {U}_{b} = {u}_{b}, \quad {V}_{b} = {v}_{b} = 0, \quad {\varPsi}_{b} = {\psi}_{b},\end{equation}

while, at the free surface $y=\eta (x)$,

(4.3a–c)\begin{equation} {U}_{s} = {u}_{s} + \omega h, \quad {V}_{s} = {v}_{s}, \quad {\varPsi}_{s} = {\psi}_{s} + \tfrac12 \omega h^2.\end{equation}

Note that ${\varPsi }_{s}$ is not uniform (unlike ${\psi }_{s}$) while ${\varPsi }_{b}$ is a constant.

Thus, introducing a velocity potential $\varPhi$ such that $U = \partial _x\varPhi$ and $V = \partial _y\varPhi$, the complex potential and velocity are defined as

(4.4a,b)\begin{equation} F(z) \stackrel{\rm def}{=} \varPhi + \mathrm{i} \varPsi, \quad W(z) \stackrel{\rm def}{=} U - \mathrm{i} V = \mathrm{d} F/\mathrm{d} z, \end{equation}

where $z \stackrel{\rm def}{=} x + \mathrm {i} y$ is a complex coordinate in $\varOmega$.

With these new dependent variables, the Bernoulli equation (2.6) evaluated at the free surface becomes

(4.5)\begin{equation} {U}_{s}^{2} + {V}_{s}^{2} + 2 ( g\eta - \omega h {U}_{s} ) + (\omega h)^2 = {B}_{s}, \end{equation}

while, at the bed, it yields

(4.6)\begin{equation} 2 ( {p}_{b} - gd ) + {U}_{b}^{2} = {B}_{s} - 2\omega ({\psi}_{b} - {\psi}_{s}) = \langle {U}_{b}^{2} \rangle = {B}_{b}. \end{equation}

Note that (4.5) and (4.6) can be rewritten, respectively, as

(4.7a,b)\begin{equation} {U}_{s} = \omega h - \sqrt{({B}_{s}-2g\eta)/(1 + \eta_x^{2})}, \quad {U}_{b} ={-} \sqrt{ {B}_{b} - 2 ( {p}_{b} - gd )},\end{equation}

the minus sign in front of the radicals being a consequence of the choice $c_1>0$. Since ${V}_{b} = 0$, the complex velocity can then be expressed as

(4.8)\begin{equation} W(z) ={-} \sqrt{ {B}_{b} - 2{p}_{b}(z+{\rm i}d) + 2gd }, \end{equation}

a relation that suggests the introduction of a complex pressure. Following Clamond & Constantin (Reference Clamond and Constantin2013), we introduce a ‘complex pressure’ function $\mathfrak {P}$ defined as

(4.9)\begin{equation} \mathfrak{P}(z) \stackrel{\rm def}{=} gd + \omega({\psi}_{s}-{\psi}_{b}) - \tfrac12 (W^2-{B}_{s}), \end{equation}

which is holomorphic in the fluid domain $\varOmega$. Note that the expression in (4.9) is purely real when restricted to the flat bed, with ${\mathfrak {P}}_{b} = {p}_{b}$ on $y=-d$. Accordingly, ${p}_{b}$ determines $\mathfrak {P}$ uniquely within the entire fluid domain $\varOmega$, and so ${\mathfrak P}(z)={p}_{b}(z+{\rm i}d )$. We note that, since $p$ is not a harmonic function in the fluid domain (Constantin Reference Constantin2011), it can coincide with the real part of $\mathfrak {P}$ only at $y=-d$.

Similarly, as for irrotational waves in Clamond (Reference Clamond2013, Reference Clamond2018), it is useful to introduce the holomorphic function $\mathfrak {Q}$

(4.10)\begin{equation} \mathfrak{Q}(z) \stackrel{\rm def}{=} \int_{z_0}^z [ \mathfrak{P}(z') - gd] \,\mathrm{d} z' = \tfrac{1}{2} \int_{z_0}^z [ {B}_{b} - W(z')^2 ]\,\mathrm{d} z' , \end{equation}

where $z_0\in \varOmega$ is an arbitrary constant.

