# Equilibrium States with Finite Amplitudes at Exactly and Nearly Class-I Bragg Resonances

Authors: Dali Xu; Jingyu Zhang; Hongsheng Zhang
Journal: Journal of Mathematics
 (2021)

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## Abstract

The exactly and nearly class-I Bragg resonances of strongly nonlinear waves are studied analytically by the homotopy analysis method. Two types of equilibrium states with time-independent wave spectra and different energy distributions are obtained. Effects of the incident wave height, the seabed height, and the frequency detuning on resonant waves are investigated. Bifurcation points of the equilibrium states are found and tend to greater value of relatively incident wave height for a steeper wave. The wave steepness of the whole wave system grows linearly with the seabed height. Meanwhile, the resonant peak can shift to up or down side when the near resonance is considered. This work provides us a deeper understanding on class-I Bragg resonance and enlightens further studies of higher-order wave-bottom interactions.


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## Body

## 1. Introduction

When an incident wave propagates over a monochromatically undulated and fixed seabed, a new resonant wave will be generated if the following resonance conditions are fulfilled:(1)k2=k1±kb,(2)ω2=ω1+δω,where k1 and k2 represent the vector wavenumbers of the incident and reflected (i.e., resonant) waves, respectively, kb is the vector wavenumber of the undulated seabed, And ωii=1,2 is the angular frequency related with vector wavenumber ki by the linear dispersion relation:(3)Dωi;ki=ki≡ωi−gkitanhkid=0,where d denotes the mean water depth And δω is the frequency detuning measuring how far is the near resonance δω≠0 away from the exact resonance δω=0. This resonance is called class-I Bragg resonance [1].The class-I Bragg resonance was first studied by a flume experiment [2]. The experiment verified the condition of class-I Bragg resonance proposed by Davies [3]. Multiscale perturbation method was applied to derive the analytical solution of Bragg resonance and obtained the formula of reflected and transmission coefficient at any point in the far area [4].When an evolutionary wave system was considered, namely, that wave amplitudes are assumed to be functions of time and space; periodically, energy exchanges were found between incident and reflected waves through the envelope equation for weakly nonlinear waves and small seabed undulations [4]. Mei first proposed the application of artificial sand bars to coastal protection [5]. Then, the research on Bragg resonance turned to applications. Based on the modified mild slope equation (MMSE), the Frobenius series solution and Taylor series solution were constructed, which were used to derive the Bragg resonance of the surface gravity wave on the bottom of the single-period sine and the Bragg resonance of the surface wave on the finite trapezoidal block array [6, 7]. The class-I Bragg resonance mechanism was applied to the shore protection [8]. When the incident wave obliquely propagated over a patch of corrugations which had a plane of symmetry in the incident direction, waves were deflected to the downstream and left a protected wake near shore.In addition, equilibrium states of waves at class-I Bragg resonance with constant spectra also exist [9]. The wave amplitudes and angular frequencies in such wave systems are time independent. And, bifurcations in energy distributions of such waves at class-I Bragg resonance were found using the homotopy analysis method (HAM). It should be emphasized that these equilibrium states are only for the exact resonance δω=0. Meanwhile, these waves have relatively small amplitudes. The correspondingly angular frequencies of the incident and resonant waves are also quite small. However, nonlinearity is an essential characteristic of waves, and the wave-bottom interaction of nonlinear waves is a focus of research. Previously, the studies of Bragg resonance mainly investigate linearized waves or weakly nonlinear waves (Mei et al. [4, 5], Kirby [10], and Liu and Yue [1]). Research studies on strongly nonlinear waves are infrequently studied.In this paper, we apply the amplitude-based homotopy analysis method [11] on the class-I Bragg resonant wave system with relatively larger nonlinearity. Both exact and near resonances are studied. Effects of the obliquely incident angle, relatively incident wave height, seabed height, and frequency detuning are discussed in detail.This paper is divided into the following parts. Section2 gives the mathematical description of problem and the solving procedures using HAM. Some related details of derivations are shown in the Appendix. Results including the effects of above physical parameters on the equilibrium states of resonant waves are presented and the energy distributions, nonlinear angular frequencies, and wave steepness of such waves are investigated in Section 3. Concluding remarks and discussions come in Section 4.


## 2. Problem Formulation

The fluid is assumed to be invisicid and the flow irrotational. Thus, the wave field can be described by the velocity potentiald, where t is the time, x,y,z are the Cartesian coordinates, and z-axis points upwards from the mean water level z=0. The surface gravity wave with its elevation of ζx,y,t propagates over seabed bars at z=−d+hx,y with d as the mean water depth. The seabed corrugation hx,y=bcoskb⋅r is monochromatic and has finite slope bkb, where kb=kb and r=x,y.The velocity potentialφx,y,z,t satisfies Laplace equation,(4)φxx+φyy+φzz=0,in the domain of −d+hx,y<z<ζx,y,t and subjects to the boundary conditions,(5)ζt+φxζx+φyζy=φz,(6)gζ+φt+12▽φ2=0,on the unknown free surface z=ζx,y,t and(7)φz=φxhx+φyhy,on the corrugated bottom z=−d+hx,y with no flux, where g is the gravitational acceleration, ▽=∂xi+∂yj+∂zk with i, j, and k as the unit vectors in x,y,z directions, respectively. Note that subjecting boundary condition (5) into (6) gives an alternative set of boundary conditions on the free surface as follows:(8)φtt+gφz+▽φ⋅▽φt+12▽φ⋅▽▽φ⋅▽φ=0,(9)ζ=−1gφt+12▽φ⋅▽φ.Thus, the boundary condition (8) is decoupled and only depends on the potential function φx,y,z,t. After introducing the dimensionless variables,(10)X,Y,Z,D=x,y,z,dk1,K1,Kb=k1,kbk1−1,T=tω1,Ω1,Δω=σ1,δωω1−1,ζ˜X,Y,T=ζx,y,ta1−1,φ˜X,Y,Z,T=a1−1ω1−1k1φx,y,z,t,h˜X,Y=hx,yb−1,and two phase functions(11)ξ1=k1⋅r−σ1t=Xcosα1+Ysinα1−Ω1T,ξ2=kb⋅r=KbXcosαb+KbYsinαb,the governing equation (4) and boundary conditions (7)–(9) can be nondimensionalized and rewritten in the new variables of ξ1 and ξ2 as follows:(12)▽^2ϕ=0,−ε1+ε3H<Z<ε2η,(13)Ω12ϕξ1ξ1+ϕZtanhε1−2ε2Ω1Fξ1+ε22▽^ϕ⋅▽^F=0,Z=ε2η,(14)η=Ω1tanhε1ϕξ1−ε2tanhε1F,Z=ε2η,(15)ϕZ=▽^ϕ⋅▽^−ε1+ε3H,Z=−ε1+ε3H,where(16)ε1=k1d,ε2=a1k1,ε3=bk1,ϕξ1,ξ2,Z=φ˜X,Y,Z,T,ηξ1,ξ2=ζ˜X,Y,T,Hξ2=h˜X,Y,▽^=cosα1∂ξ1+Kbcosαb∂ξ2i+sinα1∂ξ1+Kbsinαb∂ξ2j+∂ZkFξ1,ξ2,Z=ϕξ12+2Kbcosα1−αbϕξ1ϕξ2+Kb2ϕξ22+ϕZ22,where α1 and αb are the angles between k1 and kb and the x-axis, respectively, σ1 is the dimensional angular frequency in the nonlinear theory, and a1 is the dimensional amplitude of the incident wave.


