# On the Construction of Traveling Water Waves with Constant Vorticity and Infinite Boundary

Authors: Pawan Shrestha; Durga Jang KC; Ramjee Sharma
Journal: International Journal of Mathematics and Mathematical Sciences
 (2023)

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## Abstract

The issue of whether there is a closed orbit in the water waves in an infinite boundary condition is an outstanding open problem. In this work, we first discuss the various developments on the structure of water waves in the context of finite bottom conditions. We then focus on the behavior of water for the kinematic boundary for the infinite depth. We present some findings to address this issue by creating a water wave profile for the zero and constant vorticity conditions through the application of the Crandall–Rabinowitz theorem.


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## Body

## 1. Introduction

In the nineteenth century, Airy and Stokes started the mathematical study of wave theory within the framework of linear theory. Stokes water waves are nonlinear and periodic surface waves on an inviscid fluid layer of constant mean depth, which were introduced in the mid-nineteenth century. These waves are the progressive periodic waves of permanent form. These water waves, which propagate on the water surface of a sea or river, are periodic two-dimensional waves.Stokes noticed the actual water wave characteristics and then studied the nonlinear governing equations extensively for water waves [1]. The governing equations for the two-dimensional irrotational gravity water waves have the Hamiltonian structure [2–4]. The irrotational flow (zero vorticity) was considered in most of the theoretical works of surface water waves. In most of the studies, incompressible Euler equations with free boundary conditions are considered governing equations for water waves. The insight into the dynamics of waves of large amplitude is given in [5–8] with the help of a qualitative understanding of periodic traveling waves in an irrotational flow over the flatbed and also the aspects of propagation of irrotational waves of the variable bottom are discussed [9]. The authors in [10–12] dealt with the full nonlinearity of the governing equations for the water waves providing valuable quantitative information.Constantin and Villari proved that in linear periodic gravity water waves, there are no closed orbits for the water particles in the fluid and the paths are approximately elliptical, and they considered the flow up to the finite depth for the kinematic boundary condition [13].Ehrnstrom and Villari showed that for the positive vorticity, the situation resembles that of Stokes waves. For large enough vorticity, the particle trajectories are affected. Also, for the negative vorticity, there is the appearance of internal waves, and vortices and the trajectories are not closed ellipses [14]. The author in [15] presented a mathematical formulation of the water wave problem. He gave the existence result for a small amplitude solution based on the bifurcation theory. Also, he described the particle motion in the physical frame and obtained the fluid in a moving frame, and confirmed the prediction from the linear theory [14].The authors in [16] showed that the steady periodic deep-water waves are symmetric and propagate against a current with vorticity if their profile is monotone between crests and troughs. They discussed the traveling waves for water waves with zero and nonzero vorticity and showed that symmetry is valid, particularly for the irrotational waves. They also showed that vorticity distribution vanishes at the infinite depth. Constantin and Strauss [17] constructed the two-dimensional inviscid periodic traveling waves with vorticity with the consideration of a free surface under the effect of gravity over a flat bottom. These waves are symmetric and monotone between each crest and trough. They used the bifurcation theory and degree theory for the construction.The authors in [18] constructed the periodic traveling waves with vorticity, which are symmetric waves whose profiles are monotone between each crest and trough. For this, they used bifurcation theory and considered the water wave problem described by the Euler equation with a free surface over a flat bottom. Also, the authors in [19] proved that the profile of a periodic traveling wave propagating on the surface of the water above a flatbed in a flow with real analytic vorticity must be real analytic with the assumption that wave speed exceeds the horizontal fluid velocity throughout the flow.Recently with a degenerate diffusivity, Eyring–Powell viscosity term, and a Darcy– Forchheimer law in the porous medium, the authors in [20] obtained the traveling wave profile and showed the existence of the asymptotic solution using the geometric perturbation theory. They also showed the existence of an exponential profile of the solution under an asymptotic approximation. Similarly, the authors in [21] explored the solutions in the traveling wave domain with the use of geometric perturbation theory. Furthermore, the authors in [22] transformed the problem into the study of traveling wave-kind solutions. With the use of geometric perturbation theory, they confirmed the existence of an exponentially decaying rate in the traveling wave profile. Both the authors [21, 22] performed numerical simulations to validate their results.The authors in [23] explored the regularity, existence, and uniqueness of the solutions with the use of variational weak formulations. Then they transformed the Eyring –Powell equation into a traveling wave domain where they obtained the analytical solution with the use of geometric perturbation theory. Their main work was to show the existence of an exponential traveling wave tail together with a certain minimizing error critical speed.All of these existing results apply only to the finite boundary, but this problem remains open for the infinite boundary. With the ideas in [18, 19], and [17], and using the Crandall–Rabinowitz theorem, we created the water waves profile by considering the kinematic boundary condition up to infinity. We studied this as an extension of the flat bed at a finite depth to an infinite depth. Our findings will be useful for the study of the extension of the solutions from finite boundary conditions to infinite boundary conditions. Similar concepts can be used in other situations while dealing with problems in fluid dynamics with infinite boundaries.In Section2, we present the equation of motion with boundary conditions. In Section 3, the use of the Crandall–Rabinowitz theorem is proposed to guarantee the existence of traveling waves for water waves with zero and nonzero vorticity for the kinematic boundary condition up to the infinite depth. Section 4 concludes the paper.


