## Abstract

We investigate the location of the point of maximal horizontal velocity for steady periodic water waves with vorticity.

## 1. Introduction

We consider steady two-dimensional periodic water waves of finite depth that are acted upon by gravity. The waves may possess vorticity but are inviscid and without surface tension. If (*u*, *v*) denotes the velocity field and *c*>0 the horizontal speed of the wave, then a point where *u*=*c* and *v*=0 is called a *stagnation point*. Although we consider waves with *u*<*c*, we are interested in waves that approach stagnation. For a wave for which the maximum of *u* is nearly *c*, what is the location of the maximum point?

First insights into the dynamics of water waves were obtained within the framework of linear theory, starting with the work of Airy, Stokes and their contemporaries in the nineteenth century. Since the typical water waves propagating on the surface of the sea (or on a river or lake) are, as a matter of common experience, locally approximately periodic and two dimensional with the motion being identical in any direction parallel to the crest line, steady periodic waves have played a special role within hydrodynamics. Stokes (1847) was among the first to observe that actual water wave characteristics deviate significantly from the predictions of linear theory and he initiated an extensive study of the nonlinear governing equations. Remarkable success was achieved in the understanding of large-amplitude steady periodic waves within the irrotational framework, that is, for flows with no vorticity (see the review Toland 1996). Field evidence (Lighthill 1978) as well as laboratory experiments (Thomas & Klopman 1997) indicate that generally *u*<*c* throughout the fluid for a wave that is not near breaking. For irrotational flows, Stokes conjectured in 1847 that there exist periodic steady waves with stagnation points at their crests. This conjecture was proven by Amick, Fraenkel and Toland both for water of finite depth over a flat bed and for water of infinite depth (Amick & Toland 1981; Amick *et al.* 1982). For flows with vorticity, Gerstner (1809) found a family of explicit periodic steady water waves with a special non-zero vorticity distribution in water of infinite depth, the limiting case being that of waves with stagnation points at their crests (Constantin 2001).

In Constantin & Strauss (2004), we proved that, for an arbitrary vorticity distribution and for any given *c*>0 and relative mass flux *p*_{0}, there is a global continuum of steady periodic water waves travelling at speed *c* and such that *u*<*c* throughout the fluid. The continuum contains waves with *u* arbitrarily close to the wave speed *c*. We are interested in the location of the points where *u* is maximally close to *c*. In §3, we establish a number of results concerning the location of the point of maximal horizontal velocity *u*. In §4, we prove that for constant negative vorticity which is sufficiently small, the point of maximal horizontal velocity is located precisely at the wave crest. This generalizes a well-known fact for irrotational flows (Toland 1978, 1996).

## 2. Preliminaries

To describe steady periodic water waves, it suffices to consider a cross-section of the flow that is perpendicular to the crest line. Choose Cartesian coordinates (*x*, *y*) with the *y*-axis pointing vertically upwards and the *x*-axis being the direction of wave propagation, with the origin located on the mean water level. Let (*u*(*t*, *x*, *y*), *v*(*t*, *x*, *y*)) be the velocity field of the flow, let {*y*=−*d*} for some *d*>0 be the flat bed and let *S*={*y*=*η*(*t*, *x*)} be the water's free surface (see figure 1).

We consider gravity water waves: the restoring force acting on the free surface of water is gravity. A physically reasonable assumption for gravity waves is that the density is constant (Lighthill 1978), which implies the equation of mass conservation(2.1)That the flow is inviscid is also appropriate for gravity waves (Lighthill 1978), so that the equation of motion is Euler's equation(2.2)where *P*(*t*, *x*, *y*) denotes the pressure and *g* is the gravitational constant of acceleration. The free surface decouples the motion of water from that of the air, so that(2.3)must hold, where *P*_{atm} is the constant atmospheric pressure (Johnson 1997) and the surface tension is assumed to be negligible. Moreover, since the same particles always form the free surface, we have(2.4)On the flat bed, the boundary condition(2.5)expresses the fact that water cannot penetrate the rigid bed at *y*=−*d.* The governing equations for water waves (2.1)–(2.5) are of course highly nonlinear. We assume the flow is periodic, in the sense that the velocity field (*u*, *v*), the pressure *P* and the free surface *η* all have period (wavelength) 2*L* in the *x*-variable. For steady periodic waves with wave speed *c*>0, the (*x*, *t*) dependence of the free surface, the pressure and the velocity field has the form *x*−*ct*. Letbe one period of the fluid domain. For the solutions of equations (2.1)–(2.5) under consideration, we assume that *u*, *v*, *P*, *η* are all *C*^{2} in the variables (*t*, *x*, *y*) for .