### 2.1. HAM-Based Approach

The amplitude-based HAM approach is first used in the steady-state resonant waves where the wave amplitude is given and the frequency is to-be-determined [11]. Here, it is used to solve the class-I Bragg resonant waves with finite amplitudes.The first step is to establish the zeroth-order deformation equations. They are constructed by introducing the embedding parameterq∈0,1, which does not have any physical meaning at all, into the boundary conditions (13)–(14),(17)1−qℒ1qϕ^ξ1,ξ2,Z;q=qc0N1ϕ^ξ1,ξ2,Z;q,Ω^1q,(18)1−qη^ξ1,ξ2;q=qc0N2ϕ^ξ1,ξ2,Z;q,η^ξ1,ξ2;q,Ω^1q,on Z=ε2η^ξ1,ξ2;q and the boundary condition (15),(19)1−qℒ2ϕ^ξ1,ξ2,Z;q−ϕ0ξ1,ξ2,Z=qc0N3ϕ^ξ1,ξ2,Z;q,on Z=−ε1+qε3Hξ2, where c0≠0 is the convergence-control parameter and(20)ℒ1q=μu,vΩ^12q∂∂ξ12+1tanhε1∂∂Z,(21)ℒ2=∂∂Z,(22)N1ϕ^,Ω^1=Ω^12ϕ^ξ1ξ1+ϕ^Ztanhε1−2ε2Ω^1F^ξ1+ε22▽^ϕ^⋅▽^F^,(23)N2ϕ^,η^,Ω^1=η^−Ω^1tanhε1ϕ^ξ1+ε2tanhε1F^,(24)F^ξ1,ξ2,Z=ϕ^ξ12+2Kbcosα1−αbϕ^ξ1ϕ^ξ2+Kb2ϕ^ξ22+ϕ^Z22,(25)N3ϕ^=ϕ^Z−▽^ϕ^⋅▽^−ε1+ε3H.Here, the homotopy-Maclaurin series is defined as(26)ϕ^ξ1,ξ2,Z;q=ϕ0ξ1,ξ2,Z+∑n=1+∞ϕnξ1,ξ2,Zqn,(27)η^ξ1,ξ2;q=η0ξ1,ξ2+∑n=1+∞ηnξ1,ξ2qn,(28)Ω^1q=ϖ1,0+∑n=1+∞ϖ1,nqn,with the definitions of nth-order homotopy-derivative,(29)ϕnξ1,ξ2,Z=Dnϕ^ξ1,ξ2,Z;q,ηnξ1,ξ2=Dnη^ξ1,ξ2;q,ϖ1,n=DnΩ^q,and the homotopy-derivative operator,(30)Dnfq=1n!dnfqdqn|q=0.Thus, whenq varies continuously from 0 to 1, the above zeroth-order deformation equations lead to the continuous variation from equations(31)ℒ10ϕ^ξ1,ξ2,Z;0=0,(32)η^ξ1,ξ2;0=η0ξ1,ξ2=0,(33)ℒ2ϕ^ξ1,ξ2,Z;0−ϕ0ξ1,ξ2,Z=0,to(34)N1ϕ^ξ1,ξ2,Z;1,Ω^11=0,N2ϕ^ξ1,ξ2,Z;1,η^ξ1,ξ2;1,Ω^11=0,N3ϕ^ξ1,ξ2,Z;1=0,which suggests that, as long as the convergence-control parameter c0 is properly chosen, the continuous variations of ϕ^ξ1,ξ2,Z;q, η^ξ1,ξ2;q and Ω^1q occur from the initial guesses ϕ^ξ1,ξ2,Z;0=ϕ0ξ1,ξ2,Z, η^ξ1,ξ2;0=η0ξ1,ξ2=0 and Ω^10=ϖ1,0 to their exact solutions ϕ^ξ1,ξ2,Z;1=ϕξ1,ξ2,Z, η^ξ1,ξ2;1=ηξ1,ξ2, and Ω^11=Ω1, whose Mth-order approximations are(35)ϕξ1,ξ2,Z≈ϕ0ξ1,ξ2,Z+∑n=1Mϕnξ1,ξ2,Z,(36)ηξ1,ξ2≈∑n=1Mηnξ1,ξ2,(37)Ω1≈ϖ1,0+∑n=1Mϖ1,n.The aboventh-order homotopy derivatives can be obtained through the nth-order deformation equations,(38)ℒ¯1ϕn=−Un−Tn+χnSn−1+c0χnΔn−1ϕ,(39)ηn=ηn−1+c0Δn−1η,on Z=0 and(40)ℒ¯2ϕn=χn−1Vn−1+c0Δn−1b−V¯n,on Z=−ε1, which are determined by taking the nth-order homotopy-derivative on both sides of the zeroth-order deformation equations (17)–(20) or by substituting the Maclaurin series (26)–(28) into the zeroth-order deformation equations (17)–(20) and setting the coefficient of qn to be zero. The derivations of equations (38)–(40) are shown in detail in Appendix.


#### 2.1.1. Solution Expressions

The elevation and potential function of surface gravity waves governed by equations (12)–(15) can be expressed by(41)ηξ1,ξ2≈∑n=1Mηnξ1,ξ2=∑n=1M∑u=0+∞∑v=−∞+∞Anu,vcosuξ1+vξ2=∑u=0+∞∑v=−∞+∞Au,vcosuξ1+vξ2,(42)ϕξ1,ξ2,Z≈∑n=0Mϕnξ1,ξ2,Z=∑n=0M∑u=1+∞∑v=−∞+∞B1,nu,vPu,vξ1,ξ2,Z+B2,nu,vQu,vξ1,ξ2,Z=∑u=1+∞∑v=−∞+∞B1u,vPu,vξ1,ξ2,Z+B2u,vQu,vξ1,ξ2,Z,where(43)Pu,v=sinuξ1+vξ2coshKu,vZ+ε1coshKu,vε1,Qu,v=sinuξ1+vξ2sinhKu,vZKu,vcoshKu,vε1,Ku,v=u2+Kb2v2+2uvKbcosα1−αb,Au,v=∑n=0MAnu,v,B1u,v=∑n=0MB1,nu,v,B2u,v=∑n=0MB2,nu,v.Note thatAu,v, B1u,v, and B2u,v are all constants for waves with time-independent spectra. Except the amplitude of the incident wave A1,0=1 (i.e.,a1A1,0=a1), Au,v, B1u,v, and B2u,v are to be determined. We remark that the form of solution expressions (41)–(42) satisfies the governing equation (12) automatically. Thus, only boundary conditions (13)–(15) need to be solved to get the unknown variables Ω1, ηξ1,ξ2, and ϕξ1,ξ2,Z.