## 2. Equation of Motion

The equation of motion of the wave is given by the following:(1)∂tu⟶+u⟶.∇u⟶+∇P=−ge⟶y,divu⟶=0,where u is the velocity, P is the pressure, and g is the acceleration due to gravity.The related boundary conditions are as follows:(1)
Kinematic: The free surface with moving fluid is given byηt=1+∇η2u⟶⋅n⟶ where n⟶ is the unit normal to the surface(2)
Dynamic: The dynamic boundary condition is the balance of forces at the free surface and is given byP=Patm, the atmospheric pressure(3)
Bottom: At the bottom, the second component of the fluid is zero. It just moves horizontally i.e.,(2)limy→−∞ey→.u→=0.With these, the equation of motion is given byΔϕ=0 in Dη, ϕ=ψ at y=ηt,x, and(3)limy→−∞∂yϕ=0.Here,ϕ is the velocity potential function, Dη is the domain of the fluid motion, ηt,x is the free surface which describes a wave on the bottom of the fluid dependent on time t and ψt,x is the value of the potential at the free surface.


## 3. Traveling Waves for Water Waves

In this section, the water waves for the two cases with zero vorticity and constant vorticity are discussed.


### 3.1. Traveling Waves for Water Waves with Zero Vorticity

The water waves formulation [24, 25] is given by the following:(4)ηt=Gηψ,ψt=−gη−ψx22+12Gηψ+ηxψx21+ηx2,where Gηψ is a Dirichlet–Neumann operator [26, 27] and is given by(5)Gηψ=1+ηx2∂n⟶ϕ|y=ηx.Here, we are looking for the small size traveling wave periodic inx. That means we expect the solution of the formηt,x=η⌣x+ct and ψt,x=ψ⌣x+ct, where c is the phase velocity. These solutions must satisfy the equations as follows:(6)cηx=Gηψ,cψx=−gη−ψx22+12Gηψ+ηxψx21+ηx2.For this, we look for the zeros of the equation, which indicate the situation with zero vorticity(7)Fc,η,ψ=−cηx+Gηψcψx+gη+ψx22−12Gηψ+ηxψx21+ηx2.Whenη=0,ψ=0, (7) takes the form Fc,0,0=0,∀c, which is a bifurcation problem. To solve this type of bifurcation problem, the Crandall-Rabinowiz theorem [28] is applied. For this, we must have the following:(1)
