A study on nonlinear steady-state waves at resonance in water of finite depth by the amplitude-based homotopy analysis method ＊

Da-li Xu, Zeng Liu

1. College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, China

2. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Abstract: Nonlinear steady-state waves are obtained by the amplitude-based homotopy analysis method (AHAM) when resonances among surface gravity waves are considered in water of finite depth. AHAM, newly proposed in this paper within the context of homotopy analysis method (HAM) and well validated in various ways, is able to deal with nonlinear wave interactions. In waves with small propagation angles, it is confirmed that more components share the wave energy if the wave field has a greater steepness.However, in waves with larger propagation angles, it is newly found that wave energy may also concentrate in some specific components. In such wave fields, off-resonance detuning is also considered. Bifurcation and symmetrical properties are discovered in some wave fields. Our results may provide a deeper understanding on nonlinear wave interactions at resonance in water of finite depth.

Key words: Nonlinear wave, wave resonance, steady state, homotopy analysis method (HAM)

Introduction

Resonant interactions among surface gravity waves are one of the physical processes governing the evolution of ocean wave spectra. It has been studied a lot since 1960, pioneered by Phillips’ classic work[1].The resonance conditions of gravity waves with two primary components are:

where the angular frequencyωand vector wavenumberkfulfill the linear dispersion relationandddenotes water depth,uandvare integers. Exact （Δωu,v=0)and near(Δωu,v≠0) resonances are represented by the frequency detuning Δωu,v.

Note that the quartet resonance of surface gravity waves is the lowest-order resonance in deep water and water of finite depth[1]. It is natural and necessary to start from the quartet resonance when we study the evolution of ocean wave spectra. Particularly, mainly for the sake of simplicity, the quartet resonance in the degenerate case (i.e., triads where one wave is counted twice) has been studied a lot since last century. In deep water, the resonant component was found to grow linearly with time in the initial stage[1-3]and then exchange wave energy with the primary ones in relatively large time domains. These waves are unsteady, namely that they have time evolutionary wave spectra[4].

In addition to those resonant waves with time evolutionary spectra, steady-state waves with timeindependent spectra were found as well by means of HAM[5]for both exact and near resonances in deep water and water of finite depth[6-8]. HAM can successfully avoid singularities that are mathematically caused by resonances[9-10]. Meanwhile, such steady-state resonant waves were also verified by experiment observations in a wave tank[11]. When waves propagated over undulated bottoms, steadystate waves were found as well when the Bragg resonance condition of the first kind was fulfilled[12].Note that all of these waves have weak nonlinearity.The wave slope (ak) of each component is quite small and aboutak≈ 0 .01. Almost all the wave energy is distributed among the primary and resonant components.

Recently, Liu et al.[13]considered nonlinear steady-state waves at multiple resonances in deep water. They extended weakly nonlinear waves[14]to those with finite amplitudes. In addition, as the wave amplitude grew, wave energy was found to distribute among more components. But to the best of our knowledge, highly nonlinear waves at the lowest order of resonance (quartet resonance) in water of finite depth which play an important role in the energy distribution and spectra evolution of ocean waves have not been investigated yet.

In this paper, by means of the amplitude-based homotopy analysis method (AHAM) that is newly proposed and well validated, we extend the weakly nonlinear steady-state waves at quartet resonance[7-8]to the nonlinear steady-state waves with finite amplitudes. We also extend nonlinear steady-state waves at multiple resonances[13]in deep water to those in water of finite depth.

Firstly, the way of energy distribution that found by Liu et al.[13]in deep water is confirmed in water of finite depth. Specifically, when the propagation angle between the primary components is small, wave energy tends to distribute among more components of the waves with larger amplitudes.

Secondly, for steady-state waves at quartet resonance, it is found that most wave energy can also concentrate in the quartet even when the wave field has great steepness, as long as the propagation angle is relatively large.

Thirdly, exact resonance was found to be only a special case of near resonance for weakly nonlinear waves in deep water[8]. This is also confirmed for waves with great amplitudes in water of finite depth.Waves vary continuously when the frequency detuning changes around exactly zero. In addition,bifurcations are found in the obtained wave fields.Meanwhile, symmetrical behavior of the resonant component with respect to the frequency detuning is also discovered.