#### 2.1.2. Auxiliary Linear Operator

The auxiliary linear operator,(44)ℒ¯1=μu,vΩ1,02∂2∂ξ12+1tanhε1∂∂Z|Z=0,has the following property:(45)ℒ¯1Pu,v=λu,vsinuξ1+vξ2,where(46)λu,v=−u2μu,vΩ1,02+Ku,vtanhKu,vε1tanhε1.If we define(47)μu,v=Ku,vtanhKu,vε1u2tanhε1,u=u∗,v=v∗,1,others,where u∗ and v∗ satisfy the resonant conditions,(48)u∗k1−v∗kb=k2,u∗ω1=ω2+δωu∗,v∗,then we have(49)λu,v=0,u=u∗,v=v∗,−u2+Ku,vtanhKu,vε1u2tanhε1,others.Thus,(50)ℒ¯1P1,0ξ1,ξ2,Z≡0,ℒ¯1Pu∗,v∗ξ1,ξ2,Z=0.Meanwhile, the linear operatorℒ¯2 has the following property:(51)ℒ¯2Pu,vξ1,ξ2,Z=0,ℒ¯2Qu,vξ1,ξ2,Z=sinuξ1+vξ2.


#### 2.1.3. Thenth-Order Homotopy Variables

The initial guesses of the potential function and nonlinear frequency are(52)ϕ0ξ1,ξ2,Z=B1,01,0P1,0ξ1,ξ2,Z+B1,0u∗,v∗Pu∗,v∗ξ1,ξ2,Z,(53)Ω1,0=1.Whenn=1, substituting (52) into the right-hand side of (40), we have(54)ℒ2ϕ1ξ1,ξ2,Z=∑u∑vB2,1u,vsinuξ1+vξ2,where B2,1u,v is a function of B1,01,0 and B1,0u∗,v∗. Since the linear operator has the property of (50), the solution of (54) can be written as(55)ϕ1ξ1,ξ2,Z=B1,1u,vPu,vξ1,ξ2,Z+B2,1u,vQu,vξ1,ξ2,Z=ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v.Then, we substitute (52) into (39) and have(56)η1ξ1,ξ2=η1ξ1,ξ2;B1,01,0,B1,0u∗,v∗.Then, after substituting (55) into the left-hand side of (38) and (2)and (56) into the right-hand side of (38), we have(57)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1,where(58)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=∑u∑vB1,1u,v+B2,1u,vtanhε1coshKu,vε1sinuξ1+vξ2,ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1=∑u∑vCu,vsinuξ1+vξ2.Cu,v is a function of B1,01,0, B1,0u∗,v∗, and Ω1,1. Meanwhile, u,v≠1,0 and u,v≠u∗,v∗ in the coefficient of B1,1u,v.Balancing the coefficient of termsinuξ1+vξ2 on both sides of (57) gives a series of equations concerning B1,1u,v (except B1,11,0 and B1,1u∗,v∗), B1,01,0, B1,0u∗,v∗, and Ω1,1. An additional algebraic equation can be established from the boundary condition of A1,0=1. Thus, the problem is closed. Unknowns can be successfully determined. Till now, η1ξ1,ξ2, ϕ0ξ1,ξ2,Z, and Ω1,1 are obtained. And, those coefficients in (55) are all known except B1,11,0 and B1,1u∗,v∗, which will be determined when n=2.Whenn≥2, the same solving procedure is used order by order. And, finally, the Mth-order HAM approximations of (35)–(37) can be obtained.It should be mentioned that the convergent-control parameterc0 is determined when residual squares of equations (13)–(15) decrease. The optimal value of c0 corresponds to the fastest dropping of the residual squares. In this paper, the residual squares for the Mth-order HAM approximations are at least in order of 10−10.


## 2.1. HAM-Based Approach

The amplitude-based HAM approach is first used in the steady-state resonant waves where the wave amplitude is given and the frequency is to-be-determined [11]. Here, it is used to solve the class-I Bragg resonant waves with finite amplitudes.The first step is to establish the zeroth-order deformation equations. They are constructed by introducing the embedding parameterq∈0,1, which does not have any physical meaning at all, into the boundary conditions (13)–(14),(17)1−qℒ1qϕ^ξ1,ξ2,Z;q=qc0N1ϕ^ξ1,ξ2,Z;q,Ω^1q,(18)1−qη^ξ1,ξ2;q=qc0N2ϕ^ξ1,ξ2,Z;q,η^ξ1,ξ2;q,Ω^1q,on Z=ε2η^ξ1,ξ2;q and the boundary condition (15),(19)1−qℒ2ϕ^ξ1,ξ2,Z;q−ϕ0ξ1,ξ2,Z=qc0N3ϕ^ξ1,ξ2,Z;q,on Z=−ε1+qε3Hξ2, where c0≠0 is the convergence-control parameter and(20)ℒ1q=μu,vΩ^12q∂∂ξ12+1tanhε1∂∂Z,(21)ℒ2=∂∂Z,(22)N1ϕ^,Ω^1=Ω^12ϕ^ξ1ξ1+ϕ^Ztanhε1−2ε2Ω^1F^ξ1+ε22▽^ϕ^⋅▽^F^,(23)N2ϕ^,η^,Ω^1=η^−Ω^1tanhε1ϕ^ξ1+ε2tanhε1F^,(24)F^ξ1,ξ2,Z=ϕ^ξ12+2Kbcosα1−αbϕ^ξ1ϕ^ξ2+Kb2ϕ^ξ22+ϕ^Z22,(25)N3ϕ^=ϕ^Z−▽^ϕ^⋅▽^−ε1+ε3H.Here, the homotopy-Maclaurin series is defined as(26)ϕ^ξ1,ξ2,Z;q=ϕ0ξ1,ξ2,Z+∑n=1+∞ϕnξ1,ξ2,Zqn,(27)η^ξ1,ξ2;q=η0ξ1,ξ2+∑n=1+∞ηnξ1,ξ2qn,(28)Ω^1q=ϖ1,0+∑n=1+∞ϖ1,nqn,with the definitions of nth-order homotopy-derivative,(29)ϕnξ1,ξ2,Z=Dnϕ^ξ1,ξ2,Z;q,ηnξ1,ξ2=Dnη^ξ1,ξ2;q,ϖ1,n=DnΩ^q,and the homotopy-derivative operator,(30)Dnfq=1n!dnfqdqn|q=0.Thus, whenq varies continuously from 0 to 1, the above zeroth-order deformation equations lead to the continuous variation from equations(31)ℒ10ϕ^ξ1,ξ2,Z;0=0,(32)η^ξ1,ξ2;0=η0ξ1,ξ2=0,(33)ℒ2ϕ^ξ1,ξ2,Z;0−ϕ0ξ1,ξ2,Z=0,to(34)N1ϕ^ξ1,ξ2,Z;1,Ω^11=0,N2ϕ^ξ1,ξ2,Z;1,η^ξ1,ξ2;1,Ω^11=0,N3ϕ^ξ1,ξ2,Z;1=0,which suggests that, as long as the convergence-control parameter c0 is properly chosen, the continuous variations of ϕ^ξ1,ξ2,Z;q, η^ξ1,ξ2;q and Ω^1q occur from the initial guesses ϕ^ξ1,ξ2,Z;0=ϕ0ξ1,ξ2,Z, η^ξ1,ξ2;0=η0ξ1,ξ2=0 and Ω^10=ϖ1,0 to their exact solutions ϕ^ξ1,ξ2,Z;1=ϕξ1,ξ2,Z, η^ξ1,ξ2;1=ηξ1,ξ2, and Ω^11=Ω1, whose Mth-order approximations are(35)ϕξ1,ξ2,Z≈ϕ0ξ1,ξ2,Z+∑n=1Mϕnξ1,ξ2,Z,(36)ηξ1,ξ2≈∑n=1Mηnξ1,ξ2,(37)Ω1≈ϖ1,0+∑n=1Mϖ1,n.The aboventh-order homotopy derivatives can be obtained through the nth-order deformation equations,(38)ℒ¯1ϕn=−Un−Tn+χnSn−1+c0χnΔn−1ϕ,(39)ηn=ηn−1+c0Δn−1η,on Z=0 and(40)ℒ¯2ϕn=χn−1Vn−1+c0Δn−1b−V¯n,on Z=−ε1, which are determined by taking the nth-order homotopy-derivative on both sides of the zeroth-order deformation equations (17)–(20) or by substituting the Maclaurin series (26)–(28) into the zeroth-order deformation equations (17)–(20) and setting the coefficient of qn to be zero. The derivations of equations (38)–(40) are shown in detail in Appendix.