dη,ψFc∗,0,0 has the 1-dimensional kernel(2)
Imdη,ψFc∗,0,0 is closed and has codimension 1(3)
It must satisfy the traversability condition i.e.,∂c,η,ψFc∗,0,0η∗,ψ∗∉R where η∗,ψ∗∈Kerdη,ψFc∗,0,0 where Ker d is the kernel of d and R is the image set.By linearizing (7) at η=0,ψ=0, we have(8)dη,ψFc,0,0η^,ψ^=−cη^x+G0ψ^cψ^x+gη^.We assume the series representation forη^ and ψ^ as(9)η^x=∑n≥0ηncosnx,ψ^x=∑n≥0ψnsinnx.Which are even and odd, respectively, together with the norms(10)ηs2=∑n2sηn2,ψs2=∑n2sψn2,which are the norms defined in the homogeneous Sobolev space Hs.With these series representations, we have(11)dη,ψFc,0,0η^,ψ^=∑n≥1cnηn+nψnsinnxcnψn+gηncosnx+0gη0.We define(12)f=∑n≥0fncosnx,g=∑n≥0gnsinnx.For thesegandf; we solve, dη,ψFc,0,0η^,ψ^=gfForn ≥ 1, we have(13)cnngcnηnψn=gnfn.We note that detMnc=0 iff c=±g/n, where Mn=cnngcn. For fixed n0 and choosing cn0=g/n0, the n0-matrix has kernel. Now for cn0, we have(14)cn0n0n0gcn0n0ηn0ψn0=00⇔ηn0ψn0=cn0−gn0.Here, we note that ifn0=0, then the inverse of that matrix cn0n0n0gcn0n0 does not exist. So, the matrix becomes singular, and hence the solution is not possible.Therefore,Kerdη,ψFcn0,0,0=cn0cosn0x−g/n0sinn0xwhich is of dimension 1.Again(15)0−gη0=0f0,cnngcnηnψn=gnfn;n≥1.This implies two possibilities,n=n0 or n≠n0. In both cases, it can be shown that η,ψ∈Hs. Again for n=n0, choosing gn0=1, we have(16)R=Imdη,ψFcn0,0,0=∑n≠n0gnsinnxfncosnx+sinn0xcn0cosn0x.But, this is perpendicular toR⊥=cn0sinn0x−cosn0x. This shows that the range is closed and has codimension 1. Also, the traversability condition is satisfied as(17)∂c,η,ψFcn0,0,0η∗,ψ∗=−cn0n0sinn0x−gcosn0x∉R.As all the conditions for the Crandall–Rabinowitz theorem [28] are satisfied, we have the following theorem:Theorem 1.
Lets>5/2 and n0∈N:∃ϵ0>0 and C1 functions Cϵ:−ϵ0,ϵ0⟶R; η⌣ϵψ⌣ϵ−ϵ0,ϵ0⟶Hs×Hs such that η⌣x−ctψ⌣x−ct solves the water wave equation. Moreover, Cϵ=Cn0+0ϵ, η⌣ϵxψ⌣ϵx=ϵcn0cosn0x−g/n0sinn0x+Oϵ2.Thus, we have constructed traveling waves of the form(18)ηt,x=ϵcn0cosn0x−cϵt+Oϵ2.