1. The amplitude-based homotopy analysis method

In this section, the mathematical formulae related to AHAM are presented in detail for resonant wave interactions with great nonlinearity. The physical model is nondimensionalized firstly and solving procedures by means of AHAM come later.

1.1 Governing equations

The fluid is assumed to be inviscid and the flow irrotational. Thus the velocity field can be described by the potential functionφ(x,y,z,t), wheretis the time, (x,y,z) are the rectangular coordinates andzaxis points upwards from the mean water level(z=0). In the whole fluid domain -d＜z＜ζ(x,y,t)whereζ(x,y,t) is the free surface, the velocity potentialφ(x,y,z,t) satisfies the Laplace equation

i,jandkare unit vectors in thex,yandzdirections, respectively. Meanwhile, the velocity potentialφ(x,y,z,t) subjects to the boundary conditions on the unknown free surfacez=ζ(x,y,t):

and boundary condition on the horizontal bottom with no fluxz=-d

wheregis the gravitational acceleration. We now consider the steady-state waves at resonance containing two primary components with timeindependent wave amplitudeai, wavenumberki=and angular frequencyσi(i= 1 ,2).Note thatiσis the angular frequency in nonlinear theory and does not equal to the angular frequencyiωin linear theory.

By introducing the following dimensionless variables:

1.2 Solving procedures of AHAM

HAM is an analytic technique for highly nonlinear problems. The traditional perturbation method has been proved to be only a special case of HAM[15].Here we investigate nonlinear gravity waves at resonance based on HAM. It should be emphasized that the angular frequency of each component is unknown and to be determined while the amplitudes of primary components are given. Based on this assumption, the corresponding approach in the HAM context (i.e.,the amplitude-based HAM) is proposed below.

1.2.1Solution expressions

Now we are going to get the solutions of Eqs.(15)-(18) using AHAM when the gravity waves satisfy the resonance conditions (1)-(2). The steadystate solutions of the unknown wave elevationand velocity potentialΦ(ξ1,ξ2,Z) can be expressed by:

Au,vandBu,vare constants which should be determined, excluding the given amplitudes of the primary components

a1anda2are the dimensional wave amplitudes of the two primary components in the elevationζ(x,y,t). The governing equation (15) and the bottom boundary condition (18) are satisfied automatically by the form of Eq. (23). Thus, in order to get the unknown wave elevationη(ξ1,ξ2) and velocity potentialin the form of Eqs.(22) and (23) as well as the unknown wave frequencies, nonlinear boundary conditions (16) and(17) on the unknown free surface need to be solved.Note that the unknown frequencies include1Ωand 2Ωfor the primary components, and also include the frequenciesΩu,v=uΩ1+vΩ2for other wave components.

1.2.2 Continuous variations

1.2.3The auxiliary linear operator and initial guess of the velocity potential

2. Validations of AHAM

In this section, in order to test and build some confidence in AHAM, some validations are performed.Satisfactory agreements are found between AHAM and other theories. First of all, waves at near resonance are compared with the ones obtained by the third-order perturbation theory in finite water depth.Then, nonlinear waves at resonances in deep water are investigated to perform the validation of AHAM by Zakharov equation and by the fully nonlinear wave equations that are solved by means of FHAM. Note that the non-uniqueness of the finite-depth water-wave problem leads to some differences between the fully nonlinear wave Eqs. (3)-(7) and Zakharov equation[7].In addition, till now nonlinear waves at multiple resonances has been only obtained in deep water by FHAM[13], which is an approach based on HAM while angular frequencies of the primary components are provided in advance. Thus, deep water case is investigated to test AHAM by numerically solving Zakharov equation and by nonlinear wave equations solved using FHAM.