### 2.1.1. Solution Expressions

The elevation and potential function of surface gravity waves governed by equations (12)–(15) can be expressed by(41)ηξ1,ξ2≈∑n=1Mηnξ1,ξ2=∑n=1M∑u=0+∞∑v=−∞+∞Anu,vcosuξ1+vξ2=∑u=0+∞∑v=−∞+∞Au,vcosuξ1+vξ2,(42)ϕξ1,ξ2,Z≈∑n=0Mϕnξ1,ξ2,Z=∑n=0M∑u=1+∞∑v=−∞+∞B1,nu,vPu,vξ1,ξ2,Z+B2,nu,vQu,vξ1,ξ2,Z=∑u=1+∞∑v=−∞+∞B1u,vPu,vξ1,ξ2,Z+B2u,vQu,vξ1,ξ2,Z,where(43)Pu,v=sinuξ1+vξ2coshKu,vZ+ε1coshKu,vε1,Qu,v=sinuξ1+vξ2sinhKu,vZKu,vcoshKu,vε1,Ku,v=u2+Kb2v2+2uvKbcosα1−αb,Au,v=∑n=0MAnu,v,B1u,v=∑n=0MB1,nu,v,B2u,v=∑n=0MB2,nu,v.Note thatAu,v, B1u,v, and B2u,v are all constants for waves with time-independent spectra. Except the amplitude of the incident wave A1,0=1 (i.e.,a1A1,0=a1), Au,v, B1u,v, and B2u,v are to be determined. We remark that the form of solution expressions (41)–(42) satisfies the governing equation (12) automatically. Thus, only boundary conditions (13)–(15) need to be solved to get the unknown variables Ω1, ηξ1,ξ2, and ϕξ1,ξ2,Z.


### 2.1.2. Auxiliary Linear Operator

The auxiliary linear operator,(44)ℒ¯1=μu,vΩ1,02∂2∂ξ12+1tanhε1∂∂Z|Z=0,has the following property:(45)ℒ¯1Pu,v=λu,vsinuξ1+vξ2,where(46)λu,v=−u2μu,vΩ1,02+Ku,vtanhKu,vε1tanhε1.If we define(47)μu,v=Ku,vtanhKu,vε1u2tanhε1,u=u∗,v=v∗,1,others,where u∗ and v∗ satisfy the resonant conditions,(48)u∗k1−v∗kb=k2,u∗ω1=ω2+δωu∗,v∗,then we have(49)λu,v=0,u=u∗,v=v∗,−u2+Ku,vtanhKu,vε1u2tanhε1,others.Thus,(50)ℒ¯1P1,0ξ1,ξ2,Z≡0,ℒ¯1Pu∗,v∗ξ1,ξ2,Z=0.Meanwhile, the linear operatorℒ¯2 has the following property:(51)ℒ¯2Pu,vξ1,ξ2,Z=0,ℒ¯2Qu,vξ1,ξ2,Z=sinuξ1+vξ2.


### 2.1.3. Thenth-Order Homotopy Variables

The initial guesses of the potential function and nonlinear frequency are(52)ϕ0ξ1,ξ2,Z=B1,01,0P1,0ξ1,ξ2,Z+B1,0u∗,v∗Pu∗,v∗ξ1,ξ2,Z,(53)Ω1,0=1.Whenn=1, substituting (52) into the right-hand side of (40), we have(54)ℒ2ϕ1ξ1,ξ2,Z=∑u∑vB2,1u,vsinuξ1+vξ2,where B2,1u,v is a function of B1,01,0 and B1,0u∗,v∗. Since the linear operator has the property of (50), the solution of (54) can be written as(55)ϕ1ξ1,ξ2,Z=B1,1u,vPu,vξ1,ξ2,Z+B2,1u,vQu,vξ1,ξ2,Z=ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v.Then, we substitute (52) into (39) and have(56)η1ξ1,ξ2=η1ξ1,ξ2;B1,01,0,B1,0u∗,v∗.Then, after substituting (55) into the left-hand side of (38) and (2)and (56) into the right-hand side of (38), we have(57)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1,where(58)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=∑u∑vB1,1u,v+B2,1u,vtanhε1coshKu,vε1sinuξ1+vξ2,ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1=∑u∑vCu,vsinuξ1+vξ2.Cu,v is a function of B1,01,0, B1,0u∗,v∗, and Ω1,1. Meanwhile, u,v≠1,0 and u,v≠u∗,v∗ in the coefficient of B1,1u,v.Balancing the coefficient of termsinuξ1+vξ2 on both sides of (57) gives a series of equations concerning B1,1u,v (except B1,11,0 and B1,1u∗,v∗), B1,01,0, B1,0u∗,v∗, and Ω1,1. An additional algebraic equation can be established from the boundary condition of A1,0=1. Thus, the problem is closed. Unknowns can be successfully determined. Till now, η1ξ1,ξ2, ϕ0ξ1,ξ2,Z, and Ω1,1 are obtained. And, those coefficients in (55) are all known except B1,11,0 and B1,1u∗,v∗, which will be determined when n=2.Whenn≥2, the same solving procedure is used order by order. And, finally, the Mth-order HAM approximations of (35)–(37) can be obtained.It should be mentioned that the convergent-control parameterc0 is determined when residual squares of equations (13)–(15) decrease. The optimal value of c0 corresponds to the fastest dropping of the residual squares. In this paper, the residual squares for the Mth-order HAM approximations are at least in order of 10−10.


