Faculty of Mathematics and Computer Science, Babeș-Bolyai University, Cluj-Napoca, 400084, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, 400110, Romania

Symmetric water waves with surface tension: traveling wave behavior and maximal horizontal velocity

Abstract

We study spatially periodic solutions to the two-dimensional water wave problem with surface tension. Assuming symmetry of the wave profile and of the horizontal velocity component at the free surface, we show that the solution necessarily corresponds to a traveling wave. This result relies on maximum principles for subharmonic functions and the structural properties of the governing equations. As a further consequence, in the case of positive constant vorticity, we establish that the maximum of the horizontal velocity remains invariant in time.

keywords:

Symmetry of the free surface; wave-breaking; traveling wave profile

pacs:

[

AMS Subject Classification]35Q31, 35Q35, 35Q86, 76B15, 35R35

1 Introduction

The study of periodic traveling waves is of interest as they provide a simple yet effective model for free-surface water flows, and their structure facilitates the analysis of various flow properties. The relationship between symmetry and the traveling wave profile has been explored in the literature (see, e.g., [2, 3, 13, 16, 8, 11]). In [8], the authors show that for a two-dimensional free-surface water flow over a flat bed, symmetry in velocity, pressure, and the free surface implies that the flow must be a traveling wave. In [11], it is shown that the same conclusion holds if symmetry is assumed only for the horizontal velocity at the free surface and on the free surface itself.

We note that the results in [8] and [11] do not account for surface tension. Therefore, in Sections 3 and 4, inspired by these two papers, we aim to demonstrate that the same conclusions remain valid when the surface tension is included.

Building on these results and assuming positive vorticity constant, we further investigate the behavior of the maximal horizontal velocity. We show that, even in the presence of surface tension, the maximal horizontal velocity remains constant in time. Related discussions can be found in [4, 14, 12] and the references therein.

The novelty of our contribution lies in two directions:

[(a)]
(a)

We show that, in the presence of surface tension, symmetry implies that the flow has a traveling wave profile.
(b)

Using the symmetry of the traveling wave profile, one deduces that, in the presence of surface tension, the maximum of the horizontal velocity is preserved, provided the free surface can be represented as the graph of a function.

The study of the maximal horizontal velocity is important for understanding the onset of wave breaking. Although this phenomenon has been extensively investigated, its timing and intensity remain difficult to predict due to the strongly nonlinear governing equations and boundary conditions. As shown in field observations [7, 10], numerical simulations [1], and theoretical studies [4, 14], wave breaking is typically associated with a loss of symmetry and, most relevant for our purposes, with an increase in the horizontal fluid velocity.

2 The governing equations

In this paper we consider a two-dimensional, inviscid, and incompressible fluid. Its motion is captured by the Euler equations

	$\displaystyle u_{t}+uu_{x}+vu_{y}$	$\displaystyle=-P_{x},$		(1a)
	$\displaystyle v_{t}+uv_{x}+vv_{y}$	$\displaystyle=-P_{y}-g,$		(1b)

subject to the incompressibility condition

u_{x}+v_{y}=0.

(1c)

Boundary conditions are imposed at the rigid flat bed and the free surface. The kinematic boundary condition on the flat bed $y=-d$ is

v=0,

(1d)

while on the free surface $y=\beta(t,x)$ it takes the form

v=\beta_{t}+\beta_{x}u.

(1e)

Also, on the free surface we have the dynamic boundary condition

P=P_{\text{atm}}-\sigma\nabla\cdot n=P_{\text{atm}}-\sigma\frac{\beta_{xx}}{(1+\beta_{x}^{2})^{3/2}},

(1f)

where $\sigma$ is the surface tension coefficient, $n$ is the outward unit normal to the free surface, and $P_{\text{atm}}$ denotes the constant atmospheric pressure.

As shown in [6], if the normal derivative of the pressure at the free surface is strictly negative, that is, if there exists a constant $C<0$ such that

\frac{\partial P(t,x,\beta(t,x))}{\partial n}=P_{y}(t,x,\beta(t,x))-\beta_{x}(t,x)P_{x}(t,x,\beta(t,x))\leq C,

(1)

then there exists $T>0$ such that the governing equations (1a)–(1f) admit a unique smooth solution on $[0,T]$ . The precise smoothness depends on whether $\sigma$ is zero or not; however, in both cases, the components of the solution are at least in $H^{3}$ , with respect to the spatial variables.