5. Equations for the surface recovery

Integrating (4.10) along the free-surface path, with the origin located at the crest (i.e. $z_0=\mathrm {i} a$, $a\stackrel{\rm def}{=} \eta (0)$ being the wave amplitude), one gets

(5.1)\begin{equation} {\mathfrak{Q}}_{s}(x) = \int_0^x [{\mathfrak{P}}_{s}(x') - gd](1 + \mathrm{i}\eta_x') \,\mathrm{d}\kern0.7pt x', \end{equation}

where ${\mathfrak {P}}_{s}(x) = gd +({B}_{b} - {U}_{s}^2 + {V}_{s}^2)/2 + \mathrm {i}{U}_{s}{V}_{s}$. With (2.4a), (4.3a,b) and (4.7a), and splitting real and imaginary parts, (5.1) yields after some algebra

(5.2)\begin{align} {\mathfrak{Q}}_{s}(x) &= \int_{0}^{x} [ \operatorname{Re}\{{\mathfrak{P}}_{s}(x')\} - gd - \operatorname{Im}\{{\mathfrak{P}}_{s}(x')\} \eta_x' ]\, \mathrm{d}\kern0.7pt x' \nonumber\\ &\quad +\mathrm{i} (\eta-a)[ \hat{B} - t\tfrac12 (g+\omega^2d) (\eta+a) - \tfrac16 \omega^2(\eta^2+a\eta+a^2)] , \end{align}

where $\hat {B}\stackrel{\rm def}{=}({B}_{s}+{B}_{b}-\omega ^2d^2)/2$ is an expression involving Bernoulli constants at the surface and the bottom.

The imaginary part of (5.2) provides an implicit relation for the surface elevation expressed in terms of the holomorphic function ${\mathfrak {Q}}_{s}$,

(5.3)\begin{equation} \operatorname{Im}\{{\mathfrak{Q}}_{s}\} = (\eta-a)[ \hat{B} - \tfrac12 (g+\omega^2d) (\eta+a) - \tfrac16 \omega^2(\eta^2+a\eta+a^2)]. \end{equation}

From the differentiation of (5.1) and (5.3), one gets the differential equation

(5.4)\begin{equation} \frac{\mathrm{d}\eta}{\mathrm{d}\kern0.7pt x} = \frac{\operatorname{Im} \{{\mathfrak{P}}_{s}\}}{\hat{B} - (g+\omega^2d) \eta - \frac12 \omega^2 \eta^2 - \operatorname{Re} \{{\mathfrak{P}}_{s} \} + gd}. \end{equation}

In the special case of irrotational motion ($\omega =0$), the differential equation derived in Clamond & Constantin (Reference Clamond and Constantin2013) is recovered.

Evaluating (5.3) at the trough ($x=-{\rm \pi} / k$), bearing in mind relation (3.8), one obtains an expression for $\hat {B}$ as

(5.5)\begin{align} \hat{B} &= {B}_{s} + \tfrac12 \omega^2 \langle \eta^2 \rangle - \omega\left\langle h \sqrt{({B}_{s}-2g\eta)(1+\eta_x^2)} \right\rangle, \nonumber\\ &= \tfrac12 (g+\omega^2d) (a-b) + \tfrac16 \omega^2 (a^2-ab+b^2) - (a+b)^{{-}1} \operatorname{Im}\{{\mathfrak{Q}}_{s}(-{\rm \pi} /k)\} , \end{align}

where $b\stackrel{\rm def}{=}-\eta (-{\rm \pi} /k)$ denotes the trough height (thus $H\stackrel{\rm def}{=} a+b$ is the total wave height).

We now have algebraic expressions for the recovery of $\eta$ and $\hat {B}$, as functions of the remaining unknown parameters $a$, $b$ and $\omega$. Three relations are then needed to close our set of equations. These relations are obtained by considering ${\mathfrak {P}}_{s}=gd + \omega ({\psi }_{s}-{\psi }_{b})- \frac 12 ({W}_{s}^{2}-{B}_{s})$ at the crest, at the trough and at an intermediate point. Using the surface impermeability with the decomposition (4.1a–c), the squared complex velocity reduces to

(5.6)\begin{align} {W}_{s}^{2} &\stackrel{\rm def}{=} ({U}_{s} - \mathrm{i}{V}_{s})^2 = \omega^2 h^2 + 2\omega h {u}_{s} (1 - \mathrm{i}\eta_x) + {u}_{s}^{2}\, (1 - \mathrm{i}\eta_x)^2 , \nonumber\\ &= \omega^2 h^2 + \left[ ({B}_{s}-2g\eta)/(1 + \mathrm{i}\eta_x) - 2\omega h \sqrt{({B}_{s}-2g\eta)/(1+\eta_x^2)} \right] (1 - \mathrm{i}\eta_x) . \end{align}

Following Clamond & Constantin (Reference Clamond and Constantin2013), we substitute (5.6) in the definition of ${\mathfrak {P}}_{s}$ at the crest and the trough – together with (3.8) and (3.10) – to get the two relations

(5.7)\begin{gather} {\mathfrak{P}}_{s}(0) = g h_c + \omega^2 \frac{\langle h^2 \rangle - h_c^2}{2}- \omega \left[ \left\langle h \sqrt{({B}_{s}-2g\eta)(1+\eta_x^{2})} \right\rangle - h_c \sqrt{{B}_{s} - 2ga} \right] , \end{gather}

