2. Preliminaries

We consider two-dimensional periodic travelling waves propagating on the surface of an incompressible inviscid fluid under the restoring force of gravity. The wave is periodic in the $x$-direction, with period $L$, and the wavenumber is given by $k=2{\rm \pi} /L$. Obviously, aperiodic waves are obtained letting $k\to 0$. Thus, discussing periodic waves (for simplicity) is not a limitation for our purpose here. If $(X,Y)$ denotes horizontal and vertical coordinates in a fixed reference frame, with $(U,V)$ the respective velocity field, then the travelling wave hypothesis implies a functional dependence on the variables of the form $(X-ct,Y)$ (t the time): the Galilean change of variables

(2.1a–d)\begin{equation} x=X-ct, \quad y=Y, \quad u=U-c, \quad v=V, \end{equation}

transforms to a reference frame where the fluid motion is steady, with the resulting velocity field in the frame moving with the wave given by $(u,v)$. The wave phase velocity $c$ is an arbitrary constant as long as we have not defined what we mean by ‘fixed frame of reference’. If it is the frame where the horizontal velocity has zero mean at bottom (which corresponds to Stokes’ first definition of the wave speed), the wave phase velocity $c$ is such that $\left \langle U_{b}\right \rangle =0$ in the fixed frame, that is $\left \langle u_{b}\right \rangle =-c$ in the frame of reference moving with the wave. Here $\left \langle \boldsymbol {\cdot } \right \rangle$ is the Eulerian average operator over one wavelength and, in what follows, the subscripts ‘${s}$’ and ‘${b}$’ denote the restrictions to the free surface and at the bed, respectively. Note that $c>0$ if the wave travels toward the increasing $x$-direction; a choice that can always be made without loss of generality. Many other definitions of $c$ can of course be made, with resulting consequences for the Bernoulli constant, and the gauge condition for the velocity potential, for instance (cf. Clamond (Reference Clamond2017) for discussions on this, and related matters).

The horizontal bottom is located at $y=-d$, and the free surface is described by $y=\eta (x)$: since these boundaries are impermeable and steady, we have ${v}_{s} = \eta _x/{u}_{s}$ (with $\eta _x = {\rm d} \eta /{\rm d}\kern0.7pt x$) and ${v}_{b} = 0$. The mean water level is located at $y=0$, which implies that $\left \langle \eta \right \rangle =0$ for the ($2{\rm \pi} /k$)-periodic wave profile $\eta$, that is

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

The local spin or rotation of an infinitesimal fluid element is measured by the vorticity, which is expressed by $\omega =u_y-v_x$ for two-dimensional motion. An inherent property of inviscid flows is that the vorticity of a given fluid element is preserved by the resultant fluid motion (Constantin Reference Constantin2011), and, accordingly, a fluid mass which is initially irrotational will remain so for all further times. Therefore, assuming irrotational motion, $u_y=v_x$, allied to the incompressibility condition $\boldsymbol {\nabla } \boldsymbol {\cdot } (u,v)=0$, enables us to define a velocity potential, $\phi (x,y)$, and stream function, $\psi (x,y)$, respectively, such that $u = \phi _x = \psi _y$ and $v = \phi _y = -\psi _x$. It follows that the complex potential, defined by $f(z) = \phi (x,y)+{\rm i}\psi (x,y)$, and the complex velocity, $w={\rm d} f/{\rm d} z = u-{\rm i} v$, are holomorphic functions of the complex abscissa $z=x+{\rm i} y$ (where ${\rm i}^2=-1$). The stream function $\psi = \operatorname {Im}\{\,f\}$ is constant at $y=-d$ and at $y=\eta$ since these boundaries are impermeable and steady; hence, $f(z)$ represents a conformal mapping from the fluid domain to a fixed rectangle, the so-called conformal plane, once the non-stagnation condition (for some $\epsilon<0$)

(2.3)\begin{equation} u(x,y) \leqslant \epsilon<0 \end{equation}

holds throughout the fluid domain. The equation of motion is given by the Euler equation which, for irrotational fluid, can be expressed in terms of the following Bernoulli condition