In this paper, condition (1) is always assumed. The quadruple $(u,v,P,\beta)$ denotes the unique smooth solution of the governing equations (1a)–(1f), whose existence and uniqueness follow whenever (1) holds. By a smooth solution, we always mean a solution of the system (1a)–(1f) together with condition (1).

In the subsequent, $T$ denotes the maximal time of existence of $(u,v,P,\beta)$ . For each $t\in[0,T]$ , the fluid domain is

\Omega(t)=\left\{(x,y)\in\mathbb{R}^{2}:y\in[-d,\beta(t,x)]\right\}.

3 Symmetric water waves with surface tension are traveling waves

In this section, following closely the proof of [8, Theorem 4.1], we show that a flow symmetric with respect to the $x$ -axis corresponds to a traveling wave. We emphasize that, unlike in [8], our analysis includes the effect of surface tension, which modifies the boundary conditions accordingly.

Theorem 1.

Assume that $(u,v,P,\beta)$ is $x$ -symmetric, that is, cf. [8], there exists a function $\lambda\in C^{1}[0,T]$ such that

	$\displaystyle u(t,x,y)=u(t,2\lambda(t)-x,y),$		(2)
	$\displaystyle v(t,x,y)=-v(t,2\lambda(t)-x,y),$		(3)
	$\displaystyle P(t,x,y)=P(t,2\lambda(t)-x,y),$		(4)
	$\displaystyle\beta(t,x)=\beta(t,2\lambda(t)-x),$		(5)

for all $t\in[0,T]$ and $(x,y)\in\Omega(t)$ . Then $(u,v,P,\beta)$ has a traveling wave profile.

Proof.

Using (2) and (3) in (1a) and (1b), we obtain (recall that $\lambda\in C^{1}[0,T]$ )

	$\displaystyle(u-\dot{\lambda}(t))u_{x}+vu_{y}=-P_{x},$		(6)
	$\displaystyle(u-\dot{\lambda}(t))v_{x}+vv_{y}=-P_{y}-g,$		(7)
	$\displaystyle v=0\text{ on }y=-d,$		(8)

while substituting (5) in (1e), gives

v(t,x,y)=(u(t,x,y)-\dot{\lambda}(t))\beta_{x}(t,x).

(9)

In addition, from (4) and (5), since

\beta_{x}(t,x)=-\beta_{x}(t,2\lambda(t)-x)

and

\beta_{xx}(t,x)=\beta_{xx}(t,2\lambda(t)-x),

we find that

P(t,x,\beta(t,x))=P(t,x,\beta(t,2\lambda(t)-x)).

(10)

For some $t_{0}$ fixed, we define $c=\dot{\lambda}(t_{0})$ , and consider the functions

	$\displaystyle\bar{u}(t,x,y)=u(t_{0},x-c(t-t_{0}),y),$		(11)
	$\displaystyle\bar{v}(t,x,y)=v(t_{0},x-c(t-t_{0}),y),$		(12)
	$\displaystyle\bar{P}(t,x,y)=P(t_{0},x-c(t-t_{0}),y),$		(13)
	$\displaystyle\bar{\beta}(t,x)=\beta(t_{0},x-c(t-t_{0})).$		(14)

Simple computations yield that

	$\displaystyle\bar{u}_{t}+\bar{u}\bar{u}_{x}+\bar{v}\bar{u}_{y}=(\bar{u}-c)\bar{u}_{x}+\bar{v}\bar{u}_{y}$		(15)
	$\displaystyle\bar{v}_{t}+\bar{u}\bar{v}_{x}+\bar{v}\bar{v}_{y}=(\bar{u}-c)\bar{v}_{x}+\bar{v}\bar{v}_{y}$		(16)
	$\displaystyle\bar{\beta}_{t}+\bar{\beta}_{x}\bar{u}=(\bar{u}-c)\bar{\beta}_{x}\quad\text{ on }y=\bar{\beta}(t,x)$		(17)
	$\displaystyle\bar{v}=0\text{ on }y=-d,$		(18)

and

\bar{P}=P_{\text{atm}}-\sigma\frac{\bar{\beta}_{xx}}{(1+\bar{\beta}_{x}^{2})^{3/2}}\quad\text{ on }y=\bar{\beta}(t,x).