(5.8)\begin{gather}{\mathfrak{P}}_{s} \left(-\frac{{\rm \pi} }{k}\right) = g h_t + \omega^2 \frac{\langle h^2 \rangle - h_t^2}{2} - \omega \left[ \left\langle h \sqrt{({B}_{s}-2g\eta)(1+\eta_x^{2})} \right\rangle - h_t \sqrt{{B}_{s} + 2gb}\right], \end{gather}

where $h_t\stackrel{\rm def}{=} d-b$ and $h_c\stackrel{\rm def}{=} d+a$ are local depths under, respectively, the trough and the crest. As expected, both expressions reduce in the irrotational limit to the formulae derived in Clamond (Reference Clamond2013). With (5.7) and (5.8) we have two relations that close the problem if $\omega$ is known. However, in practice, $\omega$ is generally unknown a priori, so another independent relation must be introduced.

A last relation is obtained considering the complex pressure ${\mathfrak {P}}_{s}$ at an abscissa $x_0$ strictly between crest and trough. This point is chosen at a coordinate of median bottom pressure measurement such that ${p}_{b}(x_0)=(\max {p}_{b}-\min {p}_{b})/2$ to ensure a large distance from the crest and the trough. With $x_0$ being thus chosen, $\operatorname {Im}\{{\mathfrak {Q}}_{s}(x_0)\}$, $\operatorname {Re}\{{\mathfrak {P}}_{s} (x_0)\}$ and $\operatorname {Im}\{{\mathfrak {P}}_{s}(x_0)\}$ (together with (5.6) applied at $x_0$) provide three relations for $\eta (x_0)$, $\eta _x(x_0)$ and $\omega$ (relations not written in extenso here, for brevity).

6. Reconstruction procedure

From measurements of the bottom pressure ${p}_{b}$, the free-surface reconstruction procedure takes the following form. The first step is to choose a suitable basis of functions, generally a Fourier polynomial or elliptic functions (Clamond Reference Clamond2013; Clamond & Constantin Reference Clamond and Constantin2013). The best choice is that providing the best fit among data with a minimum of eigenfunctions. Here, we consider only Fourier polynomials for simplicity, but the procedure is identical for any basis of functions expressible in the complex plane. Thus, the wavenumber $k$ and the coefficients $\mathfrak {p}_n$ of the $N$th-order Fourier polynomials, ${p}_{b} \approx \sum _{n=-N}^N \mathfrak {p}_n \,\mathrm {e}^{\mathrm {i} nkx}$, can be determined by least-squares minimisation (Clamond & Barthélémy Reference Clamond and Barthélémy1995). From (3.4), we know that $\mathfrak {p}_0 = gd$ and, since the acceleration due to gravity $g$ is known, this relation gives an expression for the mean water depth $d$. Thus, after this first step, the parameters $g$, $d$, $k$ and $\mathfrak {p}_n$ are known explicitly and the bottom pressure can be extended everywhere in the bulk of the fluid.

From the analytic approximation of ${p}_{b}$, the holomorphic pressure $\mathfrak {P}$ is, by definition,

(6.1)\begin{equation} \mathfrak{P}(z) \approx \sum_{n={-}N}^N \mathfrak{p}_n \,\mathrm{e}^{\mathrm{i} n k (z + \mathrm{i} d)} = \sum_{n={-}N}^N \mathfrak{p}_n \,\mathrm{e}^{- n k d} \,\mathrm{e}^{\mathrm{i} n k z}, \end{equation}

so one obtains at once

(6.2)\begin{equation} {\mathfrak{Q}}_{s}(x) = \int_{0}^{x} [ {\mathfrak{P}}_{s}(x') - gd ]\, \mathrm{d}\kern0.7pt x' \approx \sum_{|n|>0}^N \frac{\mathrm{i} \mathfrak{p}_n} {n k} \frac{\mathrm{e}^{{-}nka} - \mathrm{e}^{\mathrm{i} nk(x+\mathrm{i}\eta)}}{\mathrm{e}^{nkd}} . \end{equation}

After this second step, ${\mathfrak {P}}_{s}$ and ${\mathfrak {Q}}_{s}$ are known explicitly as functions of $\eta$. The wave surface profile $\eta$ is then determined by solving the algebraic (i.e. not differential, nor integral) equation (5.3), which, in general, can only be achieved numerically. This results in $\eta$ being obtained as a function of the parameters $a$, $b$, $\omega$ and ${B}_{s}$. These parameters are then determined by solving the nonlinear equations (5.5), (5.7) and (5.8) and those at $x_0$.