2.1 Validation by the perturbation theory

Fig. 1 Residual squares of AHAM approximations when dα=α2 -α1 = 30°, ε1=k1 d=1.5and a 2 k2=0.05

The dimensionless angular frequencies of the primary components, as well as the resonant one(the subscript 3 is used to denote the physical parameters of the resonant wave component in the degenerate quartet hereinafter) are illustrated by Fig. 2 as the wave slopea1k1changes.The angular frequencies from both theories agree quite well at small wave slopea1k1. Meanwhile,AHAM can also give the higher-order corrections of the primary waves’ frequencies asa1k1increases. In addition, the angular frequency of the resonant component can be also presented explicitly by AHAM.

Fig. 2 Dimensionless angular frequencies of the primary and resonant componentsσ iωi- 1(i=1,2,3)whenα2-α1 = 30°,k1 d =1.5, a2 k 2 =0.0 5 and δ ω2,- 1 = 0.008.“PT” and “AHAM” denote the results obtained by the third-order perturbation theory and the 10th-order AHAM approximation, respectively

2.2 Validation by Zakharov equation

In this section, we consider the degenerate quartet (u=2,v=-1 in conditions (1) and (2) in exact resonance when

For this case, three different wave systems are found by AHAM and validated by solving Zakharov equation. Details of the solving procedures of Zakharov equation can be found in the work of Xu et al.[7].

The dimensionless angular frequencies of the primary and resonant componentσi/ωi(i=1,2,3), the resonant wave slopea3k3and wave steepness of the whole wave systemhsare investigated and presented in Table 1, wherehsis defined as follows[13]

2.3 Validation by FHAM

The so-called FHAM is the method within HAM context when the angular frequencies of the primary components are provided in advance as parameters. In this section, we use FHAM to validate the waves obtained by AHAM. Here we consider the case when

Table 1 Information (resonant wave slope a3 k3, wave steepness hs and dimensionless angular frequencies σ i / ωi) of three wave systems (denoted by Waves A, B and C, respectively) found from Eqs. (15 )-(18) by AH AM (symbolized by “AHAM”) and validated by numerically solving Zakharov equation (symbolized by “Z-eq”).Here k1 d= ∞ ,a1 k 1 = a 2 k 2 =0.03, α1=0, α2=30° and δ ω2,-1=0

Table 2 The obtained wave slopes of primary and resonant components a i ki ( i = 1,2 ,3) by FHAM, that are compared with the counterparts provided by AHAM when k1 d=∞, α1=0, α2=30°

which are partially same as case (56). Note that the angular frequenciesσ1/ω1andσ2/ω2(Eqs.(59)-(61)) are provided in frequency-based HAM approach. Meanwhile the wave slopes of primary(a1k1,a2k2) and resonant (a3k3) components obtained by frequency-based HAM approach are used to compare with the counterparts in AHAM, as listed in Table 2.

Excellent agreements are found between results from AHAM and FHAM for waves B and C respectively, which proves AHAM’s validation once again for highly nonlinear problems. Meanwhile,waves D and E got by FHAM are missed by AHAM while Wave A obtained by AHAM are missed as well by FHAM. The relation between the waves A-E obtained by AHAM and FHAM is illustrated by Fig. 3,where two sets of waves (sets 1, 2) show the mutually complementary property of these two approaches and they together make the physical problem understood deeply.

Fig. 3 The resonant waves in degenerated case obtained by AHAM and FHAM (Set 1: Waves A, B and C from AHAM, Set 2: Waves B, C, D and E from FHAM)

It should be emphasized that there are some advantages of AHAM. Firstly, as presented in Fig. 3,AHAM plays a complementary role to FHAM. The union of sets 1, 2 in Fig. 3 expand the scope of the solutions. This helps us see more aspects of the nonlinear wave interaction. Secondly, usually when wave interactions are studied by theories like perturbation method[9]and high-order spectral method[17], by numerical calculations from Zakharov equation[18], or by doing experiment in the wave tank[11], the wave amplitudes of primary components are known. This is consistent with AHAM. Thus, it is more convenient if we do the comparisons between results from AHAM and other approaches. Thirdly, as described in Section 1.2, since AHAM is an analytical approach, the angular frequencies can be written in forms of the primary wave amplitudes. If needed,dispersive relations of the nonlinear waves (although very complicated in form) can be given in different orders. This is quite similar to the traditional perturbation theory. However, the difference is that AHAM is valid for highly nonlinear waves while the traditional perturbation theory can only deal with the weakly nonlinear waves.