On the other hand, since

	$\displaystyle(\bar{u}-c)\bar{u}_{x}+\bar{v}\bar{u}_{y}=-\bar{P}_{x}$
	$\displaystyle(\bar{u}-c)\bar{v}_{x}+\bar{v}\bar{v}_{y}=-\bar{P}_{y}-g$
	$\displaystyle\bar{v}=(\bar{u}-c)\bar{\beta}_{x}\text{ on }y=\bar{\beta}(t,x),$

we see that $(\bar{u},\bar{v},\bar{P},\bar{\beta})$ also satisfies the same equations (1a)–(1f). Since condition (1) clearly holds for $\bar{P}$ , and $\bar{u},\bar{v},\bar{P},\bar{\beta}$ are smooth, we conclude that $(\bar{u},\bar{v},\bar{P},\bar{\beta})$ is a smooth solution. Now given the uniqueness of a smooth solution for the governing equations, we deduce that $(u,v,P,\beta)=(\bar{u},\bar{v},\bar{P},\bar{\beta})$ , as desired. ∎

4 Symmetric waves in horizontal component and free surface with surface tension are traveling waves

Using the results from Section 3, we demonstrate that for a fluid that is $L$ -periodic in the $x$ -direction with $x$ -symmetric vorticity, it suffices to impose symmetry only on the horizontal velocity component at the free surface and on the free surface itself in order to conclude that the flow under consideration consists of traveling waves.

Let

\gamma:=u_{y}-v_{x}

(19)

denote the vorticity of the fluid.

Theorem 2.

Assume that $(u,v,P,\beta)$ are $L$ -periodic ( $L>0$ ) in the $x$ -variable, and $u$ , $\beta$ , and $\gamma$ are $x$ -symmetric, i.e., there exists a function $\lambda\in C^{1}[0,T]$ such that

		$\displaystyle u(t,x,\beta(t,x))=u\bigl(t,\,2\lambda(t)-x,\,\beta(t,x)\bigr),$		(20)
		$\displaystyle\beta(t,x)=\beta\bigl(t,\,2\lambda(t)-x\bigr),$		(21)
		$\displaystyle\gamma(t,x,y)=\gamma\bigl(t,\,2\lambda(t)-x,\,y\bigr),$		(22)

for all $t\in[0,T]$ and $(x,y)\in\Omega(t)$ . Then $(u,v,P,\beta)$ defines a traveling wave.

Proof.

Our aim is to show that $v$ satisfies (3) and $P$ satisfies (4). Once this is established, Theorem 1 provides the desired conclusion.

To achieve this, we follow a similar reasoning to [11, 12]. Let $\psi$ be the stream function which is given in such a way that

\displaystyle\psi_{y}(t,x,y)=u(t,x,y),

and

\psi_{x}(t,x,y)=-v(t,x,y).

(23)

Our first goal is to show that

\psi(t,x,y)=\psi(t,2\lambda(t)-x,y).

(24)

To prove this, let us denote

\tau(t,x,y)=\psi(t,x,y)-\psi(t,2\lambda(t)-x,y).

(25)

Note that $\tau$ is $L$ -periodic in $x$ and vanishes on the flat bed $y=-d$ . Indeed, the first assertion is immediate. For the second one, from (23) we have

\psi_{x}(t,x,-d)=0,

which implies that $\psi$ is constant along the flat bed, and consequently $\tau=0$ on $y=-d$ . Simple computations yield

\Delta\psi(t,x,y)=u_{y}(t,x,y)-v_{x}(t,x,y)=\gamma(t,x,y),

and

	$\displaystyle\Delta\psi(t,2\lambda(t)-x,y)$	$\displaystyle=\psi_{xx}(t,2\lambda(t)-x,y)+\psi_{yy}(t,2\lambda(t)-x,y)$
		$\displaystyle=u_{y}(t,2\lambda(t)-x,y)-v_{x}(t,2\lambda(t)-x,y)$
		$\displaystyle=\gamma(t,2\lambda(t)-x,y).$

Thus, from the symmetry condition of the vorticity (22) and the definition of $\tau$ , it follows that $\tau$ is harmonic throughout the fluid domain. By Hopf’s maximum principle (see, e.g., [9, Section 6.4.2]), the maximum of $\tau$ must be attained either on the flat bed or on the free surface. We remark that the $L$ -periodicity of the flow in the horizontal direction excludes the possibility that the maximum occurs on the lateral boundaries. Indeed, if this were the case, one could reflect the domain periodically, leading to an interior maximum, which contradicts the maximum principle.