The numerical procedure consists in solving simultaneously the nonlinear set of equations (5.3), (5.5), (5.7) and (5.8) (and those at $x_0$) to recover the surface wave profile and related parameters. To this end, we use an iterative root-finding algorithm (the built-in function fsolve in MATLAB) supplemented with initial values given by the linear theory (Kishida & Sobey Reference Kishida and Sobey1988). When needed, $\eta _x$ is computed directly from its explicit expression (5.4).

7. Examples

In order to validate our procedure, we computed exact rotational waves with an accurate numerical algorithm similar to that of Da Silva & Peregrine (Reference Da Silva and Peregrine1988). We then obtained bottom pressures that are considered the ‘measured’ ones from which the free surface and vorticity are recovered. Throughout the procedure, we monitor the convergence of our iterative algorithm using a tolerance of $10^{-12}$. When converged, the numerical solution is compared with the exact surface wave profile $\eta ^{{ex}}$ and vorticity $\omega ^{{ex}}$.

For waves of relatively small amplitudes, as well as small $\omega$, we witness a rapid convergence of the recovery procedure described above. The numerical errors of the recovered surface and vorticity are, respectively, $\epsilon _{\eta }=\lVert \eta - \eta ^{{ex}} \rVert _{\infty }<10^{-8}$ and $\epsilon _{\omega }=\lvert \omega - \omega ^{{ex}} \rvert < 10^{-6}$. This excellent agreement is reached with as few as $N=5$ harmonics when considering the pressure Fourier expansion. However, for steady rotational waves with a steeper profile (thus departing from linear theory), the procedure naturally requires a larger number of harmonics.

Here, we illustrate our surface wave recovery procedure with numerical examples depicted in figures 1 and 2 for a domain of size $L/d=2{\rm \pi}$ (rather deep water for which surface recovery is a priori difficult). For our first example, we examine a steep wave with negative vorticity $\omega \sqrt {d/g}=-1.7$. Although the steep wave in figure 1(c) would appear to be quite challenging to compute, we still recover the correct surface profile using $N=30$ harmonics in our Fourier expansion (cf. figure 1b), with the recovered solutions showing an excellent agreement with the exact data ($\epsilon _{\eta } \approx 10^{-4}$ and $\epsilon _{\omega } \approx 10^{-4}$).

Figure 1. Example of surface recovery for $\omega \sqrt {d/g}=-1.7$, $H/d=0.14$ and $L/d=2{\rm \pi}$: (a) pressure at the bottom; (b) Fourier spectrum extracted from bottom pressure (blue squares) and fitted harmonics used for the reconstruction (red crosses); and (c) recovered wave profile (red line) versus exact surface profile (blue circles).

Figure 2. Same as figure 1 for the case $\omega \sqrt {d/g}=3$, $H/d=0.5$ and $L/d=2{\rm \pi}$.

For the second case of interest (figure 2), we take the positive vorticity $\omega \sqrt {d/g}=3$. This value is greater than that predicated by linear wave theory for the existence of critical layers. Indeed, linear waves with $c_0>0$ never allow for stagnation points when $\omega <0$, while they contain stagnation points for $\omega >0$ if and only if $\tanh (kd)/kd \leq \omega ^2 d /(g+\omega ^2 d)$ (Da Silva & Peregrine Reference Da Silva and Peregrine1988; Constantin & Varvaruca Reference Constantin and Varvaruca2011). This example is of special interest because it involves three stagnation points per wavelength: two at the bottom (about halfway between crests and troughs) and one within the fluid (under the crests) (cf. figure 14a of Da Silva & Peregrine (Reference Da Silva and Peregrine1988)). The surface recovery still works relatively well in this case; taking $N=20$ Fourier modes, errors are $\epsilon _{\eta } \approx 10^{-3}$ and $\epsilon _{\omega } \approx 10^{-4}$ (see figure 2).

We observe from figure 1(a) that the bottom pressure distribution for our choice of wave configuration displays a monotonic increase between trough and crest, which is not matched by that illustrated in figure 2(a). This is an artefact not solely of the difference in signs of the vorticities, but also their magnitudes. An examination of explicit linear solutions for water waves with constant vorticity (Brink-Kjær Reference Brink-Kjær1976) illustrates the richness in behaviour of the pressure fluctuations even for small-amplitude waves, with regard to both its monotonicity properties and the location of its extrema. For nonlinear waves, the precise qualitative behaviour of the bottom pressure fluctuations due to wave motion (and the location of its extrema) was only recently rigorously established for irrotational periodic waves in (Constantin Reference Constantin2016), and there are presently no similarly rigorous mathematical results for nonlinear waves with constant vorticity. It is clear from figures 1(a) and 2(a) that considerable insight into this matter can be gained from a numerical approach.