## 2.1.1. Solution Expressions

The elevation and potential function of surface gravity waves governed by equations (12)–(15) can be expressed by(41)ηξ1,ξ2≈∑n=1Mηnξ1,ξ2=∑n=1M∑u=0+∞∑v=−∞+∞Anu,vcosuξ1+vξ2=∑u=0+∞∑v=−∞+∞Au,vcosuξ1+vξ2,(42)ϕξ1,ξ2,Z≈∑n=0Mϕnξ1,ξ2,Z=∑n=0M∑u=1+∞∑v=−∞+∞B1,nu,vPu,vξ1,ξ2,Z+B2,nu,vQu,vξ1,ξ2,Z=∑u=1+∞∑v=−∞+∞B1u,vPu,vξ1,ξ2,Z+B2u,vQu,vξ1,ξ2,Z,where(43)Pu,v=sinuξ1+vξ2coshKu,vZ+ε1coshKu,vε1,Qu,v=sinuξ1+vξ2sinhKu,vZKu,vcoshKu,vε1,Ku,v=u2+Kb2v2+2uvKbcosα1−αb,Au,v=∑n=0MAnu,v,B1u,v=∑n=0MB1,nu,v,B2u,v=∑n=0MB2,nu,v.Note thatAu,v, B1u,v, and B2u,v are all constants for waves with time-independent spectra. Except the amplitude of the incident wave A1,0=1 (i.e.,a1A1,0=a1), Au,v, B1u,v, and B2u,v are to be determined. We remark that the form of solution expressions (41)–(42) satisfies the governing equation (12) automatically. Thus, only boundary conditions (13)–(15) need to be solved to get the unknown variables Ω1, ηξ1,ξ2, and ϕξ1,ξ2,Z.


## 2.1.2. Auxiliary Linear Operator

The auxiliary linear operator,(44)ℒ¯1=μu,vΩ1,02∂2∂ξ12+1tanhε1∂∂Z|Z=0,has the following property:(45)ℒ¯1Pu,v=λu,vsinuξ1+vξ2,where(46)λu,v=−u2μu,vΩ1,02+Ku,vtanhKu,vε1tanhε1.If we define(47)μu,v=Ku,vtanhKu,vε1u2tanhε1,u=u∗,v=v∗,1,others,where u∗ and v∗ satisfy the resonant conditions,(48)u∗k1−v∗kb=k2,u∗ω1=ω2+δωu∗,v∗,then we have(49)λu,v=0,u=u∗,v=v∗,−u2+Ku,vtanhKu,vε1u2tanhε1,others.Thus,(50)ℒ¯1P1,0ξ1,ξ2,Z≡0,ℒ¯1Pu∗,v∗ξ1,ξ2,Z=0.Meanwhile, the linear operatorℒ¯2 has the following property:(51)ℒ¯2Pu,vξ1,ξ2,Z=0,ℒ¯2Qu,vξ1,ξ2,Z=sinuξ1+vξ2.


## 2.1.3. Thenth-Order Homotopy Variables

The initial guesses of the potential function and nonlinear frequency are(52)ϕ0ξ1,ξ2,Z=B1,01,0P1,0ξ1,ξ2,Z+B1,0u∗,v∗Pu∗,v∗ξ1,ξ2,Z,(53)Ω1,0=1.Whenn=1, substituting (52) into the right-hand side of (40), we have(54)ℒ2ϕ1ξ1,ξ2,Z=∑u∑vB2,1u,vsinuξ1+vξ2,where B2,1u,v is a function of B1,01,0 and B1,0u∗,v∗. Since the linear operator has the property of (50), the solution of (54) can be written as(55)ϕ1ξ1,ξ2,Z=B1,1u,vPu,vξ1,ξ2,Z+B2,1u,vQu,vξ1,ξ2,Z=ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v.Then, we substitute (52) into (39) and have(56)η1ξ1,ξ2=η1ξ1,ξ2;B1,01,0,B1,0u∗,v∗.Then, after substituting (55) into the left-hand side of (38) and (2)and (56) into the right-hand side of (38), we have(57)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1,where(58)ℒ¯1ϕ1ξ1,ξ2,Z;B1,1u,v,B2,1u,v=∑u∑vB1,1u,v+B2,1u,vtanhε1coshKu,vε1sinuξ1+vξ2,ℛξ1,ξ2;B1,01,0,B1,0u∗,v∗,Ω1,1=∑u∑vCu,vsinuξ1+vξ2.Cu,v is a function of B1,01,0, B1,0u∗,v∗, and Ω1,1. Meanwhile, u,v≠1,0 and u,v≠u∗,v∗ in the coefficient of B1,1u,v.Balancing the coefficient of termsinuξ1+vξ2 on both sides of (57) gives a series of equations concerning B1,1u,v (except B1,11,0 and B1,1u∗,v∗), B1,01,0, B1,0u∗,v∗, and Ω1,1. An additional algebraic equation can be established from the boundary condition of A1,0=1. Thus, the problem is closed. Unknowns can be successfully determined. Till now, η1ξ1,ξ2, ϕ0ξ1,ξ2,Z, and Ω1,1 are obtained. And, those coefficients in (55) are all known except B1,11,0 and B1,1u∗,v∗, which will be determined when n=2.Whenn≥2, the same solving procedure is used order by order. And, finally, the Mth-order HAM approximations of (35)–(37) can be obtained.It should be mentioned that the convergent-control parameterc0 is determined when residual squares of equations (13)–(15) decrease. The optimal value of c0 corresponds to the fastest dropping of the residual squares. In this paper, the residual squares for the Mth-order HAM approximations are at least in order of 10−10.


## 3. Results

Here, we study the equilibrium states of waves at class-I Bragg resonance. From the point of view of energy distribution, waves can be classified into two types. For Type 1, the incident and resonant wave components have the same wave energy, and they together share nearly the whole wave energy. In other words, each of them has about half of the total wave energy. However, in Type 2, the incident and resonant components contain different wave energy. Effects of physical parameters on these two types of waves are studied in detail.