Assume now that $\tau\not\equiv 0$ . Since $\tau=0$ along the flat bed $y=-d$ , the maximum or the minimum must occur on the free surface $y=\beta(t,x)$ . Assuming the maximum is attained at $(x_{0},\beta(t,x_{0}))$ , one has

	$\displaystyle 0$	$\displaystyle=\frac{d}{dx}\tau(t,x,\beta(t,x))\|_{x=x_{0}}$
		$\displaystyle=\tau_{x}(t,x_{0},\beta(t,x_{0}))+\beta_{x}(t,x_{0})\tau_{y}(t,x_{0},\beta(t,x_{0}))$
		$\displaystyle=-v(t,x_{0},\beta(t,x_{0}))-v(t,2\lambda(t)-x_{0},\beta(t,x_{0}))$
		$\displaystyle\quad+\beta_{x}(t,x_{0})\left(u(t,x_{0},\beta(t,x_{0}))-u(t,2\lambda(t)-x_{0},\beta(t,x_{0}))\right).$

On the other hand, by the Hopf’s maximum principle, the derivative of $\tau$ in the outward normal direction at the free surface must be strictly positive, that is,

	$\displaystyle 0$	$\displaystyle<\left.\frac{\partial\tau}{\partial n}\right\|_{(x,y)=(x_{0},\beta(t,x_{0}))}$
		$\displaystyle=-\beta_{x}(t,x_{0})\tau_{x}(t,x_{0},\beta(t,x_{0}))+\tau_{y}(t,x_{0},\beta(t,x_{0}))$
		$\displaystyle=\beta_{x}(t,x_{0})\left(-v(t,x_{0},\beta(t,x_{0}))-v(t,2\lambda(t)-x_{0},\beta(t,x_{0}))\right)$
		$\displaystyle\quad+u(t,x_{0},\beta(t,x_{0}))-u(t,2\lambda(t)-x_{0},\beta(t,x_{0})).$

From (20), we deduce that

-v(t,x_{0},\beta(t,x_{0}))=v(t,2\lambda(t)-x_{0},\beta(t,x_{0})),

(26)

which, together with (20), implies

\tau_{x}(t,x_{0},\beta(t,x_{0}))=\tau_{y}(t,x_{0},\beta(t,x_{0}))=0.

Hence,

\left.\frac{\partial\tau}{\partial n}\right|_{(x,y)=(x_{0},\beta(t,x_{0}))}=0,

(27)

which contradicts the conclusion of Hopf’s maximum principle. If the minimum is attained on the free surface, similar arguments show that

0>\left.\frac{\partial\tau}{\partial n}\right|_{(x,y)=(x_{0},\beta(t,x_{0}))},

where $(x_{0},\beta(t,x_{0}))$ is the point of the minimum. This also leads to a contradiction with (27). Consequently, relation (24) holds. Differentiating this with respect to $x$ and $y$ , we obtain that

u(t,x,y)=u(t,2\lambda(t)-x,y),

(28)

and

\displaystyle v(t,x,y)=-v(t,2\lambda(t)-x,y).

(29)

Our final aim is to show the symmetry of the pressure. Letting

\Psi(t,x,y)=P(t,x,y)-P(t,2\lambda(t)-x,y),

we observe that $\Psi$ is harmonic and vanishes on the free surface, i.e.,

\Delta\Psi=0\quad\text{ and }\quad\Psi=0\text{ on }y=\beta(t,x).

Indeed, from the symmetry of the free surface we observe that the mapping

(t,x,y)\mapsto\frac{\beta_{xx}}{1+(\beta_{x})^{2}},

is also symmetric, thus the second assertion follows. For the first one, from the governing equations (1a) and (1b), we have

	$\displaystyle-\Delta P$	$\displaystyle=\partial_{x}(u_{t}+uu_{x}+vu_{y})+\partial_{y}(v_{t}+uv_{x}+vv_{y})$
		$\displaystyle=u_{x}^{2}+u_{y}^{2}+2v_{x}u_{y}.$

Next, relation (19) yields

	$\displaystyle 2v_{x}u_{y}$	$\displaystyle=v_{x}u_{y}+u_{y}v_{x}$
		$\displaystyle=v_{x}(v_{x}+\gamma)+u_{y}(u_{y}-\gamma)$
		$\displaystyle=v_{x}^{2}+u_{y}^{2}+\gamma(v_{x}-u_{y})$
		$\displaystyle=v_{x}^{2}+u_{y}^{2}-\gamma^{2},$

hence,

-\Delta P=u_{x}^{2}+u_{y}^{2}+v_{x}^{2}+u_{y}^{2}-\gamma^{2}.