### 3.2. Traveling Waves for Water Waves with Constant Vorticity

The water waves formulation [24, 25] is given by(19)ηt=Gηψ+γηηx,ψt=−gη−ψx22+ηxψx+Gηψ221+ηx2+γηψx+γ∂x−1Gηψ,where Gηψ is a Dirichlet–Neumann operator [26, 27].Here, we are looking for the small size traveling waves periodic inx. That means we expect the solution of the form ηt,x=η⌣x+ct and ψt,x=ψ⌣x+ct, where c is the phase velocity.These solutions must satisfy the equations:(20)cηx=Gηψ+γηηx,cψx=−gη−ψx22+12Gηψ+ηxψx21+ηx2+γηψx+γ∂x−1Gηψ.For this, we look for the zeros of the equation, which indicate the situation with constant vorticity(21)Fc,η,ψ=−cηx+Gηψ+γηηxcψx+gη+ψx22−12Gηψ+ηxψx21+ηx2−γηψx−γ∂x−1Gηψ.Whenη=0,ψ=0, (21) takes the form Fc,0,0=0;∀c, which is a bifurcation problem. Here also, the Crandall–Rabinowitz theorem [28] is applied to solve this type of bifurcation problem. For this, we must have the following:(1)
dη,ψFc∗,0,0 has 1 dimensional kernel(2)
Imdη,ψFc∗,0,0 is closed and has codimension 1(3)
It must satisfy the traversability condition i.e.∂c,η,ψFc∗,0,0η∗,ψ∗∉R where η∗,ψ∗∈Kerdη,ψFc∗,0,0 where Ker d is the kernel of d and R is image set.By linearizing equation (21) at η=0,ψ=0, we have(22)dη,ψFc,0,0η^,ψ^=−cη^x+G0ψ^cψ^x+gη^−γ∂x−1Dψ.We assume the series representation forη^ and ψ^ as(23)η^x=∑n≥0ηncosnx,ψ^x=∑n≥0ψnsinnx,which are even and odd, respectively, together with the norms(24)ηs2=∑n2sηn2,ψs2=∑n2sψn2.These are the norms defined on homogeneous Sobolev spaceHs.With these series representation, we have(25)dη,ψFc,0,0η^,ψ^=∑n≥1cnηn+nψnsinnxcn+γψn+gηncosnx+0cn0+γψ0+gη0.Assuming(26)g=∑n≥0gnsinnx,f=∑n≥0fncosnx.For theseg and f; we solve dη,ψFc,0,0η^,ψ^=gf.Then, we havecnngcn+γηnψn=gnfn.We note that detMnc=0 if c=−γ/2n±γ2/4n2+g/n, where Mn=cnngcn+γ. For fixed n0 and choosing cn0=−γ/2n0±γ2/4n02+g/n0, the n0-matrix has kernel.Now,(27)cn0n0n0gcn0n0+γηn0ψn0=00⇔ηn0ψn0=cn0+γn0−gn0.Therefore,Kerdη,ψFcn0,0,0=cn0+γ/n0cosn0x−g/n0sinn0x,which is of dimension 1.Again(28)0−gη0=0f0,cnngcn+γηnψn=gnfn;n≥1.This implies two possibilitiesn=n0 or n≠n0. In both the cases, it can be shown that η,ψ∈Hs. Again for n=n0 and choosing gn0=1, we have,(29)R=Imdη,ψFcn0,0,0=∑n≠n0gnsinnxfncosnx+sinn0xcn0+γn0cosn0x,which is perpendicular toR⊥=−cn0+γ/n0sinn0xcosn0x.This shows that the range is closed and has codimension 1. Also the traversability condition is satisfied as(30)∂c,η,ψFcn0,0,0η∗,ψ∗=−cn0+γ/n0n0sinn0x−gcosn0x∉R.With the help of the Crandall–Rabinowitz theorem [28], we have obtained the following theorem:Theorem 2.
Lets>5/2 and n0∈N:∃ϵ0>0 and C1 functions Cϵ:−ϵ0,ϵ0⟶R ; η⌣ϵψ⌣ϵ−ϵ0,ϵ0⟶Hs×Hs such that η⌣x−ctψ⌣x−ct solves the water wave equation.Moreover,Cϵ=Cn0+0ϵ,η⌣ϵxψ⌣ϵx=ϵcn0+γ/n0cosn0x−g/n0sinn0x+Oϵ2.Thus, we have constructed traveling waves of the form(31)ηt,x=ϵcn0+γn0cosn0x−cϵt+Oϵ2.Here, we have created the water profile for the zero and constant vorticity cases. We use Newtonian fluid. A similar result may be useful for the non-Newtonian fluid. Also, a similar result may be extended to the positive or negative, or variable viscosity cases.