(2.4)\begin{equation} 2p + 2gy + u^2 + v^2= B, \quad x \in \mathbb{R},-d \leqslant y \leqslant \eta, \end{equation}

where g is the acceleration due to gravity, $B$ is the Bernoulli constant, and $p=p(x,y)$ is the pressure (relative to the atmospheric pressure) divided by the (constant) density. At the free surface, surface tension is neglected and the pressure is zero, ${p}_{s} = 0$. From (2.2) and (2.4), we get

(2.5)\begin{equation} B=\left\langle u_{s}^{2} + v_{s}^{2} \right\rangle = \left\langle u_{b}^{2} \right\rangle > 0, \end{equation}

where the second equality in (2.5) follows from an application of the divergence theorem applied to the vector field $(-2uv,u^2-v^2)$, cf. Clamond & Constantin (Reference Clamond and Constantin2013). Note that $B=c^2$ in deep water ($d\to \infty$), and for solitary waves ($k\to 0$), since in these settings $(u,v)\to (-c,0)$ as $y\to -\infty$, and $|x|\rightarrow \infty$, respectively. Finally we observe that relations (2.4) and (2.5) yield

(2.6)\begin{equation} \left\langle {\mathscr{p}} \right\rangle = gd, \end{equation}

where ${\mathscr {p}}(x)\stackrel {\rm def}{=}p(x,-d)={p}_{b}(x)$ is the pressure at the bed. This is a useful formula for determining the mean fluid depth $d$ from pressure measurements at the flat bed.

3. Stokes extreme wave

In this paper, by ‘Stokes wave’ we mean a periodic, irrotational water wave, with a strictly monotone symmetric free-surface profile – increasing from trough-to-crest and decreasing from crest-to-trough – which moves with a constant speed and with a constant form over a body of water on a flat bed. For such waves, the functions $u$, $\eta$, $p$ are even functions and $v$ is an odd function with respect to the $x$-variable; we assume the wave crest is located at $x=0$, and the wave troughs are located at $x={\pm }{\rm \pi} /k$. It was shown by Stokes (Reference Stokes1880), using formal methods, that regular wave solutions exist for which the strict inequality (2.3) holds throughout the fluid domain. Given this inequality, it can be rigorously proven that a continuously differentiable free surface must a priori be a symmetric, real-analytic graph of a function: surveys of rigorous analytical results for Stokes waves can be found in Buffoni & Toland (Reference Buffoni and Toland2003), Constantin (Reference Constantin2011), Constantin (Reference Constantin and Constantin2016) and Toland (Reference Toland1996).

Stokes also posited, using heuristic arguments, that a limiting wave solution exists for which (2.3) fails and the equality $u=0$ is attained precisely at the wave crest, with the associated breakdown in regularity of the wave profile at the crest manifested in the form of an angular point which has a containing angle of $2{\rm \pi} /3$. In the conformal plane, the curve $\eta (\phi )$ presents a cusp at the crest where $\eta \propto |\phi |^{2/3}$ locally, while in the physical plane the curve $\eta (x)$ presents an angular point where $\eta \propto |x|$ locally. The existence of Stokes waves is rigorously established by way of highly technical bifurcation-theory methods and, indeed, it was only in recent decades that a number of fundamental questions concerning the existence of an extreme Stokes wave, and properties of the associated wave profile, were rigorously settled (cf. Toland Reference Toland1996; Buffoni & Toland Reference Buffoni and Toland2003; Constantin Reference Constantin and Constantin2016). For this limiting wave ‘of greatest height’, or extreme Stokes wave, it can be shown that the free surface is real-analytic everywhere except for the stagnation point at the crest, where it is merely continuous. We note that the wave crest is an ‘apparent’ stagnation point, with fluid particles only coming to rest there instantaneously, cf. Constantin (Reference Constantin2012a).