The incompressibility condition (2.1) enables us to introduce a stream function *ψ*(*x*, *y*) on , defined up to a constant by(2.6)Thus,(2.7)where *ω*=*v*_{x}−*u*_{y} is the vorticity. Euler's equation (2.2) implies that *ψ* and *ω* have parallel gradients and the same level sets and are functionally dependent. If the steady wave satisfies the condition *u*<*c* throughout , this functional dependence is global, that is, there exists a *C*^{1} function *γ*, called the vorticity function, such that(2.8)throughout the fluid (Constantin & Strauss 2004). Under assumption (2.8), the governing equations can be reformulated in the moving frame as(2.9)where(2.10)is the relative mass flux and is independent of *x* (Constantin & Strauss 2004). The nonlinear boundary condition on the free surface *y*=*η*(*x*) is an expression of Bernoulli's law, the constant *Q* being given by *Q*=2(*E*+*gd*), where *E* is the hydraulic head of the flow (Constantin & Strauss 2004). We assume that(2.11)

Since in equation (2.9) the function *ψ* is constant on the free surface *y*=*η*(*x*) as well as on the flat bed *y*=−*d*, it is natural to introduce the new independent variables(2.12)(see figure 2). They have the effect of transforming the free boundary problem into the boundary value problem(2.13)in the fixed rectangle for *h*(*q*, *p*)=*y*+*d* even and of period 2*L* in the *q*-variable. The height *h* above the flat bed satisfies(2.14)The coordinate transformation (*x*, *y*)→(*q*, *p*) is global if equation (2.11) holds. The evenness of *h* reflects the requirement that *u* and *η* are symmetric, while *v* is antisymmetric around the line *x*=0 located strictly below the wave crest (Constantin & Strauss 2004). Note that any solution of equations (2.9)–(2.11) with a free surface which is monotone between crests and troughs has to be symmetric (Constantin & Escher 2004*a*,*b*; Hur 2007). For further use, we recall the formulae (Constantin & Strauss 2004)(2.15)

Working with the reformulation (2.13) of the water-wave problem (2.1)–(2.5) and using bifurcation and degree theory, and given *p*_{0}<0, *c*>0 and *γ*∈*C*^{2}, we proved in Constantin & Strauss (2004) the existence of a global connected set of smooth solutions. This continuum contains waves with *u* arbitrarily close to the wave speed *c*. Each solution in has certain properties which we now specify. Let us introduce the line segmentswhich represent the four sides (top, bottom, left and right) of the boundary (with the exception of the four corners) of the open rectangle *R*=(0, *L*)×(*p*_{0}, 0), which has half the size of . We proved in Constantin & Strauss (2004) that for every wave in , we have the strict inequalities(2.16)(2.17)on the four sides. At the bottom corners we have(2.18)and at the top corners we have(2.19)The inequalities (2.16)–(2.18) were proven in Constantin & Strauss (2004). We also showed in Constantin & Strauss (2004) that at the top left corner (0, 0) we have either *h*_{qq}<0 or *h*_{qqp}>0, while at the top right corner (*L*, 0) we have the opposite inequalities.

We now prove the more precise statement (2.19) for the top corners. Pick, for instance, the top left corner (0, 0). Since *h* is even, we have *h*_{q}(0, *p*)=*h*_{qqq}(0, *p*)=0 and hence *h*_{qp}(0, *p*)=0 for *p*_{0}≤*p*≤0. Since *h*_{q}(*q*, 0)<0 on *R*_{t} and *h*_{q}(0, 0)=0, we have *h*_{qq}(0, 0)≤0. Now differentiating the nonlinear boundary condition in equation (2.13) twice with respect to *q*, we obtainandEvaluation of this equation at (0, 0) yieldssince *h*_{q}=*h*_{pq}=0 at the corner. If we supposed that *h*_{qq}(0, 0)=0, then only a single term would be left in this expression. Since both *h*_{p}(0, 0) and 2*gh*(0, 0)−*Q* are non-zero in view of equations (2.11), (2.13) and (2.14), we would infer that *h*_{pqq}(0, 0)=0. But then both *h*_{qν}(0, 0)=0 and *h*_{pνν}(0, 0)=0, where *ν* is a vector that strictly exits the open rectangle. But *h*_{q}<0 in *R* and *h*_{q}(0, 0)=0 which contradicts Serrin's edge-point lemma (Fraenkel 2000). We conclude that *h*_{qq}(0, 0)<0.