### 3.1. Oblique Incidence

We focus on the oblique incidence, namely, that the angleα=α1−αb between vector wavenumbers k1 and kb is nonzero, i.e., α≠0.Figure1 shows the energy distributions of incident a12/Π and resonant a22/Π components in both Type 1 and 2 when k1d = 2.5, bk1 = 0.05, and δω = 0, where Π is the sum of squares of all wave amplitudes. As α decreases, resonant wave energy gradually grows until it is equal to the incident part. Bifurcations are found between Type 1 and 2.Figure 1
Wave energy distribution (a12/Π and a22/Π) versus α of the class-I Bragg resonant wave system in the case of k1d = 2.5, bk1 = 0.05, and δω = 0. Solid line: a1k1 = 0.01; dashed line: a1k1 = 0.02; dotted line: a1k1 = 0.03; dash-dotted line: a1k1 = 0.04; dash-double-dotted line: a1k1 = 0.05; square: a1k1 = 0.06; circle: a1k1 = 0.07; triangle: a1k1 = 0.08; cross: a1k1 = 0.09; diamond: a1k1 = 0.1.For type 2, as the incident wave slopea1k1 increases, the values of α corresponding to the bifurcation points tend to a fixed one, i.e., α=63.7∘, as shown in Figure 1 and Table 1. Thus, this second type of equilibrium state of waves at class-I Bragg resonance exists in the range of 63.7∘<α<90∘. Without loss of generality, we choose α=3π/8=67.5∘ as the obliquely incident angle when the effects of other physical parameters on the class-I Bragg resonant wave system are studied.Table 1
Bifurcation points of incident angleα versus incident wave steepness a1k1 of the class-I Bragg exact resonant wave system in the case of k1d = 2.5, bk1 = 0.05, and δω = 0.
a1k1α (degree)0.0170.000.0265.200.0364.300.0463.950.0563.820.0663.750.0763.720.0863.710.0963.700.1063.70


### 3.2. Exact Resonance

According to the solving procedures shown in Section2, the resonant component in the wave elevation should contain the term of sinξ1−ξ2, where(59)ξ1−ξ2=k1−kb⋅r−σ1t=k2⋅r−σ2t.Thus, here in this paper, the nonlinear frequencies of incident and resonant waves are always equal to each other, i.e.,(60)σ1=σ2,so are the dimensionless nonlinear frequencies(61)σ1ω1=σ2ω1=σ2ω2,when the conditions of the class-I Bragg exact resonance (1), (2), and (60) with δω=0 are satisfied.


#### 3.2.1. Effect of Relatively Incident Wave Heighta1/d

Now, we consider the effect of relatively incident wave heighta1/d on exact resonant wave systems. As mentioned before, two types of equilibrium states can be found. We assume that the seabed is infinitely extended in space. Full interactions between sea and bottom result in the equilibrium state of Type 1, where the incident and resonant waves equally share the whole wave energy. Now, we investigate the effects of relatively incident wave height a1/d on the equilibrium state of Type 2.Whena1/d grows, less waves would be reflected by the seabed. An extreme case is that when a1/d tends to 0 or the mean water depth d tends to infinity, no reflection occurs since the surface waves cannot feel the seabed. We take the case of a1k1=0.15 in Figure 2(a) as an example. The energy of incident and resonant waves are represented by a12/Π and a22/Π, respectively. When a1/d grows, more wave energy flows to the resonant component, while the incident one has less energy. This situation ends at a1/d=0.083. And, after that, this second type of equilibrium state merges into the first type. The point of a1/d=0.083 is called the bifurcation point.Figure 2
Energy distributions (a12/Π and a22/Π), dimensionless frequencies of the incident σ1/ω1 and resonant wave components σ2/ω2, and wave steepness hs versus a1/d in the wave systems at class-I Bragg exact resonance when bk1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Dimensionless angular frequencies of Type 2. (c) Equivalent wave height of Type 2.
(a)(b)(c)Meanwhile, various values of the incident wave slopea1k1 are also presented in Figure 2(a). For those with smaller wave slopes a1k1, bifurcations occur at smaller a1/d. On the contrary, the equilibrium state of Type 2 with greater nonlinearity combines with Type 1 at shallower area with larger a1/d. Thus, as shown by Figure 2(a) and Table 2, the bifurcation points shift to greater a1/d when the incident wave becomes more and more steep.Table 2
Values of relative wave height at bifurcation points for incident waves with different slopes in the wave systems at class-I Bragg exact resonance whenbk1 = 0.15, α = 3π/8, and δω = 0.
a1k1a1d0.100.0450.110.0530.120.0600.130.0660.140.0740.150.083It is mentioned by Stiassnie [12] that a single wave tends to break when its wave steepness ak exceeds 0.2. However, for a wave system which is consisted by several wave components, the correspondingly wave steepness at break point has not been clearly given. From the nonlinearly angular frequency σi/ωii=1,2 and the wave steepness hs of the wave system shown by Figures 2(b) and 2(c), wave may break when a1k1=0.16 and 0.17. Accordingly, no bifurcation point is obtained in Figure 2(a) for these two cases. Here, the wave steepness hs of the wave system is defined same as Liu [13], i.e.,(62)hs=kdηmax−ηmin2,where kd is the wavenumber of the dominant wave. In addition, for a wave system with some fixed a1k1, it becomes steeper at greater a1/d according to the nonlinear angular frequencies and wave steepness, as depicted in Figures 2(b) and 2(c).


#### 3.2.2. Effect of Seabed Heightb/d

Figure3(a) shows the energy distributions as b/d changes when k1d=2.0, a1k1=0.15, α=3π/8, and δω=0. When b/d=0, the bottom is flat. There will be no resonant wave. As b/d grows, the resonant wave component has more and more energy, while the incident one gets less and less. And, finally, these two components share nearly the whole wave energy equally. The bifurcation point of these two types of wave systems is at about b/d=0.105.Figure 3
Wave energy distribution (a12/Π and a22/Π), dimensionless incident angular frequency σ1/ω1, and wave steepness hs versus b/d of the class-I Bragg exact resonant wave system in the case of k1d = 2.0, a1k1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Angular frequency and wave steepness in Type 2.
(a)(b)The corresponding angular frequenciesσi/ωii=1,2 and wave steepness hs are presented in Figure 3(b). As b/d grows, both the frequency and wave height have turning points. These turning points locate exactly at the above bifurcation point b/d=0.105.Whenb/d<0.105, the dimensionless frequencies σi/ωii=1,2 have approximately linear growth with the relative seabed height b/d and so is the equivalent wave height hs. When b/d>0.105, the wave height hs does not increase further. Meanwhile, the angular frequencies σi/ωi even decrease. Since hs>0.3 at b/d=0.125, wave breaking may also occur.


### 3.3. Near Resonance

Waves propagating over a flat seabed in various water depths with time-independent spectra have been studied by homotopy analysis method when the near resonance occurs [13–15]. In this section, we consider the linearly near resonance of class-I Bragg which satisfies conditions (1), (2), and (60) with δω≠0. Only the equilibrium-state waves of Type 2 are found.The peak of wave steepness at near resonance shifts from the exact resonanceδω=0, as shown by Figure 4. Both up and down shifts of the peaks are found. In other words, resonances occur in both domains of positive and negative frequency detunings. For example, the resonance occurs in the domain of δω<0 when a1k1=0.17, while in the domain of δω>0, when a1k1=0.19 and 0.20. Meanwhile, incident waves with different amplitudes are considered. From the cases of a1k1=0.14 to 0.20, it is clear that the peak shifts more away from δω=0 for a steeper incident wave. In addition, more than one local peaks are found when a1k1=0.16, 0.18, and 0.20. Moreover, the effects of positive and negative detunings on the wave system are not symmetric.Figure 4
Wave steepnesshs versus δω of the class-I Bragg nearly resonant wave system in the case of k1d = 2.0, bk1 = 0.15, and α = 3π/8.