All technical complications arising in the rigorous analysis of extreme Stokes waves essentially stem from the lack of regularity at the stagnation point, with the primary impediment being a breakdown in the conformality of the mapping $f(z)$ due to the failure of condition (2.3) at the wave crest. Nevertheless, a detailed asymptotic description of the wave profile, and velocity field, in the neighbourhood of the wave crest is possible. This work originated with Stokes (Reference Stokes1880), who postulated that the wave of maximum amplitude has an included crest angle of $2{\rm \pi} /3$, the crest being a stagnation point which, in the conformal plane, has a power $2/3$ singularity. It was shown by Grant (Reference Grant1973) that higher-order expansions in the vicinity of the crest involve non-algebraic powers (see also Norman Reference Norman1974); this work was placed on a rigorous footing in Amick & Fraenkel (Reference Amick and Fraenkel1987) and McLeod (Reference McLeod1987). Performing the same local analysis (around the crest at $x = 0$ where $\eta = a$) in the physical plane, one obtains after some algebra

(3.1)\begin{gather} \eta =a - 3^{-1/2}\,|x| + \beta\kappa^{\nu-1}|x|^{\nu} + O\left(|x|^{2\nu-1}\right), \end{gather}

(3.2)\begin{gather}w^2 = g(a+{\rm i} z) + 2^{-\nu}\sqrt{3^{1+\nu}/(1+\nu+\nu^2)}\beta g\kappa^{\nu-1} (a+{\rm i} z)^{\nu} + O \left((a+{\rm i} z)^{2\nu-1}\right), \end{gather}

where $\beta$ is a strictly positive (dimensionless) constant (McLeod Reference McLeod1987), $\kappa$ is a (freely choosable) characteristic wavenumber (e.g. $\kappa =k$ or $\kappa =g/B$) and $\nu \approx 2.20401861065003478$ is the smallest positive root of the transcendental equation (Grant Reference Grant1973)

(3.3)\begin{equation} \sqrt{3} \tan (\tfrac{1}{3}\nu{\rm \pi}) = -1- 2 \nu^{-1}. \end{equation}

The next positive root of (3.3) is $\nu _2\approx 5.36>2\nu -1$. There are an infinite number of real positive roots $\nu _j$ of (3.3) such that $\nu _j\sim 3j-1/2-\sqrt {3}/2{\rm \pi} j$ as $j\to \infty$. There are also two complex roots, approximately $-1/2\pm {\rm i}\times 1.0714$, that do not appear in the extreme wave (as they would yield an unbounded free surface); they do play a role in near extreme waves, however (Longuet-Higgins & Fox Reference Longuet-Higgins and Fox1977).

In order to derive expression (3.2), we use – cf. equation (3.3) in Clamond & Constantin (Reference Clamond and Constantin2013) – the relation

(3.4)\begin{equation} w_{s}^{2}= 2g(a-\eta)(1-{\rm i}\eta_x)/(1+{\rm i}\eta_x), \end{equation}

and taking $\eta$ of the form (3.1), for some $\nu >1$, (3.4) yields

(3.5)\begin{equation} \frac{w_{s}^{2}}{g}= \frac{|x|}{\sqrt{3}} + {\rm i} x + \beta |\kappa x|^{\nu-1} \frac{(3\nu-2) |x| - {\rm i} (\nu+2) \sqrt{3} x}{2} + O \left(|x|^{2\nu-1}\right). \end{equation}

Then, seeking a complex velocity of the form $w^2=g(a+{\rm i} z)+A(a+{\rm i} z)^{\nu }+O ((a+ {\rm i} z)^{2\nu -1})$, A being unknown, comparison with (3.5) leads to $A$ as defined by (3.2) and the relation (3.3). Note the simplicity of this approximation, in particular the leading approximation for $w^2\approx g(a+{\rm i} z)$, which yields at once (from the Bernoulli equation) the following approximation for the pressure near the wave crest:

(3.6)\begin{equation} p(x,y) \approx g (a-y)- \tfrac{1}{2} g \sqrt{x^2 + (a-y)^2}. \end{equation}

This relation will play an important role in § 5 below. To the best of our knowledge, relations (3.2) and (3.6) have not appeared before in the literature. A study of the qualitative behaviour of the velocity field for the extreme Stokes wave near the wave crest has been explored in Constantin (Reference Constantin2012a), where it is shown that the velocity field exhibits the limiting behaviours

(3.7a,b)\begin{equation} \lim_{x\to 0} |x|^{-1} u^2(x,\eta(x)) = \tfrac{1}{2} 3^{1/2}g, \quad \lim_{x\to 0} |x|^{-1} v^2(x,\eta(x)) = \tfrac{1}{2} 3^{-1/2}g, \end{equation}

which accord with the limits given by (3.2) and (3.5).