The inequalities (2.19) are actually essential for the proof of lemma 5.2 in Constantin & Strauss (2004), because equations (2.16)–(2.19) define an open set in our function space *X* that consists of the *C*^{3+α} functions *h*(*q*, *p*), such that *h*=0 on the bed and *h* is periodic and even in the variable *q*. However, the inequalities (5.1)–(5.3) in Constantin & Strauss (2004) do not define an open set. We thank E. Wahlen for the latter observation. Thus, the validity of the arguments made in §5 of Constantin & Strauss (2004) is justified by relations (2.19). To see that eqns (5.1)–(5.3) in Constantin & Strauss (2004) do not define an open set, consider, for instance, the family of functions(2.20)The function *h* defined by equation (2.20) satisfies equations (2.16)–(2.18) if and only if *k*(*p*)>0 for *p*∈(*p*_{0}, 0) and *k*′(*p*)>0. Sincewe see that a function *h* of type (2.20) with of class *C*^{4} satisfies eqns (5.1)–(5.3) of Constantin & Strauss (2004) if and only if *k*(*p*)>0 for *p*∈(*p*_{0}, 0) and *k*′(*p*_{0})>0 and either *k*(0)>0 or *k*′(0)<0 (see figure 3). But in any *C*^{4} neighbourhood of such a function *k*_{0} with *k*_{0}(0)=0, there are functions that do not belong to the class specified above: it suffices to translate the graph of *k* downwards. This shows that the inequalities (5.1)–(5.3) in Constantin & Strauss (2004) do not define an open set.

On the other hand, the inequalities (2.16)–(2.19) do define an open set. Indeed, on *R*_{l}∪*R*_{r}, we have *h*_{q}=*h*_{qp}=0, while on *R*_{b} we have *h*_{q}=*h*_{qq}=*h*_{qqq}=0, so thatwhileThe openness therefore follows using Taylor's expansion in a neighbourhood of *R*.

## 3. Points of maximal horizontal velocity

In this section, we prove some results concerning the location of the points of maximal horizontal velocity for waves with non-zero vorticity. We consider waves with the properties indicated above; this includes all the waves in . To avoid trivialities, we exclude the waves with completely flat surfaces *η*′(*x*)≡0. The vorticity function *γ* is arbitrary unless otherwise stated.

*There is no interior point of maximal horizontal velocity if γ*′≥0.

From Δ*ψ*=−*γ*(*ψ*), we deduce that(3.1)Since *γ*′≥0 and *ψ*_{y}=*u*−*c*<0, the maximum principle (Fraenkel 2000) asserts that sup_{Ω}{*ψ*_{y}} is attained on the boundary of *Ω* and not inside.

*If a point of maximal horizontal velocity lies on the bed, then it is the point directly below the crest*.

Let(3.2)where for *p*∈[*p*_{0},0]. Then,so thatin view of equations (2.6) and (2.2). Therefore, equation (2.15) yields(3.3)throughout the fluid. We also have(3.4)Indeed, we use equation (2.15) to writesince *ψ*_{y}=*u*−*c* and *v*_{x}−*u*_{y}=*ω*=*γ*(*ψ*).

Now, on *R*_{b}⊂{*y*=−*d*}, we have *v*=0 andby equations (2.14)–(2.16) and (2.5). Thus, on *R*_{b}, we have . Since *ψ*=−*p*_{0} on *y*=−*d*, from the monotonicity of *f* we infer that (*c*−*u*)(*x*,−*d*) strictly increases for *x*∈[0,*L*].

The function *f* defined by equation (3.2) has certain monotonicity properties.1 Let us denote by *A* the wave crest (0, *η*(0)), by *B* the point (0, −*d*) located on the bed directly below the wave crest and by *C* the point (*L*, −*d*) located on the bed directly below the wave trough *D*=(*L*, *η*(*L*)) (see figure 4).

*The function f is strictly increasing on the segments from A to B, B to C and C to D*.