Finally, the symmetry of $\gamma$ , together with relations (28) and (29), implies that $\Delta\Psi=0$ throughout the fluid.

Next, we show that $\Psi$ vanishes on the flat bed. Assuming the contrary, by Hopf’s principle, the normal derivative of $\Psi$ on $y=-d$ must be nonzero. However, the derivative of $\Psi$ in the direction of the outward normal is given by

\nabla\Psi\cdot(0,1)=\partial_{y}\Psi(t,x,-d)=P_{y}(t,x,-d)-P_{y}(t,2\lambda(t)-x,-d).

Since $v=0$ on $y=-d$ (relation (1d)), from (1b) one has $P_{y}=-g$ on $y=-d$ , whence,

\nabla\Psi\cdot(0,1)=P_{y}(t,x,-d)-P_{y}(t,2\lambda(t)-x,-d)=0,

which is a contradiction. Consequently $\Psi$ must vanish on the flat bed.

Given that $\Psi$ is harmonic, and since both its maximum and minimum values must be attained either on the flat bed or on the free surface where $\Psi=0$ (as established above), we deduce that $\Psi$ vanishes everywhere. Hence, $P$ is $x$ -symmetric.

∎

5 Evolution in time of the maximal horizontal velocity $u$

Using the results established in Section 4, we now show that, in the case of nonegative constant vorticity, the maximum of the horizontal velocity component $u$ remains constant as long as the flow retains its symmetry.

Throughout this section, we assume that the vorticity $\gamma$ is a nonnegative constant ( $\gamma\geq 0$ ) and that $(u,v,P,\beta)$ are $L$ -periodic ( $L>0$ ) in the $x$ -variable. We remark that the $L$ -periodicity in the $x$ component yields that the fluid domain $\Omega(t)$ can be taken as

\Omega(t)=\left\{(x,y)\in\mathbb{R}^{2}:x\in[0,L],\;y\in[-d,\beta(t,x)]\right\}.

For each $t\in[0,M]$ , we define

M(t):=\max_{(x,y)\in\Omega(t)}u(t,x,y).

As shown in [4, 14], the maximum $M(t)$ is attained at the free surface, i.e.,

M(t)=\max_{x\in[0,L]}u(t,s,\beta(t,x)).

Indeed, since $(x,y)\mapsto u(t,x,y)$ is harmonic, and $u$ is nonconstant, the maximum is attained either on the free surface or on the flat bed. If $M(t)$ were attained on the flat bed, Hopf’s maximum principle would give $u_{y}(t,x,-d)<0$ . On the other hand, because $v_{x}=0$ on $y=-d$ , by the definition of $\gamma$ we obtain

0\leq\gamma=u_{y}-v_{x}=u_{y}<0,

a contradiction. Hence $M(t)$ must be attained on the free surface.

One easily sees that the above arguments are independent of surface tension, so the conclusion remains valid in this case as well. We also note that, within our framework, the following result from [4, Section 3] (see also [14]), which relates the time derivative of $M(t)$ to the pressure gradient, also holds.

Theorem 3.

The function $t\mapsto M(t)$ is absolutely continuous and hence differentiable almost everywhere. Moreover, for almost every $t\in(0,T)$ , one has

M^{\prime}(t)=-P_{x}(t,x(t),y(t)),

(30)

where $(x,y)$ is any point at which $u(t,x,y)$ attains its maximum.

In the next result, we show that the mapping $t\mapsto M(t)$ remains constant as long as the free surface is symmetric and can be represented as a graph. A similar result, under more general assumptions—where the free surface is given in parametric form and admits overhanging profiles—was established in [4].

Theorem 4.

Assume that $u$ and $\beta$ are $x$ -symmetric, i.e., (20) and (21) hold. Then the maximal horizontal fluid velocity remains constant on the time interval of existence (the mapping $M(t)$ is constant on $t\in[0,T]$ ).

Proof.

Since the vorticity is constant, clearly relation (22) holds. Thus, we may apply Theorem 2 to conclude that $(u,v,P,\beta)$ defines a travelling wave. From this, the first relation of Euler’s equations (1a) can be rewritten as

\displaystyle-P_{x}=(u-c)u_{x}+vu_{y}\quad\text{ on }\quad y=\beta(t,x).