## 3.1. Traveling Waves for Water Waves with Zero Vorticity

The water waves formulation [24, 25] is given by the following:(4)ηt=Gηψ,ψt=−gη−ψx22+12Gηψ+ηxψx21+ηx2,where Gηψ is a Dirichlet–Neumann operator [26, 27] and is given by(5)Gηψ=1+ηx2∂n⟶ϕ|y=ηx.Here, we are looking for the small size traveling wave periodic inx. That means we expect the solution of the formηt,x=η⌣x+ct and ψt,x=ψ⌣x+ct, where c is the phase velocity. These solutions must satisfy the equations as follows:(6)cηx=Gηψ,cψx=−gη−ψx22+12Gηψ+ηxψx21+ηx2.For this, we look for the zeros of the equation, which indicate the situation with zero vorticity(7)Fc,η,ψ=−cηx+Gηψcψx+gη+ψx22−12Gηψ+ηxψx21+ηx2.Whenη=0,ψ=0, (7) takes the form Fc,0,0=0,∀c, which is a bifurcation problem. To solve this type of bifurcation problem, the Crandall-Rabinowiz theorem [28] is applied. For this, we must have the following:(1)
dη,ψFc∗,0,0 has the 1-dimensional kernel(2)
Imdη,ψFc∗,0,0 is closed and has codimension 1(3)
It must satisfy the traversability condition i.e.,∂c,η,ψFc∗,0,0η∗,ψ∗∉R where η∗,ψ∗∈Kerdη,ψFc∗,0,0 where Ker d is the kernel of d and R is the image set.By linearizing (7) at η=0,ψ=0, we have(8)dη,ψFc,0,0η^,ψ^=−cη^x+G0ψ^cψ^x+gη^.We assume the series representation forη^ and ψ^ as(9)η^x=∑n≥0ηncosnx,ψ^x=∑n≥0ψnsinnx.Which are even and odd, respectively, together with the norms(10)ηs2=∑n2sηn2,ψs2=∑n2sψn2,which are the norms defined in the homogeneous Sobolev space Hs.With these series representations, we have(11)dη,ψFc,0,0η^,ψ^=∑n≥1cnηn+nψnsinnxcnψn+gηncosnx+0gη0.We define(12)f=∑n≥0fncosnx,g=∑n≥0gnsinnx.For thesegandf; we solve, dη,ψFc,0,0η^,ψ^=gfForn ≥ 1, we have(13)cnngcnηnψn=gnfn.We note that detMnc=0 iff c=±g/n, where Mn=cnngcn. For fixed n0 and choosing cn0=g/n0, the n0-matrix has kernel. Now for cn0, we have(14)cn0n0n0gcn0n0ηn0ψn0=00⇔ηn0ψn0=cn0−gn0.Here, we note that ifn0=0, then the inverse of that matrix cn0n0n0gcn0n0 does not exist. So, the matrix becomes singular, and hence the solution is not possible.Therefore,Kerdη,ψFcn0,0,0=cn0cosn0x−g/n0sinn0xwhich is of dimension 1.Again(15)0−gη0=0f0,cnngcnηnψn=gnfn;n≥1.This implies two possibilities,n=n0 or n≠n0. In both cases, it can be shown that η,ψ∈Hs. Again for n=n0, choosing gn0=1, we have(16)R=Imdη,ψFcn0,0,0=∑n≠n0gnsinnxfncosnx+sinn0xcn0cosn0x.But, this is perpendicular toR⊥=cn0sinn0x−cosn0x. This shows that the range is closed and has codimension 1. Also, the traversability condition is satisfied as(17)∂c,η,ψFcn0,0,0η∗,ψ∗=−cn0n0sinn0x−gcosn0x∉R.As all the conditions for the Crandall–Rabinowitz theorem [28] are satisfied, we have the following theorem:Theorem 1.
Lets>5/2 and n0∈N:∃ϵ0>0 and C1 functions Cϵ:−ϵ0,ϵ0⟶R; η⌣ϵψ⌣ϵ−ϵ0,ϵ0⟶Hs×Hs such that η⌣x−ctψ⌣x−ct solves the water wave equation. Moreover, Cϵ=Cn0+0ϵ, η⌣ϵxψ⌣ϵx=ϵcn0cosn0x−g/n0sinn0x+Oϵ2.Thus, we have constructed traveling waves of the form(18)ηt,x=ϵcn0cosn0x−cϵt+Oϵ2.