On the line segment *AB*, we have *x*=0, *v*=0 andby equations (2.14), (2.15) and (2.17), since along this segment *v*=0 as *v* is an odd function of *x*. Similarly along the line segment *CD* where *x*=*L*, we have .

Along the line segment *BC* for which *y*=−*d* and *x*∈(0, *L*), we have *v*=0 andas in the proof of proposition 3.2.

*Any point J of maximal horizontal velocity on the top surface S satisfies the inequality*

By equation (2.13), any point on the free surface satisfies the boundary condition . At the crest *A*, *h*_{q}(*A*)=0 soTherefore, on the free surfaceFor the point *J* with maximal *u*, we have *u*(*J*)≥*u*(*A*), so that andThus, by equation (2.14)

*The trough can never be a point of maximal horizontal velocity (unless the free surface is flat)*.

At the trough *D*, *v*=0 so that by lemma 3.2, *η*(*D*)=*η*(*A*) and the surface is flat. A second proof follows directly from lemma 3.1 since *f*(*D*)>*f*(*A*), so that *u*(*D*)>*u*(*A*).

By equation (2.4), the inequality in lemma 3.2 can be written asThis indicates that if *J*≠*A* is almost a stagnation point, then the slope of the profile is very large at *J*.

*Consider the restriction of c*−*u to the vertical segment AB directly below the crest*.

*If γ*≤0,*then c*−*u is strictly monotone on AB with its minimum at the crest A*.*If γ*>0,*then c*−*u has a local minimum at the point B on the bed*.*If γ*>0*is sufficiently small and if the solutions depend continuously on γ, then c*−*u has a local minimum at the crest A as well as B*.

Along

*AB*we havesince 0>*h*_{qq}=*v*_{x}/(*u*−*c*) as in the proof of lemma 3.1.We evaluate the partial differential equation (2.13) at the point

*B*on the bed. At that point*h*=*h*_{q}=*h*_{qq}=0, so that . Since*h*_{p}>0, we have*h*_{pp}(*B*)<0. By equations (2.14) and (2.15), this is equivalent to (*c*−*u*)_{y}>0 at*B*.In case

*γ*=0, we have (*c*−*u*)_{y}=−*v*_{x}<0 at the crest*A*. By assumption, if*γ*is sufficiently small, the same must be true, that is, (*c*−*u*)_{y}(*A*)<0.

An immediate consequence of proposition 3.2 and proposition 3.3(i) is the following result.

*If γ*≤0, *then no point of maximal horizontal velocity can lie on the bed*.

We also have the following.

*If the pressure throughout the fluid is larger than atmospheric pressure and if there exists a point of maximal horizontal velocity on the free surface, then the crest must be such a point*.

Since *P*=*P*_{atm} on *S*={*y*=*η*(*x*)} and the profile *η* is monotone decreasing from crest to trough, the hypothesis implies that *P*_{x}≤0 on *S* from crest to trough. By equation (3.3), we have *∂**f*/∂*q*≥0. But *ψ*=0 on *S*, hence (*c*−*u*)^{2} is increasing along *S* from crest to trough by equation (3.2). Since *c*−*u*>0 throughout the fluid, *u* is decreasing from crest to trough.

We now present an example for which proposition 3.4 is applicable.

Assume that *γ*′≤0 and *Γ*≥0 everywhere, as well as (*c*−*u*)*γ*≥−*g* on the bed. Then the pressure throughout the fluid is larger than atmospheric pressure and therefore if there exists a point of maximal horizontal velocity on the free surface, then the crest must be such a point.

Note that *γ*′≤0 and *Γ*≥0 imply *γ*(−*p*_{0})≤0. We have(3.5)by equations (2.2), (2.7) and (2.8). Therefore, *P*−*Γ*(−*ψ*) is superharmonic and attains its minimum over either on the flat bed {*y*=−*d*} or on the free surface *S*={*y*=*η*(*x*)}.

If min[*P*−*Γ*(−*ψ*)] were attained on the bed, then at that point we would have 0<∂_{y}[*P*−*Γ*(−*ψ*)] by Hopf's maximum principle (Fraenkel 2000). But by Bernoulli's law, the expression 1/2[(*c*−*u*)^{2}+*v*^{2}]+*gy*+*P*−*Γ*(−*ψ*) is constant throughout the fluid (Constantin & Strauss 2004), so that(3.6)because on the bed *v*=*v*_{x}=0 and *γ*(−*p*_{0})=*γ*(*ψ*)=*ω*=*v*_{x}−*u*_{y} by equation (2.8). By hypothesis, the expression on the right-hand side of equation (3.6) would be negative, which is a contradiction.

