On the Fourier approximation method for steady water waves

38ZHAO Hongjun et al. Acta OCceanol. Sin, 2014, Vol.33, No. 5, P 37-47tions for steady water waves, during which several interesting of stream function, the inverse plane method and the streamnonlinear wave phenomena have been discovered. One of the function method are only applicable for 2-D waves. Therefore, inbest known numerical methods for steady water waves is the the present study, we are still interested in developing the Fou-Fourier approximation method. This method was first present- rier approximation method in the physical plane on the basised by Chappelear (1961) on the basis of potential theory in the of potential theory. With the free surface and potential functionphysical plane by expanding the free surface and velocity com-approximated by the finite Fourier series, Section 2 presents aponents in finite Fourier series, and determining the Fourier set of nonlinear algebraic equations for the Fourier cofficientcofficients by minimizing the square errors of the free surface from the free surface boundary conditions. These nonlinear al-boundary conditions. Later, Dean (1965) improved this method gebraic equations are subsequently solved thorough Newton'sby solving the stream function instead of the velocity compo- method with a relaxation technology. In Section 3, the accuracynents. In Dean's formulation, the primary specified conditions and convergence of the Fourier approximations are examined.are the mean water depth, the total head, and the wave period. The properties of steady water waves in the finite depth are nu-However, as shown by Cokelet (1977), the total head does not merically analyzed. It is shown that the spectrum of the waveincrease monotonically with the wave height, but has the maxi- surface is broader than that of the potential function. This fur-mum for waves slightly lower than the highest. This means that ther leads us to explore a modification for the present method inthe primary specification of the total head would lead Dean's Section 4. Conclusions are summarized at the end.method into inexact solutions for the highly nonlinear waves.Thisshortcoming has been overcome by Chaplin (1980) through 2 Steady water wave equations and numnerical methodintroducing a Schmidt orthogonalization process. Rieneckerand Fenton (1981) and Fenton (1988) introduced a method that 2.1 Governing equations and Fourier approximationsconsistently used the Fourier approximation, allowing for theThe problem to be considered is that of the steadily pro-condition of deep water and for the current circumstance. In gressing periodic water wave train of wavelength L, crest-to-their methods, the Newton's iterative scheme was used to solvetrough wave height H, and wave period T. This problem can bethe nonlinear algebraic equations. Their results, even for the described using an upward vertical z-axis originating from thehighly nonlinear waves, agreed well with the numerical solu- still water level (SWL) and x-axis in the wave propagation direc-tions of Cokelet (1977). With Newton's method solving the Fou-tion. Figure 1 shows the one cycle wave sketch and some impor-rier cofficients in arbitrary precision computer arithmetic, Lu- tant physical quantities.komsky et al. (2002) obtained more accurate solutions for steadywater waves in the infinite depth. Their method was improvedto be more rapidly convergent by Lukomsky and Gandzha (2003)through the fractional Fourier approximations. For deep waterz. 1waves, Liao and Cheung (2003) developed a homotopy analysisMEL一Vmethod to determine the Fourier components.MwL二立二平later extended to the finite water depth by Tao et al. (2007).HParallel to the Fourier approximation method in the physicalSWLo≥plane, series attempts have been made to develop the inverseplane method. The inverse plane method originated from Stokes(1880). Since the boundary conditions in the inverse plane areinitially known, numerical computations for Stokes wave solu-tions are usually performed in this plane, of which Schwartz(1974) finished the leading works. He first expanded the spatial_bottomvariables as Fourier series in terms of the potential and streamfunction, and then perturbed the