On the crest line $x=0$ (i.e. for $\operatorname {Re}\{z\}=0$ and $\operatorname {Im}\{z\}\leqslant a$), we have $\operatorname {Re}\{w^2\}=u^2\geqslant 0$ and $\operatorname {Im}\{w^2\}=0$; the relation (3.2) then implies at once that $\beta$ is necessarily positive. This result was first obtained by McLeod (Reference McLeod1987) with much more convoluted arguments. However, McLeod (Reference McLeod1987) also proved that $\beta \neq 0$, the possibility $\beta =0$ being not easily ruled out here (that would require more than just a local analysis around the crest). From the inequality $\beta >0$, the semi-derivatives at the crest are given by

(3.8a–c)\begin{equation} \lim_{x\to0^{\pm}} \eta_{x} = {\mp}3^{-1/2}, \quad \lim_{x\to0^{\pm}} \eta_{xx} = 0^{+}, \quad \lim_{x\to0^{\pm}} \eta_{xxx} = {\pm}\infty, \end{equation}

in comparison with the corresponding semi-derivatives in the conformal plane (where, for example, $\eta _{\phi }$ is infinite at the crest since $\eta \propto |\phi |^{2/3}$ locally).

4. Wave surface recovery: regular waves

The function ${\mathscr {P}}$ defined by

(4.1)\begin{equation} {\mathscr{P}}(z)\, \stackrel{\rm def}{=}\, \tfrac{1}{2} B + gd- \tfrac{1}{2} w^2(z)= \tfrac{1}{2} B + gd- \tfrac{1}{2} (u^2-v^2) + {\rm i} u v, \end{equation}

is holomorphic, and its restriction at the flat bed coincides with the real-valued function ${\mathscr {p}}(x)$, since ${{\mathscr {P}}}_{b} = {\mathscr {P}}(x-{\rm i}d) = g/d+(B-u_{b}^{2})/2 = {\mathscr {p}}(x)$. Accordingly ${\mathscr {P}}(z) = {\mathscr {p}}(z+{\rm i}d)$ and, since $p$ is not a harmonic function (Constantin & Strauss Reference Constantin and Strauss2010; Constantin Reference Constantin2011; Henry Reference Henry2011), the flat bed is the only location where ${\mathscr {p}}$ coincides with $\operatorname {Re}{\mathscr {P}}$. It can be shown (cf. Clamond Reference Clamond2013; Clamond & Constantin Reference Clamond and Constantin2013) that

(4.2)\begin{equation} g \eta (1-{\rm i} \eta_x) + {\rm i} B \eta_x= ({{\mathscr{P}}}_{s}- gd) (1+{\rm i} \eta_x), \end{equation}

with ${{\mathscr {P}}}_{s} = {\mathscr {P}}({z}_{s}) = {\mathscr {P}}(x+{\rm i}\eta (x))$. The real and imaginary parts of (4.2) give the two equations

(4.3)\begin{gather} g \eta+ \operatorname{Re}\{{\mathscr{P}}_{s}\}- gd- \eta_x \operatorname{Im}\{{\mathscr{P}}_{s}\}= \tfrac{1}{2}(B - u_{s}^{2}-v_{s}^{2}), \end{gather}

(4.4)\begin{gather}(B- g\eta) \eta_x+ (\operatorname{Re}\{\mathscr{P}_{s}\}-gd) \eta_x + \operatorname{Im}\{\mathscr{P}_{s}\}= \tfrac{1}{2} (B + u_{s}^{2} + v_{s}^{2}) \eta_x. \end{gather}

where the second equalities are obtained using (4.1) and ${v}_{s} = \eta _x/{u}_{s}$. From the viewpoint of determining the wave amplitude $a$, for regular Stokes waves we have $\eta _x(0)=0$ at the crest $z={\rm i} \eta (0)={\rm i} a$, and so (4.3) reduces to a simple implicit equation which can be iteratively solved to find the amplitude $a$ as solution of the equation