3. Nonlinear steady-state waves in exact/near resonances in water of finite depth

In this section, steady-state waves at resonances with finite wave amplitudes are obtained by AHAM in water of finite depth. The energy distribution and frequency detuning in these wave fields are investigated.

3.1 Energy distributions in nonlinear waves at multiple resonances

Recently, Liuet al.[13]obtained steady-state waves with multiple resonances in deep water using FHAM. As the angular frequencies of primary components grew, more wave components would share the whole wave energy. In these waves,nonlinearity was represented by the primary components’ frequencies, which were provided in advanced in FHAM.

In this paper, using AHAM illustrated in the previous section, we first extend these nonlinear steady-state waves at multiple resonances in deep water[13]to the ones in water of finite depth. Note that the nonlinearity is now represented by the wave steepnesshs, as defined by Eq. (57).

It is confirmed that in water of finite depth the wave energy would distribute into more wave components ashsgrows. In other words, the energy distribution that found by Liu et al.[13]in deep water is still correct in water of finite depth. We demonstrate this energy distribution by considering the following case:

where multiple resonances are considered and fulfill the following conditions:

namely thatδω2,-1=0,δω3,-2= -2.9× 1 0-4andδω4,-3= - 1.1× 1 0-3. Seven nonlinear steady-state wave systems are found for this case. Their discrete energy spectra are plotted in Fig. 4, whereσi,jrepresents the nonlinear angular frequency of the wave component whose vector wavenumber iski,j.The data corresponding to Fig. 4 are listed in detail in Table 3. In the wave with steepness ofhs=0.05,which is marked by stars in the figure, almost 99%wave energy is distributed in the degenerate quartet.For other steady-state waves with larger wave steepness, more wave components share the energy.This confirms Liu et al.’s[13]conclusion in deep water for multiple resonances.

3.2 Quartet resonance in the degenerate case

In this section, weakly nonlinear steady-state waves at quartet resonance in water of finite depth[7]are extended to waves with great nonlinearity. The greatest steepness reacheshs=0.26, which is thought to have large enough nonlinearity[4,13,16]. In addition, it is found that wave energy can still concentrate in the resonant quartet and do not distribute intoother components even for nonlinear waves with great steepness, as long as the propagation angle between primary componentsdα=α2-α1is relatively large.This kind of energy distribution is similar with the one in weakly nonlinear waves[6-7].

Table 3 Energy distribution of the component with vector wavenumber k in waves at multiple resonances i, j when-1.1× 1 0-3, where. Only those components whose energy is greater than 0.1% are presented.denotes the energy of the degenerate quartet where

Table 3 Energy distribution of the component with vector wavenumber k in waves at multiple resonances i, j when-1.1× 1 0-3, where. Only those components whose energy is greater than 0.1% are presented.denotes the energy of the degenerate quartet where

h (1,0) (0,1) (2,-1) (1,-2) (3,-1) (3,-2) (2,-3) (4,-2) (4,-3) (3,-4) (5,-3) (5,-4) (4,-5) ∑Q s 0.050 46.62 41.23 10.82 - - 1.06 - - - - - - - 98.67 0.066 36.84 41.14 1.83 2.18 - 14.32 2.39 - 0.95 - - 0.16 - 79.81 0.067 32.75 36.57 12.9 15.60 - 1.77 0.12 - - - - - - 82.22 0.075 33.30 29.45 10.17 6.98 - 18.91 0.43 - 0.54 - - - - 72.92 0.089 21.05 18.62 24.15 32.44 - 2.60 0.48 - 0.16 - - 0.11 0.10 63.82 0.110 10.19 11.38 63.77 6.20 - 5.70 1.37 0.35 0.72 0.11 - - - 85.34 0.130 13.65 15.24 22.26 0.25 0.14 27.84 13.09 1.84 - 0.21 5.2 0.15 51.15