Fourier coefficients as powerseries of the wave steepness. The Pade approximation was alsoFig.1. Sketch for the periodic steady water wave train.introduced into the perturbation expansions as a summationtechnology. In that way, very accurate solutions of highly non-If the motion is irrotational, the potential function ρ shouldlinear waves were obtained. In Schwartz (1974), the perturba- satisfy the Laplace's equation throughout the fluid domain.tion parameter is not known ab initio. This was improved byLonguet-Higgins (1975) and Cokelet (1977), where the itallyPx +9_ =0, -h≤z≤η,(1)known parameters were chosen as the perturbation ones. With-out introducing the perturbation expansions, Tanaka (1983) pre-where the subscripts denote partial differentiation; h is the stllsented a more acurate inverse plane method. In Tanaka (1983)，water depth; and n is the free surface.the free surface boundary conditions were first transformedThe boundary conditions to be satisfied are the free surfaceinto an integral equation, and then the equation was solved kinetic boundary condition:iteratively. The variable transformation was also introduced toconcentrate the points near the crest, thus giving more accurate9.=η,+9,nx，z=n;(2)solutions for the highly nonlinear waves.It should be noted that in the inverse plane method, the the free surface dy中国煤化工complex mathematical transformation from the inverse planefYHCNMHGto the physical plane is inevitable, thus making it very com-plex for the practical applications. In addition, due to the use9.+;(o? +φ?2)+gn=87o，z=7;(3)ZHAO Hongjun et al. Acta Oceanol. Sin, 2014, Vol. 33, No. 5, P37-4739and the bottom boundary condition:The free surface function can also be approximated by thefollowing Fourier series as:φ:=0, z=-h.4)η=ηo+ En。cos(n),(12)In Eqs (2)-(4), gis the gravitational acceleration, and Zo is thetotal head that is measured from the SWL.The potential function and the free surface should also sat-where nn is the Fourier cofficients of the free surface and”o isthe set-up term that is generated due to the self-interaction ofisfty the following periodic conditions:the first harmonic component. The physical significance of thea(x,z,1)=p(x-ct,z),5)set-up term lies in describing the mean water level (MWL) beinghigher than the SWL (Song et al, 2013).It should be well noted that, as Rienecker and Fenton (1981)andand Fenton (1988), the free surface η can also be used as itselfn(x,1)=n(x-c),6)without introducing the Fourier approximations when solvingEqs (10) and (11). In this paper, the objective of introducing Eq.(12) is to directly calculate the amplitudes of the free surfacewhere c is the wave celerity.harmonic components, including the set-up term n1g.As suggested by previous steady water wave theories, theTruncating the infinite Fourier expansions in Eqs (7) andpotential function o, that satisfies the governing Eq. (1), the bot-(12) into the finite series, Eqs (10) and (11) will give:tom boundary condition Eq. (4), and the periodic condition Eq.(5), can be approximated by the fllowing Fourier series:.0......)=.(13)φ=之%. can(5)5i(),7)kn- Se, sinh[ nk(h + n)]cos(n0)=0,where k=2r/L is the wavenumber; 0=kx-ot is the phase angle; andσ=2π/ T is the wave frequency; and φn is the dimensionless po-......e.<.=..tential cofficients.(14)Substituting Eq. (7) into the free surface kinetic Eq. (2), and3(7-3)=0,applying the following manipulations一:= 82x0 andin which,n= aa--o的on the re surface”, we obain:i= Eno, cosh[ nk(h+ n)]cos(n,),(15)的二Eno。sinh[ nk(h +n)]sin(nB) .(16)mp, sinh[ nk(r+n)]in(nO)}=k28)η=η% +之n。cos(n0),(17)Integrating Eq. (8) with respect to 0 gives:above equations. To solve the Fourier coefficients no, ... IN, 0,So, sinh[ nk(h+n)]cs(n0)= kn+C,9)... PN together with the wave celerity C, wavenumber k and thetotal head Zo, we should search 2N+4 equations, among which,where C is an integral constant independent with 0. For the pri-2N+2 equations can be found by satisfying Eqs (13) and (14) atmary no wave circumstance, C is 0. Thus, the integral constantN+1 points equally spaced in half wavelength. This procedureC is always to be 0. Hence, Eq. (9) gives:is the same as that of Rienecker and Fenton (1981) and Fenton(1988), and the so determined equations are as fllows:kn= Zo,sinh[ nk(h + n)]cos(n0),10)f=. F(0.......)