(4.5)\begin{equation} a= g^{-1}\,\operatorname{Re}\{\mathscr{P}({\rm i} a)\}- d. \end{equation}

The convergence of fixed-point iterations of (4.5) was studied in Clamond & Constantin (Reference Clamond and Constantin2013) considering the functional iterations (convergent for all regular waves)

(4.6)\begin{equation} y = \hat{F}(y)\, \stackrel{\rm def}{=}\, y + g^{-1}p(0,y) = g^{-1}\operatorname{Re}\{\mathscr{P}({\rm i} y)\}- d. \end{equation}

With the wave amplitude $a$ so determined, $\eta$ can subsequently be obtained by re-expressing (4.4) as the ordinary differential equation (Clamond & Constantin Reference Clamond and Constantin2013)

(4.7)\begin{equation} \eta_x= \operatorname{Im}\{\mathscr{P}_{s}\}/\left(B- g\eta- \operatorname{Re}\{\mathscr{P}_{s}\} + gd\right), \end{equation}

with initial data $\eta (0)=a$, which can iteratively be solved. In Clamond & Constantin (Reference Clamond and Constantin2013), it was rigorously proven that these recovery procedures converge.

An alternative, more direct approach to the wave-profile recovery problem was instigated in Clamond (Reference Clamond2013) by considering the holomorphic function $\mathscr {Q}$ defined by

(4.8)\begin{equation} \mathscr{Q}(z)\, \stackrel{\rm def}{=}\, \int_{z_0}^z\left[\mathscr{P}(z') - gd\right]{\rm d} z' = \int_{z_0}^z\tfrac{1}{2}[B - w(z')^2]\,{\rm d} z', \end{equation}

where $z_0$ is an arbitrary constant. Taking, as in Clamond (Reference Clamond2013), $z_0={\rm i} a$ at the origin of the free surface, and integrating along the surface, leads to the expression

(4.9)\begin{equation} {\mathscr{Q}}_{s}(x)= \int_0^xg \eta(x')\,{\rm d}\kern0.7pt x' + {\rm i}\left[\eta(x) - a\right][B - \tfrac{1}{2} g a - \tfrac{1}{2} g \eta(x)]. \end{equation}

The imaginary part of (4.9) yields an implicit relation defining $\eta$ of the form

(4.10)\begin{equation} \eta= g^{-1}\left[B- \sqrt{(B-ga)^2- 2g \operatorname{Im}\{\mathscr{Q}_{s}\}}\right], \end{equation}

which is a local algebraic (i.e. neither differential nor integral) equation for the free surface $\eta$, and it is easily shown that (4.10) is an exact implicit solution of (4.7). An explicit expression for $\eta$ can be obtained numerically via fixed-point iterations, whose convergence is determined as follows. We define functional iterations (for $y\in [-d,\eta (x)]$)

(4.11)\begin{equation} y= F(y)\, \stackrel{\rm def}{=}\, g^{-1}\left[B- \sqrt{(B-ga)^2- 2 g \operatorname{Im}\{{\mathscr{Q}}(x+{\rm i} y)\}}\right], \end{equation}

for a known holomorphic function ${\mathscr {Q}}$ at a fixed abscissa $x$ and for a given $B$. Iterations of (4.11) converge if we can show that $F$ is a contraction, that is, if $-1<F_y<1$. The definition of $F$ gives

(4.12)\begin{equation} F_y = \frac{\left[\operatorname{Im}\{{\mathscr{Q}}\}\right]_y} {\sqrt{(B-ga)^2- 2 g \operatorname{Im}\{{\mathscr{Q}}\}}} = \frac{p + g y + v^2}{B- g y} = \frac{p + g y + v^2}{2 p + g y + u^2 + v^2}, \end{equation}

where we have used definitions (4.1), (4.8) and the Bernoulli equation (2.4). The lower bound $F_y>-1$ yields the inequality $p+v^2+B>0$ that is obviously always satisfied. The upper bound for contraction condition $F_y<1$ yields the inequality $p+u^2>0$ that is always satisfied when (2.3) holds. Therefore, for regular Stokes waves, $F$ is a contraction everywhere in the bulk and at the free surface, and the iterations (4.11) converge to a unique solution.