Fig. 4 Discrete energy spectra for waves at multiple resonances when k1d =1.5,a1 k 1 = a 2 k2=0.03, α1=0° and α2=3°,δω2,-1 =0, δ ω3, -2 = - 2.9× 1 0-4 and δ ω4, -3 = -1.1× 10-3. Stars: Wave energy concentrated in the degenerate quartet,Points: The degenerate quartet share wave energy with other wave components

3.2.1 Exact quartet resonance

In order to demonstrate this energy distribution in nonlinear waves at quartet resonance, we first consider the same case as Eqs. (62) whenδω2,-1=0 with different2α. The discrete energy spectra are plotted in Fig. 5 with increasing2αin the domain ofα2∈ [ 4°,5 0 °]. Multiple waves are found for each propagation angle, namely that 8, 6, 5, 5, 3, 3 and 3 steady-state waves are found forα2=4°, 6°, 8°, 10°,20°, 30°, 40°and 50°, respectively. Their wave steepnesshsis listed in Table 4. In addition, the energy distributions of the degenerate quartet are listed in Table 5, where cases are marked bold if the quartet has more than 99% energy of the whole wave system.

Thus, it is confirmed once again that for relatively small propagation angles, wave energy will distribute into more components in waves with larger steepness. But for greater propagation angles, the whole wave energy can still concentrate in the resonant quartet even when the wave steepness is large enough.

3.2.2 Near quartet resonance

In this section, weakly nonlinear steady-state waves at near quartet resonance in deep water[8]are extended to strongly nonlinear waves in water of finite depth. The frequency detuning of the quartet resonance in the degenerate caseδ2,-1is studied in detail whenα1=0°,α2=30°,k1d=1.5 and. Three groups of waves (denoted by waves (I), (II) and (III)) are obtained for each different frequency detuning. The wave steepnesshsand angular frequencies of the primary and resonant componentsσi/ωi(i=1,2 ,3) are investigated and plotted in Figs. 6, 7.

Fig. 5 (Color online) Discrete energy spectra for waves with different propagation angles when k1 d =1.5, a 1 k 1= a 2 k2 =0.03,δω2 ,- 1 =0, α1=0°

Table 4 The wave steepness hs for waves with different propagation angles when k1 d =1.5, a 1 k 1= a 2 k2 =0.03, δω2,-1=0, α1=0°

Table 5 Wave energy of the degenerate quartet , where0.03, δ ω2 ,-1 =0, α1=0°

Table 5 Wave energy of the degenerate quartet , where0.03, δ ω2 ,-1 =0, α1=0°

2α 4°6°8°10°20°30°40°50°1 99.51 99.85 99.88 99.90 99.79 99.75 99.70 99.63 2 85.91 96.44 99.17 99.62 99.63 99.64 99.61 99.54 3 87.15 95.15 98.09 99.02 99.00 99.16 99.18 99.14 4 47.33 96.75 99.15 99.59 - - - -5 83.71 88.90 91.45 94.55 - - - -6 54.37 54.92 - - - - - -7 86.60 - - - - - - -8 53.34 - - - - - - -

Fig. 6 (Color online ) Wave steepness of three nonlinear steadystate waves in exact/near resonance when k1 d =1.5,a1 k 1 = a 2 k 2 =0.03, α1=0° and α2=30°

From both figures, it is easy to see that exact resonance (δω2,-1=0) corresponds to an ordinary point on the curves ofhsandσi/ωiwhen the angular frequency in the resonance conditions detunes continuously. From this point of view, the exact resonance is only a special case of the near resonance.Liao et al.[8]found that there is nothing significantly different between exact and near resonances for weakly nonlinear waves in deep water. Now this conclusion is confirmed for strongly nonlinear waves in water of finite depth.