## 3.2. Traveling Waves for Water Waves with Constant Vorticity

The water waves formulation [24, 25] is given by(19)ηt=Gηψ+γηηx,ψt=−gη−ψx22+ηxψx+Gηψ221+ηx2+γηψx+γ∂x−1Gηψ,where Gηψ is a Dirichlet–Neumann operator [26, 27].Here, we are looking for the small size traveling waves periodic inx. That means we expect the solution of the form ηt,x=η⌣x+ct and ψt,x=ψ⌣x+ct, where c is the phase velocity.These solutions must satisfy the equations:(20)cηx=Gηψ+γηηx,cψx=−gη−ψx22+12Gηψ+ηxψx21+ηx2+γηψx+γ∂x−1Gηψ.For this, we look for the zeros of the equation, which indicate the situation with constant vorticity(21)Fc,η,ψ=−cηx+Gηψ+γηηxcψx+gη+ψx22−12Gηψ+ηxψx21+ηx2−γηψx−γ∂x−1Gηψ.Whenη=0,ψ=0, (21) takes the form Fc,0,0=0;∀c, which is a bifurcation problem. Here also, the Crandall–Rabinowitz theorem [28] is applied to solve this type of bifurcation problem. For this, we must have the following:(1)
dη,ψFc∗,0,0 has 1 dimensional kernel(2)
Imdη,ψFc∗,0,0 is closed and has codimension 1(3)
It must satisfy the traversability condition i.e.∂c,η,ψFc∗,0,0η∗,ψ∗∉R where η∗,ψ∗∈Kerdη,ψFc∗,0,0 where Ker d is the kernel of d and R is image set.By linearizing equation (21) at η=0,ψ=0, we have(22)dη,ψFc,0,0η^,ψ^=−cη^x+G0ψ^cψ^x+gη^−γ∂x−1Dψ.We assume the series representation forη^ and ψ^ as(23)η^x=∑n≥0ηncosnx,ψ^x=∑n≥0ψnsinnx,which are even and odd, respectively, together with the norms(24)ηs2=∑n2sηn2,ψs2=∑n2sψn2.These are the norms defined on homogeneous Sobolev spaceHs.With these series representation, we have(25)dη,ψFc,0,0η^,ψ^=∑n≥1cnηn+nψnsinnxcn+γψn+gηncosnx+0cn0+γψ0+gη0.Assuming(26)g=∑n≥0gnsinnx,f=∑n≥0fncosnx.For theseg and f; we solve dη,ψFc,0,0η^,ψ^=gf.Then, we havecnngcn+γηnψn=gnfn.We note that detMnc=0 if c=−γ/2n±γ2/4n2+g/n, where Mn=cnngcn+γ. For fixed n0 and choosing cn0=−γ/2n0±γ2/4n02+g/n0, the n0-matrix has kernel.Now,(27)cn0n0n0gcn0n0+γηn0ψn0=00⇔ηn0ψn0=cn0+γn0−gn0.Therefore,Kerdη,ψFcn0,0,0=cn0+γ/n0cosn0x−g/n0sinn0x,which is of dimension 1.Again(28)0−gη0=0f0,cnngcn+γηnψn=gnfn;n≥1.This implies two possibilitiesn=n0 or n≠n0. In both the cases, it can be shown that η,ψ∈Hs. Again for n=n0 and choosing gn0=1, we have,(29)R=Imdη,ψFcn0,0,0=∑n≠n0gnsinnxfncosnx+sinn0xcn0+γn0cosn0x,which is perpendicular toR⊥=−cn0+γ/n0sinn0xcosn0x.This shows that the range is closed and has codimension 1. Also the traversability condition is satisfied as(30)∂c,η,ψFcn0,0,0η∗,ψ∗=−cn0+γ/n0n0sinn0x−gcosn0x∉R.With the help of the Crandall–Rabinowitz theorem [28], we have obtained the following theorem:Theorem 2.
Lets>5/2 and n0∈N:∃ϵ0>0 and C1 functions Cϵ:−ϵ0,ϵ0⟶R ; η⌣ϵψ⌣ϵ−ϵ0,ϵ0⟶Hs×Hs such that η⌣x−ctψ⌣x−ct solves the water wave equation.Moreover,Cϵ=Cn0+0ϵ,η⌣ϵxψ⌣ϵx=ϵcn0+γ/n0cosn0x−g/n0sinn0x+Oϵ2.Thus, we have constructed traveling waves of the form(31)ηt,x=ϵcn0+γn0cosn0x−cϵt+Oϵ2.Here, we have created the water profile for the zero and constant vorticity cases. We use Newtonian fluid. A similar result may be useful for the non-Newtonian fluid. Also, a similar result may be extended to the positive or negative, or variable viscosity cases.


## 4. Conclusion

In this work, we studied the various development of the water waves and related facts such as the wave profile and path of the wave in the finite bottom conditions. We have then extended the finite depth condition to the infinite depth condition. In infinite depth conditions, we have constructed the water wave profile for the zero and constant vorticity conditions with the use of the Crandall–Rabinowitz theorem by using an analytical approach. While we focused on the zero and constant vorticity cases, similar work may be established on the positive and negative vorticity cases for the construction of wave profiles. Here we did our work for the Newtonian fluid. This work may be extended for the non-Newtonian fluid.



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