Therefore, the function *P*−*Γ*(−*ψ*) attains its minimum on the free surface *S*, where its value is *P*_{atm} in view of equation (2.3). Since by hypothesis *Γ*≥0 throughout the fluid, we deduce that at any point in *Ω*, we have *P*≥*P*−*Γ*(−*ψ*)≥*P*_{atm}.

Considerations similar to those made in the proof of this example can be used to prove certain stability properties for water waves with vorticity (Constantin & Strauss in press) within a variational framework (Constantin *et al.* 2006).

## 4. The case of small constant negative vorticity

The aim of this section is to prove the following.

*Given p*_{0}<0, *assume that γ*≤0 *is a constant that satisfies*(4.1)*Then, for every wave in the continuum of solutions* , *the point of maximum horizontal velocity must be the wave crest*.

As shown above, from equation (3.5), the function *P*+*γψ* is superharmonic and therefore attains its minimum either on the bed {*y*=−*d*} or on the free surface *S*. Let us first show that the minimum could not occur at any point *M* on the flat bed. If it did, then Hopf's maximum principle at *M* would yield ∂_{y}[*P*+*γψ*]>0. Thus, by equation (3.6),(4.2)at *M*. Clearly, equation (4.2) is impossible in the irrotational case *γ*=0. In case *γ*<0, we proceed as follows to invalidate equation (4.2). LetThen (Constantin & Strauss 2004, p. 524)(4.3)and(4.4)where *A*=(0, *η*(0)) is the wave crest and *λ*_{0}>0 is the unique solution of the equationSince the function is strictly convex for *λ*>0 with its minimum value over (0,*∞*) attained at *λ*_{0}, we deduce from equation (4.4) thatIn combination with equation (4.3), this yields(4.5)along the continuum . On the other hand, since *v*=0 at the wave crest and the wave trough, the nonlinear boundary condition in equation (2.9) yields(4.6)with *C*=(*L*, −*d*) and *D*=(*L* ,*η*(*L*)) as in §3, the last inequality coming from lemma 3.1 on the segment *CD*. Combining equations (4.5) and (4.6), we getThe first term on the left is positive. Using lemma 3.1 on the segment *BC* together with the evenness of *c*−*u*, we obtain(4.7)Combining equations (4.2) and (4.7), we get exactly the opposite of inequality (4.1). Therefore, we deduce that the minimum of *P*+*γψ* is attained on the free surface.

By Hopf's maximum principle applied to the function *P*+*γψ*, which equals the constant *P*_{atm} on *S*, we must therefore have(4.8)all along the free surface *S*. Thus, *P*_{y}<−*γ*(*u*−*c*)≤0 on *S*. Next, we compute *P*_{y} on the surface.

*On the free surface S, we have* .

Using Euler's equation (2.2) and changing variables to (*q*, *p*) using equation (2.15), we get(4.9)on *S* by a key cancellation. On the other hand, differentiating the nonlinear boundary condition in equation (2.13) with respect to *q*, we obtainUsing this equation to replace *h*_{qq} in the expression (4.9), we obtain(4.10)all along *S*. ▪

We conclude the proof of the theorem as follows. Proposition 3.1 and corollary 3.2 ensure that any point *J* of maximal horizontal velocity must lie on the free surface. Now, *h*_{p}=1/(*c*−*u*)>0 attains its minimum along *S* at *J*, so that *h*_{pq}(*J*)=0. We showed above that *P*_{y}(*J*)≠0. Therefore, equation (4.10) implies that *h*_{q}(*J*)=0. By equation (2.16), this can occur only at the crest and the trough. By corollary 3.1, the point *J* cannot be the trough. Therefore, *J* is the crest. ▪

Since *p*_{0}<0, the inequality (4.1) holds for *γ*≤0 if |*γ*| is small enough. In the irrotational case *γ*=0, we recover the classical result (Toland 1978, 1996) that was originally obtained by complex analytical methods specific to irrotational flows.

## Acknowledgments

W.S. was supported in part by NSF grant DMS 0405066.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

↵In the irrotational case (

*γ*=0), these properties are very useful in the description of the particle trajectories (Constantin 2006).- © 2007 The Royal Society