## 3.1. Oblique Incidence

We focus on the oblique incidence, namely, that the angleα=α1−αb between vector wavenumbers k1 and kb is nonzero, i.e., α≠0.Figure1 shows the energy distributions of incident a12/Π and resonant a22/Π components in both Type 1 and 2 when k1d = 2.5, bk1 = 0.05, and δω = 0, where Π is the sum of squares of all wave amplitudes. As α decreases, resonant wave energy gradually grows until it is equal to the incident part. Bifurcations are found between Type 1 and 2.Figure 1
Wave energy distribution (a12/Π and a22/Π) versus α of the class-I Bragg resonant wave system in the case of k1d = 2.5, bk1 = 0.05, and δω = 0. Solid line: a1k1 = 0.01; dashed line: a1k1 = 0.02; dotted line: a1k1 = 0.03; dash-dotted line: a1k1 = 0.04; dash-double-dotted line: a1k1 = 0.05; square: a1k1 = 0.06; circle: a1k1 = 0.07; triangle: a1k1 = 0.08; cross: a1k1 = 0.09; diamond: a1k1 = 0.1.For type 2, as the incident wave slopea1k1 increases, the values of α corresponding to the bifurcation points tend to a fixed one, i.e., α=63.7∘, as shown in Figure 1 and Table 1. Thus, this second type of equilibrium state of waves at class-I Bragg resonance exists in the range of 63.7∘<α<90∘. Without loss of generality, we choose α=3π/8=67.5∘ as the obliquely incident angle when the effects of other physical parameters on the class-I Bragg resonant wave system are studied.Table 1
Bifurcation points of incident angleα versus incident wave steepness a1k1 of the class-I Bragg exact resonant wave system in the case of k1d = 2.5, bk1 = 0.05, and δω = 0.
a1k1α (degree)0.0170.000.0265.200.0364.300.0463.950.0563.820.0663.750.0763.720.0863.710.0963.700.1063.70


## 3.2. Exact Resonance

According to the solving procedures shown in Section2, the resonant component in the wave elevation should contain the term of sinξ1−ξ2, where(59)ξ1−ξ2=k1−kb⋅r−σ1t=k2⋅r−σ2t.Thus, here in this paper, the nonlinear frequencies of incident and resonant waves are always equal to each other, i.e.,(60)σ1=σ2,so are the dimensionless nonlinear frequencies(61)σ1ω1=σ2ω1=σ2ω2,when the conditions of the class-I Bragg exact resonance (1), (2), and (60) with δω=0 are satisfied.


### 3.2.1. Effect of Relatively Incident Wave Heighta1/d

Now, we consider the effect of relatively incident wave heighta1/d on exact resonant wave systems. As mentioned before, two types of equilibrium states can be found. We assume that the seabed is infinitely extended in space. Full interactions between sea and bottom result in the equilibrium state of Type 1, where the incident and resonant waves equally share the whole wave energy. Now, we investigate the effects of relatively incident wave height a1/d on the equilibrium state of Type 2.Whena1/d grows, less waves would be reflected by the seabed. An extreme case is that when a1/d tends to 0 or the mean water depth d tends to infinity, no reflection occurs since the surface waves cannot feel the seabed. We take the case of a1k1=0.15 in Figure 2(a) as an example. The energy of incident and resonant waves are represented by a12/Π and a22/Π, respectively. When a1/d grows, more wave energy flows to the resonant component, while the incident one has less energy. This situation ends at a1/d=0.083. And, after that, this second type of equilibrium state merges into the first type. The point of a1/d=0.083 is called the bifurcation point.Figure 2
Energy distributions (a12/Π and a22/Π), dimensionless frequencies of the incident σ1/ω1 and resonant wave components σ2/ω2, and wave steepness hs versus a1/d in the wave systems at class-I Bragg exact resonance when bk1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Dimensionless angular frequencies of Type 2. (c) Equivalent wave height of Type 2.
(a)(b)(c)Meanwhile, various values of the incident wave slopea1k1 are also presented in Figure 2(a). For those with smaller wave slopes a1k1, bifurcations occur at smaller a1/d. On the contrary, the equilibrium state of Type 2 with greater nonlinearity combines with Type 1 at shallower area with larger a1/d. Thus, as shown by Figure 2(a) and Table 2, the bifurcation points shift to greater a1/d when the incident wave becomes more and more steep.Table 2
Values of relative wave height at bifurcation points for incident waves with different slopes in the wave systems at class-I Bragg exact resonance whenbk1 = 0.15, α = 3π/8, and δω = 0.
a1k1a1d0.100.0450.110.0530.120.0600.130.0660.140.0740.150.083It is mentioned by Stiassnie [12] that a single wave tends to break when its wave steepness ak exceeds 0.2. However, for a wave system which is consisted by several wave components, the correspondingly wave steepness at break point has not been clearly given. From the nonlinearly angular frequency σi/ωii=1,2 and the wave steepness hs of the wave system shown by Figures 2(b) and 2(c), wave may break when a1k1=0.16 and 0.17. Accordingly, no bifurcation point is obtained in Figure 2(a) for these two cases. Here, the wave steepness hs of the wave system is defined same as Liu [13], i.e.,(62)hs=kdηmax−ηmin2,where kd is the wavenumber of the dominant wave. In addition, for a wave system with some fixed a1k1, it becomes steeper at greater a1/d according to the nonlinear angular frequencies and wave steepness, as depicted in Figures 2(b) and 2(c).


### 3.2.2. Effect of Seabed Heightb/d

Figure3(a) shows the energy distributions as b/d changes when k1d=2.0, a1k1=0.15, α=3π/8, and δω=0. When b/d=0, the bottom is flat. There will be no resonant wave. As b/d grows, the resonant wave component has more and more energy, while the incident one gets less and less. And, finally, these two components share nearly the whole wave energy equally. The bifurcation point of these two types of wave systems is at about b/d=0.105.Figure 3
Wave energy distribution (a12/Π and a22/Π), dimensionless incident angular frequency σ1/ω1, and wave steepness hs versus b/d of the class-I Bragg exact resonant wave system in the case of k1d = 2.0, a1k1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Angular frequency and wave steepness in Type 2.
(a)(b)The corresponding angular frequenciesσi/ωii=1,2 and wave steepness hs are presented in Figure 3(b). As b/d grows, both the frequency and wave height have turning points. These turning points locate exactly at the above bifurcation point b/d=0.105.Whenb/d<0.105, the dimensionless frequencies σi/ωii=1,2 have approximately linear growth with the relative seabed height b/d and so is the equivalent wave height hs. When b/d>0.105, the wave height hs does not increase further. Meanwhile, the angular frequencies σi/ωi even decrease. Since hs>0.3 at b/d=0.125, wave breaking may also occur.