Meanwhile, the bifurcation point atδω2,-1=0.0009 is found among these three waves. Specifically, as the frequency detuningδω2,-1decreases in the domain ofδω2,-1∈ [0 .0009,0.006], only Wave (I)is found. Onceδω2,-1decreases further, two more waves (II, III) appear. In addition, it is interesting to find that generally positive (δω2,-1＞0) and negative(δω2,-1＜0) detunings do not have symmetrical effects on the waves, except the resonant wave frequencyσ3/ω3in Wave (I) (see Fig. 7(c)).

The nonlinearity is represented by the wave steepnesshsor by the dimensionless angular frequencyσi/ωi(i=1,2,3). As shown by Figs. 6, 7,when the resonant frequency is tuned appropriately,waves with large nonlinearity can be found.Meanwhile, these waves with relatively small frequency detuning appear around the exact resonance.In addition, the bifurcation, which is a symbol property of a nonlinear system, also shows the effect of nonlinearity on the resonant wave system.

Fig. 7 (Color online) Dimensionless angular frequencies of three nonlinear steady-state waves in exact/near resonance whenk1 d =1.5, a 1 k 1 = a 2 k2 =0.03, α1=0°and α2=30°

4. Discussions and concluding remarks

In this paper, by introducing homotopyparameterqinto the auxiliary linear operator and letting the angular frequencies be unknown, we have proposed an improved approach (AHAM) that is able to treat resonant wave interactions of great nonlinearity. In particular, angular frequencies of primary components have been expanded with respect to the homotopy-parameter. AHAM is perfectly validated by the third-order perturbation theory, by numerically solving Zakharov equation and by the fully nonlinear wave eqations (3)-(7) solving by means of the FHAM. In addition, except some intersection of waves obtained from AHAM and FHAM, mutually complementary property has been found for these two approaches. AHAM is hence helpful to provide a complete and deeper understanding on resonant waves.

Secondly, by means of AHAM, weakly nonlinear steady-state waves at quartet resonance are extended to strongly nonlinear waves in water of finite depth.Meanwhile, the nonlinear waves at multiple resonances in deep water are also extended to the ones in water of finite depth.

The energy distributions of nonlinear waves at multiple resonances, namely that more wave components will share the energy as the wave steepness grows, is confirmed in water of finite depth.Furthermore, it is found that wave energy can also concentrate in the quartet even when the wave steepness is great enough, as long as the propagation angle between the primary components is relatively large. In this case, except the resonant component,higher-order wave components do exist in the wave system. But their amplitudes are quite small compared with the primary and resonant components. This property of energy distribution means a lot if a wave energy converter (WEC) is designed to capture power from ocean waves where three-dimensional spectra with different propagation angles are common.Usually the size of a WEC is designed according to the frequency of the local wave spectrum[19-22]. If the quartet is tuned by changing their propagation angles to have most energy of the wave system, for example cases in Section 4.2, the corresponding WEC could capture quite a large part of wave energy. Now WECs are usually built close to the coast. This makes our investigation for resonant waves with larger amplitudes in finite water depth have a more practical sense.

Finally, the bifurcation among nonlinear waves is discovered when the angular frequency is detuned from the exact resonance. It is also confirmed in water of finite depth that exact resonance is nothing significantly different with near resonance.

Our results for strongly nonlinear waves at resonance in water of finite depth are encouraging.And AHAM should be validated in other nonlinear water wave problems with resonances. It should have a great potential in dealing with strongly nonlinear waves at resonance that propagate over undulated bottoms. Future work can focus on strongly nonlinear waves at Bragg resonances where wave-bottom interactions occur. Besides, AHAM can also be helpful for nonlinear wave interactions in layered fluids.

Some experiment data indeed can be found for wave resonance in deep water[11]. But unfortunately,these data are not suitable to be used to validate the AHAM results in this paper for the quartet resonance in degenerate case in water of finite depth. Meanwhile,a method for measuring the surface wave elevation in a wave basin is proposed recently[23]. Thus it is possible and worth doing the experiment for nonlinear waves at quartet resonance in water of finite depth in the future.

Acknowledgements

The author Da-li Xu would like to thank Prof.Michael Stiassnie (Technion-Israel Institute of Technology) for the discussions on Zakharov equation.