=0.= ..+.1,.Substituting Eq. (7) into the free surface dynamic boundaryft =+.....9... ,k,n)=0.,-1...N+1，(19condition (3) gives:where 0;=(i-1)x/N, is the equally spaced computational points-c2 Eno, cosh[ nk(h+ n)]cos(n0)+ .in half wavelength.In order to close Eqs (18) and (19), another two equationsshould also be specifiedL Onepou1atign ran ho obtained from2no。cosh[nk (h+ n)]co(n0)}(1) the rlationship be中国煤化工he amplitudesof the free surfaceMHCN MH +2+..号ins.sin[ ()in(n0)[ +g7=g.denoted as:240ZHAO Hongjun et al. Acta OCceanol. Sin, 2014, Vol.33, No. 5, P 37-47Theoretically speaking, Eq. (25) is a quadratic convergent2M+s=台(7.+173 +.++..-1=/0(20) scheme, which is sufficient to satisfy the deep water wave sys-tem with five iterations (Lukomsky et al, 2002). However, forand the other equation can be found from the relationship shallow water waves, it usually fails to be convergent due to theamong the wave celerity, the wavenumber and the wave period, overlow of hyperbolic functions. For example, when the thirdi.e., kc=2π/ T, denoted as:case (L/h =24.4 and H/h=0.39) of Le Mehaute et al. (1968) is com-puted, the above divergent phenomena would occur. In orderto get a series of convergent iterations, a relaxation technologyfsxw=(2]c2-1=0.is further introduced into Eq. (25), thus forming a new iterativeThe above 2N+4 equations include 2N+4 variables, formingscheme as:a closed system. Owing to their nonlinear associations, these2z+'=r(z" +δz" )+(1-r)z"，(26)equations will be solved numerically through Newton's iterativemethod.where r is the relaxation parameter ranging from 0 to 1. When r2.2 Newton's method with relaxation technologyis set to be unit, Eq. (26) would reduce to Eq. (25).For simplicity, we denote Eqs (18)-(21) as:With the third case of Le Mehaute et al. (1968) used as thespecial case, we inspected the iterative behavior of Eq. (26) inf(z)=0, 1....N+..Fig. 2. It is shown that although the mean relative errors for dif-ferent relaxation parameters all exhibit oscillatory, even the vi-brations for the smaller relaxation parameter are much largerwhere -=..). is the unknown vector with 7, 1 than those for the larger ones, the terative scheme of r=0.1 and(.-=..N+1); ZN+1+f=9;(=1... N; ZN+2=c2; Z2N+3-k; and Z2N+4=Zo.r=0.5 can produce convergent solutions while r=1.0 cannot.If the approximate solution vector after mth iteration is z”, and Inspecting the nearly convergent regions, it seems that the it-that of the next iteration is 2z"+1 , then from the Taylor series ex- erative process with r=0.1 is more stable than that with r=0.Numerical tests show that r=0.1 is small enough to produce thepansion, we can getstable iterations, even for the highly nonlinear waves in the shal-4+f(z" )low water depth (if the iteration with r=0.1 cannot produce the .(=)=(*)+ 2(23) convergent solutions, tests show that the selection of r smallerthan 0.1 seems to be invalid in general).Therefore, in all the fol-According to Eq. (23), the following matrix equation can belowing calculations, the relaxation parameter r is set to be 0.1.obtained:3.07M8z"=-尸"，(24)H=7.07 cmm-=3.063mwhereM=[M"]，is the coefficient matrix with the el-2.0ements expressed as M;"时(三"):8z"=z"-z"，isgthe residual error vector of the unknown variables; and二r=0:3**(=({(=.1-..2N+4). is the error vector. The matix1.0-- r=0.1equation (24) is a linear system that can be solved directly bythe complete Gaussian pivoting elimination method. After ob-taining the solution of δ2" in mth iteration, the approximate.hiwhhsolution of the unknown vector for the next iteration can be ob-50100150tained as:z"+=z" +δz".(25)Fig.2. Mean relative error e, for Eq, (26) with dfferentrelaxation parameter r against the iterative number M.The mean relative error for the m+1h iteration can be de-The third case of Le Mehaute et al. (1968) is chosen asthe case study. Noting that for r=1.0, convergent solu-fined as e,"= 202. If e,"≤10-' is satisfied, then thetions cannot be obtained.iterative process is terminated. The initial values for the un-For waves in the shallow water, there is the problem thatknowns can be specified according to the linear wave theory,is Eq. (26) might converge jnto a wronp snluntion, where everyi.e,; values for all the unknowns can be set as 0, except z2=H/2,third Fourier coeff中国煤化工ons have beenzin2=k.H sinh(k(h)/2, ziNu2=gk tanh(k,h)/k。