Since $F_y$ is independent of $a$ (and so independent of the choice of $z_0$ in the definition of $\mathscr {Q}$), the convergence proof for the iterations (4.11) is valid wherever $z_0$ is chosen in the fluid. Furthermore, the analysis above for the wave-profile recovery procedure is valid for all regular waves, periodic or not, with one or more crests and troughs per period, symmetric or not. It is not contingent on the waves being Stokes waves.

5. Wave surface recovery: extreme waves

The above considerations may breakdown for the extreme waves, where (2.3) fails to hold at the stagnation point at a wave crest. In this case, the wave crest located at $(x,y)=(0,a)$ takes an angular form with containing angle $2{\rm \pi} /3$, with an asymptotic description provided by (3.1) (so in particular $\eta_x(0^{\pm})\neq 0$). The analysis of the previous section, see also (Clamond Reference Clamond2013; Clamond & Constantin Reference Clamond and Constantin2013), then needs to be refined.

5.3. Surface determination

The convergence of the functional iterations (4.11) is proven above (see Clamond (Reference Clamond2013) for more details) everywhere except at an angular crest where $F_y(a)=1$: this corresponds to a so-called positively neutral fixed point (Bair & Haesbroeck Reference Bair and Haesbroeck1997). To determine if the iterations converge where $F_y=1$, we can consider the second derivative of $F$:

(5.2)\begin{equation} F_{yy}= \frac{g F_y + p_y + g + 2 v v_y}{2 p + g y + u^2 + v^2}= \frac{g F_y + v v_y - u v_x}{B - g y}= \frac{g F_y + \operatorname{Im}\{w\,w_z\}}{B - g y}. \end{equation}

In order to estimate $F_{yy}$ at angular crests, we use the expansions (3.1) and (3.2) to find that $F_{yy}=3/2a>0$ at the angular crests. Hence, the fixed point $y=a$ is monotonously semi-stable from below (Bair & Haesbroeck Reference Bair and Haesbroeck1997), which means that the iterations are convergent when the fixed point $y=a$ is approached from the fluid side $y<a$, and divergent when approached from above the surface $y>a$. For practical applications, this result suggests that basic fixed-point iterations of (4.10) converge very slowly in the vicinity of an angular crest. More sophisticated iterations (such as Newton and Levenberg–Marquardt methods) should then be used instead. It is not our purpose here to investigate these possibilities, however.

5.4. Trough determination

At the wave trough, where $\eta =-b\leqslant 0$ and so $a+b$ is the total wave height, for all Stokes waves including the extreme wave, we have $\eta _x({\pm }{\rm \pi} /k)=0$. Equation (4.3) then reduces to the implicit equation

(5.3)\begin{equation} b= d- g^{-1}\operatorname{Re}\{\mathscr{P}(-{\rm i} b)\}. \end{equation}

In Clamond (Reference Clamond2013), presented numerical evidence suggests that iterations of (5.3) converge to $b$; however, a mathematically rigorous proof of this result is still lacking. Indeed, as shown in Clamond (Reference Clamond2013), convergence is guaranteed if $-2g<p_y<0$ on the trough line $y\in [-d,-b]$. The upper bound is ensured by the inequality $p_y\leqslant -g$ on the trough line, which can be rigorously established for nonlinear Stokes waves of all amplitudes (Constantin Reference Constantin2006; Constantin & Strauss Reference Constantin and Strauss2010; Henry Reference Henry2011; Lyons Reference Lyons2016) by way of maximum principles, using the fact that the pressure distribution is a super-harmonic function with respect to the physical variables, cf. Constantin (Reference Constantin2011). However, a rigorous proof of the lower bound for $p_y$ is currently lacking, being unachievable by this approach. This is not a problem for extreme waves since $b$ can be computed without relying on (5.3). Indeed, $a$ being given by (4.5), the Bernoulli constant is $B=2ga$ and $b$ is obtained directly by computing (4.10) at a trough.