(31)

Using (31) and the surface kinematic condition in the case of travelling waves

v=(u-c)\beta_{x}\quad\text{ on }\quad y=\beta(t,x),

and arguing as in [5, pg. 86], we obtain

\displaystyle-P_{x}

\displaystyle=(u-c)\left(u_{x}+\beta_{x}u_{y}\right)=0\quad\text{ on }\quad y=\beta(t,x),

where the last equality holds because $(x,\beta(t,x))$ is a point of maximum for the function $x\mapsto u(t,x,\beta(t,x))$ , i.e.,

\displaystyle 0

\displaystyle=\frac{d}{dx}u(t,x,\beta(t,x))=u_{x}(t,x,\beta(t,x))+\beta_{x}(t,x),u_{y}(t,x,\beta(t,x)).

Consequently, from Theorem 3, we see that $M^{\prime}(t)=0$ . Since $M$ is absolutely continuous, by integration, we conclude that it remains constant. ∎

Remark 1.

In the absence of surface tension, it follows immediately that the maximum of the horizontal fluid velocity is attained at a wave crest or at a trough. Indeed, since $P_{x}=0$ , the condition (1) implies that $P_{y}\neq 0$ . However, differentiating the pressure condition on the free surface (1f) yields

P_{x}+P_{y}\beta_{x}=0,

which can hold only if $\beta_{x}=0$ ; that is, the point lies at a wave crest or at a trough.

6 Discussions

The relevance of the results lies in the simple observation that certain assumptions—such as symmetry—imply specific wave behaviors. Using this finding, we show that the maximum of the horizontal velocity $M(t)$ remains constant. An immediate consequence is that any variation in $M(t)$ (as occurs at the onset of wave breaking) necessarily requires a loss of symmetry. We mention that a similar conclusion was obtained in [4, Section 4] and [14, Proposition 4.2] in the absence of surface tension, though by a different method. Also, in this paper, the free surface has a simpler representation (a graph), whereas in [4, 14] the free surface is given in a general parametric form suitable for describing overhanging profiles.

The inclusion of surface tension prevents us, within the current framework, from rigorously determining the location on the free surface where the horizontal velocity attains its maximum. Although it is expected that this maximum still occurs at a wave crest (as in the case of zero surface tension), our analysis does not confirm this, leading to the following open question: In the setting of our analysis, is the maximum horizontal velocity attained at a crest when the surface tension coefficient is nonzero?

The location of $M(t)$ in case of constant negative vorticity ( $\gamma<0$ ) still remains an open problem, although advances in this direction can be found in the recent paper by C. Martin [15]. We remark that since $u$ is harmonic, $M(t)$ must be attained either on the free surface or on the flat bed. However, it is natural to expect that $M(t)$ cannot be attained on the bottom, as the horizontal velocity is expected to decrease with respect to the depth.

Acknowledgements

The author acknowledges the support provided by the project ”Nonlinear Studies of Stratified Oceanic and Atmospheric Flows”, funded by the European Union through the Next Generation EU initiative and the Romanian Government under the National Recovery and Resilience Plan for Romania. The project is contracted under number 760040/23.05.2023, cod PNRR-C9-I8-CF 185/22.11.2022, through the Romanian Ministry of Research, Innovation, and Digitalization, within Component 9, Investment I8. Also, the author would like to thank Prof. Calin Martin and Radu Precup for valuable discussions and suggestions.

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Symmetric Water Waves with Surface Tension: Traveling Wave Behavior and Maximal Horizontal Velocity

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Symmetric water waves with surface tension: traveling wave behavior and maximal horizontal velocity

Abstract

keywords:

pacs:

1 Introduction

2 The governing equations

3 Symmetric water waves with surface tension are traveling waves

Theorem 1.

Proof.

4 Symmetric waves in horizontal component and free surface with surface tension are traveling waves

Theorem 2.

Proof.

5 Evolution in time of the maximal horizontal velocity $u$

Theorem 3.

Theorem 4.

Proof.

Remark 1.

6 Discussions

Acknowledgements

References

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Symmetric Water Waves with Surface Tension: Traveling Wave Behavior and Maximal Horizontal Velocity

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

Symmetric water waves with surface tension: traveling wave behavior and maximal horizontal velocity

Abstract

keywords:

pacs:

1 Introduction

2 The governing equations

3 Symmetric water waves with surface tension are traveling waves

Theorem 1.

Proof.

4 Symmetric waves in horizontal component and free surface with surface tension are traveling waves

Theorem 2.

Proof.

5 Evolution in time of the maximal horizontal velocity uu

Theorem 3.

Theorem 4.

Proof.

Remark 1.

6 Discussions

Acknowledgements

References

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5 Evolution in time of the maximal horizontal velocity $u$