and z2iwns-kg ,wherereported by Dalryic N M H CFenton (988，and can be avoide., and steppingko is the wavenumber determined by the linear theory.upwards in the wave height (Fenton, 1988).ZHAO Hongjun et al. Acta Oceanol. Sin, 2014, Vol. 33, No. 5, P37-47413 Numerical solutionsing the calculations of hyperbolic functions cosh[nk(h+n)] and3.1 Accuracy and convergence of Fourier approximated solu-sinh[n/<(h+n)]. In spite of this fluctuation, the present methodwith N=80 can give three significant figures even for the verytionslong waves with the maximum celerity. The numerical compu-Rienceker and Fenton (1981) presented the Fourier approxi- tations for the cases in Table 2 are all performed up to N=80.mated solutions for steady water waves on the basis of streamHowever, for waves in the depth exp(-kh)=0.l, the approxima-function method, and their solutions were proven to have the tions finer than N=41 cannot give convergent solutions due tosame accuracy as those of Cokelet (1977). All of the character- the overflow of hyperbolic functions. Despite this, the approxi-istics of Rienecker and Fenton's (1981) numerical results can be mations with N=41 can give solutions with four significant fig-obtained by the present method. Besides, since the free surfaceures. Therefore, it is known that the present Fourier approxima-is simultaneously expanded into the finite Fourier series, thetion method can give numerical solutions agreeing well withfree surface ampltudes, including "o .... "Iv can be calculated those of Cokelet (1977), and can be used to study the propertiesdirectly. Results of such series can provide us some insight into of steady water waves in the finite water depth.the harmonics of the steady water waves.A detailed comparison between the present results and 3.2 Properties ofsteady water wavesthose of Cokelet (1977) is presented in Table 1. The comparisonsIn order to study the properties of steady water waves, theare given for the non-dimensional set-up kno at various wave present Fourier approximation method with N=32 is used toheights. The still water depth is exp(-kh)=0.5, corresponding to calculate the waves in the finite depth. This is done for the waterthe wavelength to water depth ratio L/h of about 9.1. It is shown depth exp(-kh)=0.1, .... 0.7 (corresponding to L/h ranging fromthat the Fourier approximation with N=64 can give the same ac- 2.7 to 17.6), and the wave steepness kH/2=0, ...98% of the high-curate solutions as those of Cokelet (1977), even for a wave as est, with 100 increasing wave steepness.high as 98% of the largest. The approximation with N=32 alsoFigure 3 shows the plots of the non- dimensional total headeffectively predicted results that are comparable to those of/H and the non-dimensional set-up no/H against the waveN=64, with only slight deviations not exceeding 0.4% occurring steepness kH/2. We see that the total head and the set-up arefor a wave 98% size of the highest. An interesting phenomenon small magnitudes of the wave height, and they both exhibitis that although the coarsest Fourier approximation, N=8, does non-monotonic behaviors with the wave steepness. The non-not predict the maximum kano as accurately as the finer ap- monotonic behavior of the set-up also causes the mean wa-proximations, it can give convergent solutions that cannot be ter depth to become a non-monotonic variable with the waveobtained by the finer one (N=16) for a wave that is 98% of the steepness. Therefore, the use of the mean water depth shouldhighest, and the maximum deviation from the accurate one of be avoided as the given conditions for computing the highlyCokelet (1977) does not exceed 2.2%.The convergence of the Fourier approximations is inspect-Figure 4 shows the non-dimensional parameter (a7-n)/Hed in Table 2 for waves in the relative deep water exp(-kh)=0. 