## 3.2.1. Effect of Relatively Incident Wave Heighta1/d

Now, we consider the effect of relatively incident wave heighta1/d on exact resonant wave systems. As mentioned before, two types of equilibrium states can be found. We assume that the seabed is infinitely extended in space. Full interactions between sea and bottom result in the equilibrium state of Type 1, where the incident and resonant waves equally share the whole wave energy. Now, we investigate the effects of relatively incident wave height a1/d on the equilibrium state of Type 2.Whena1/d grows, less waves would be reflected by the seabed. An extreme case is that when a1/d tends to 0 or the mean water depth d tends to infinity, no reflection occurs since the surface waves cannot feel the seabed. We take the case of a1k1=0.15 in Figure 2(a) as an example. The energy of incident and resonant waves are represented by a12/Π and a22/Π, respectively. When a1/d grows, more wave energy flows to the resonant component, while the incident one has less energy. This situation ends at a1/d=0.083. And, after that, this second type of equilibrium state merges into the first type. The point of a1/d=0.083 is called the bifurcation point.Figure 2
Energy distributions (a12/Π and a22/Π), dimensionless frequencies of the incident σ1/ω1 and resonant wave components σ2/ω2, and wave steepness hs versus a1/d in the wave systems at class-I Bragg exact resonance when bk1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Dimensionless angular frequencies of Type 2. (c) Equivalent wave height of Type 2.
(a)(b)(c)Meanwhile, various values of the incident wave slopea1k1 are also presented in Figure 2(a). For those with smaller wave slopes a1k1, bifurcations occur at smaller a1/d. On the contrary, the equilibrium state of Type 2 with greater nonlinearity combines with Type 1 at shallower area with larger a1/d. Thus, as shown by Figure 2(a) and Table 2, the bifurcation points shift to greater a1/d when the incident wave becomes more and more steep.Table 2
Values of relative wave height at bifurcation points for incident waves with different slopes in the wave systems at class-I Bragg exact resonance whenbk1 = 0.15, α = 3π/8, and δω = 0.
a1k1a1d0.100.0450.110.0530.120.0600.130.0660.140.0740.150.083It is mentioned by Stiassnie [12] that a single wave tends to break when its wave steepness ak exceeds 0.2. However, for a wave system which is consisted by several wave components, the correspondingly wave steepness at break point has not been clearly given. From the nonlinearly angular frequency σi/ωii=1,2 and the wave steepness hs of the wave system shown by Figures 2(b) and 2(c), wave may break when a1k1=0.16 and 0.17. Accordingly, no bifurcation point is obtained in Figure 2(a) for these two cases. Here, the wave steepness hs of the wave system is defined same as Liu [13], i.e.,(62)hs=kdηmax−ηmin2,where kd is the wavenumber of the dominant wave. In addition, for a wave system with some fixed a1k1, it becomes steeper at greater a1/d according to the nonlinear angular frequencies and wave steepness, as depicted in Figures 2(b) and 2(c).


## 3.2.2. Effect of Seabed Heightb/d

Figure3(a) shows the energy distributions as b/d changes when k1d=2.0, a1k1=0.15, α=3π/8, and δω=0. When b/d=0, the bottom is flat. There will be no resonant wave. As b/d grows, the resonant wave component has more and more energy, while the incident one gets less and less. And, finally, these two components share nearly the whole wave energy equally. The bifurcation point of these two types of wave systems is at about b/d=0.105.Figure 3
Wave energy distribution (a12/Π and a22/Π), dimensionless incident angular frequency σ1/ω1, and wave steepness hs versus b/d of the class-I Bragg exact resonant wave system in the case of k1d = 2.0, a1k1 = 0.15, α = 3π/8, and δω = 0. (a) Wave energy distributions of Type 1 and 2. (b) Angular frequency and wave steepness in Type 2.
(a)(b)The corresponding angular frequenciesσi/ωii=1,2 and wave steepness hs are presented in Figure 3(b). As b/d grows, both the frequency and wave height have turning points. These turning points locate exactly at the above bifurcation point b/d=0.105.Whenb/d<0.105, the dimensionless frequencies σi/ωii=1,2 have approximately linear growth with the relative seabed height b/d and so is the equivalent wave height hs. When b/d>0.105, the wave height hs does not increase further. Meanwhile, the angular frequencies σi/ωi even decrease. Since hs>0.3 at b/d=0.125, wave breaking may also occur.


## 3.3. Near Resonance

Waves propagating over a flat seabed in various water depths with time-independent spectra have been studied by homotopy analysis method when the near resonance occurs [13–15]. In this section, we consider the linearly near resonance of class-I Bragg which satisfies conditions (1), (2), and (60) with δω≠0. Only the equilibrium-state waves of Type 2 are found.The peak of wave steepness at near resonance shifts from the exact resonanceδω=0, as shown by Figure 4. Both up and down shifts of the peaks are found. In other words, resonances occur in both domains of positive and negative frequency detunings. For example, the resonance occurs in the domain of δω<0 when a1k1=0.17, while in the domain of δω>0, when a1k1=0.19 and 0.20. Meanwhile, incident waves with different amplitudes are considered. From the cases of a1k1=0.14 to 0.20, it is clear that the peak shifts more away from δω=0 for a steeper incident wave. In addition, more than one local peaks are found when a1k1=0.16, 0.18, and 0.20. Moreover, the effects of positive and negative detunings on the wave system are not symmetric.Figure 4
Wave steepnesshs versus δω of the class-I Bragg nearly resonant wave system in the case of k1d = 2.0, bk1 = 0.15, and α = 3π/8.


## 4. Concluding Remarks and Discussions

We study the equilibrium states of waves at both exact and near Bragg resonances of the first kind. Using the amplitude-based homotopy analysis method, waves with time-independent spectra are considered. The waves are classified into two types. The incident and resonant waves have the same energy in Type 1, while in Type 2 these two wave components have different energy. In both types, the incident and resonant components share almost the whole wave energy. In addition, waves with strong nonlinearities are obtained. The wave steepnesshs of some resonant wave system is up to 0.3, which is believed to be large enough for a nonlinear wave.Effects of relatively incident wave heighta1/d and seabed height b/d on the equilibrium states of wave systems at exact resonance are studied when the incident wave with different wave slopes propagate over the seabed obliquely. Bifurcations of the two types of waves are found. The bifurcation point shifts to greater value of a1/d for a steeper incident wave. Meanwhile, the wave steepness hs grows linearly with the seabed height b/d.When linearly near resonance is considered, both up and down shifts of the resonant peak are found. And, more than one local peaks are obtained for some resonances, which implies that multiple resonances occur although the lower order of components are found to have tiny wave energy and the whole wave energy is still occupied by the incident and resonant component in the leading order.This work on class-I Bragg resonance can be extended to class II, III, and even higher-order resonances. Moreover, interactions of internal waves among multilayer fluids over undulated seabed can also be studied.



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