1 against the wave steepness. As shown by Jonsson and Arneborgand in the relative shallow water exp(-kh)=0.9, corresponding (1995), the total head z0 determines a constant level which isrespectively to the values of L/h of about 2.7 and 59.6. The non- called a mean energy level MEL). The MEL was first introduceddimensional wave celerity kc-/g is chosen as the inspection by Lundgren (1963) for pure waves, and the vertical distancequantity. It is shown that the Fourier approximation method is from the MWL to the MEL is called a set-down. Figure 4 displaysmore rapidly convergent for the smaller waves than for the larg- that the set-down is also a non-monotonic quantity, with theer ones, and predicts more accurate solutions for the shorterlargest value reaching maximum in an intermediate depth. Thewaves than for the longer ones. For the computations of highly maximum set-down for the present cases is about 1.5% of thenonlinear waves, the numerical results exhibit some fluctua- wave height that occurs when exp(-kh)=0.6.tions. This phenomenon also exists in the Stokes solutions ofSchwartz (1974) studied the variations of the first four har-Cokelet (1977), and may be caused by the loss of accuracy dur- monic amplitudes of the free surface against the wave steep-Table 1. Results of the non-dimensional set-up k7o when exp( kh)=0.5I)kn。kH/2Cokelet (1977)N=8N=16N=32N=640.06020760.002 9050.002905 020.088 31270.005 9970.0059970.005997 190.134 1910.012 6300.01263050.176 1770.019 4700.019 4640.0194640.019 46370.2163840.025 3400.025 2050.025 2030.025 20320.2343490.027 0260.0265470.0265410.026 5410.026541 10.241 8320.0272420.0263190.026426中国煤化工02643760.24571)0.025928CNMH G261400Notes: I The maximum wave steepness in Column 1 is as high as 98% of the largest given y Lokele(1911). - aenores no convergentsolution obtainable.42ZHAO Hongjun et al. Acta OCceanol. Sin, 2014, Vol.33, No. 5, P 37-47Table 2. Results of the non. dimensional wave celerity kc-/g against the Fourier series truncation N1)exp(-kh)=0.1exp(kh)=0.9kH/2=0.361 984kH/2=0.425941)kH/2=0.0188kH/2=0.0421)1.1253451.176956 .0.1422680.197 9621.125 3381.1747620.140 6850.194 9051.1253371.173 4400.139 0070.191 558111.125 3361.172 7000.188 5981.1723070.136 4990.185711131.125 336.1.1721110.135 6790.1832671.172 0250.1350710.181 019151.1720010.134 7050.179 154161.1720110.1344730.177 493171.1720380.1343550.176121181.1720730.134 2920.17491710.134 2680.1739141.1253361.172 1480.134 2590.1730351.172 1830.1342600.1722911.1722160.1342630.171 6342:1.1722450.1342670.171 0641.1722710.1342710.170554251.172 2940.1342730.170 101201.1723150.1342750.169 688221.172 3330.1342760.169313281.1723480.168 967291.172 3620.168 6463(1.1723740.168345311.172 3840.168 064321.172 3930.167 7971.172413 .0.165 919640.163 9620.164036Cokelet (1977)1.125 341.172450.134 3810.1645Notes: ) Waves with the larger steepness in each depth column correspond to the ones with the maximum celerity given by Cokelet(1977) (see Appendix Al and A9 in Cokelet, 1997). - denotes no convergent solutions obtainable.0.10] bexp(kh)=0.10.08十0.4_ 0.3 0.20.08.2。0.06 -0.04 t0.041 0.7h0.02 :0.02十0.1 0220.3 0.4 0.50.1中国煤化工FYHCNMHGFig.3. Variations of the non dimensional total head z/H and the non dimensional set-up 1np/H against the wave steepness k:H/2 fordifferent still water depth kh.ZHAO Hongjun et al. Acta Oceanol. Sin, 2014, Vol. 33, No. 5, P37-470.018-0.20 -0.6 0.5ex:(kh)=0.10.15 -0.012-0.豆0.10--k0.006 -0.05.<、0.2一knkH/20.20.10.2 03 0.4 0.5KH/2Fig.5. The first four non-dimensional Fourier coffi-cients of the free surface kn; against the wave steepnessFig.4. Varation of the non-dimensional set -down (20o-kH/2, when exp(-kh)=0.5.no)/H against the wave steepness kH/2 for different stillwater depth kh.manner with the index number n (Dean, 1968). This is also theness for deep water waves. He found that all of them reachedcase for the variations of the free surface coefficients n/,/1 fortheir maximum before the maximum wave height is achieved.waves with a smaller amplitude in greater depth (Fig. 7a). How-However, the method of Schwartz (1974) and the extended oneever, for waves with the larger amplitude, the free surface coef-of Cokelet (1977), are both performed in the inverse plane, it is .ficients usually deviate from this exponential manner, with theinevitable to tackle the complex mathematical transformationsfirst few cofficients above or below it depending on the depth.from the inverse plane to the physical plane if we want to obtainChaplin (1980) reported this, and presented a modified expo-the free surface harmonics. Rienecker and Fenton (1981) andnential expression for the relationship among the free surfaceFenton (1988) presented the Fourier approximated solutions forcoefficients. Another piece of information shown in Figs 6 and 7the steady water waves in the physical plane. However, if theindicates that the Fourier spectrum of the free surface is muchbroader than that of the potential function, especially for wavesfree surface amplitudes of each harmonics are to be computed, in the deeper depth. Take the wave with the maximum celer-the Fourier analysis should also be performed. The advantageity in a depth of exp(- kh)=0.1 for example, when q,/q=10-5, theof the present method is that the harmonic components can benumber of the Fourier terms for the potential function is only 3,calculated directly. Figure 5 shows the variations of the first four while when n,/n=10-5, the number of the Fourier terms for thenon-dimensional Fourier cofficients kn; with the wave steep- free surface is at least above 40. Fenton (1985) also detected thisness. From this figure, we see the same tendency as Schwartz intesting phenomenon. His fifth order solution for the stream(1974): each kn; reaches the maximum before the highest wave, function in the deep water limit only contains the first three .and the locations of the maximum moves to the right with iin- harmonic components. The phenomenon that the Fouriercreasing.spectrum of the free surface is much broader that of the poten-In order to further study the behaviors of the harmonic tial function gives us a clue to modifty the present method. Thecontents of nonlinear waves in the finite depth, results of the question is whether more accurate solutions can be obtained ifFourier cofficients for waves in Table 2 are shown in Figs 6 and the free surface and the potential function are approximated by7. It is seen from Fig. 6 that the non- dimensional Fourier co- different Fourier series, with the truncation of the former higherefficients of the potential function φ ,/φ1 vary in an exponential than that of the latter. This will be studied in Section 4.exp(kh)=0.1ex(-kh)=0.9kH/2= 0.361 984101kH/2 = 0.0188。kH/2=0.425 941。kH/2 = 0.0421010*-中国煤化工101 T104gYHCNMH GFig.6. The non-dimensional Fourier cofficients of the potential function.ZHAO Hongjun et al. Acta OCceanol. Sin, 2014, Vol.33, No. 5, P 37-47ex(-h)=0.1exp(kh)-0.9101 .kH/2= 0.361 984kH/2= 0.0188、kH/2 = 0.425 941kH/2 = 0.042r-102.10-3 .102104-10-51030 400102030405060Fig.7. The non- dimensional Fourier cofficients of the free surface.4 Modification of the present numerical methodFor Eq. (34) 0=(i-1)r/N, i1.,... N+l, and for Eq. (35)We approximate the free surface and potential function with 0=(i- 1)π/N2, i=1.... N2+1. Equations (34) and (35), togetherdifferent Fourier series as follows: .with Eqs (20) and (21), form a closed nonlinear system that canbe solved iteratively. Since the Fourier spectrum of the free sur-η=7% + En, cos(n0),(27)face is broader than that of the potential function, N; should belarger than N2. If N=N2, the above system would reduce to thatin Section 2.In order to examine the efectiveness of the present modifi-φ=° _o, cosh[ n()sin(n)(28) cation, the relative errors of the free surface kinetic boundarycondition (2) and the dynamic boundary condition (3) are de-Substituting Eqs (27) and (28) into the free surface boundaryfined, respectively, as:conditions (2) and (3), we can get:e=-[9.-n, -0.7.小, z=7,(36).......)=.(29) andkn- S%, sinh[nk(h + n)]cs(n0)=0,ande=1→φ+gn - 87.+(.+?)，2=7. (37)0..>.-... ,k,o)=g(7-20)(30)The waves with the maximum celerity in the relatively deepwater exp(-kh)=0.l and in the relatively shallow water exp(-ckh)=0.9 in Table 2 are selected for the special case studies. If themodified method can effectively predict results for the two cas-where,es, there is no doubt for it to be applicable for waves with loweramplitudes throughout this depth region. Table 3 shows the rel-; = Eno, cosh[ nk(h+n)]cos(n),(31) ative errors for the free surface boundary conditions, in which,间isthe mean rlaive eror defined as网-=IDlel0; lel...的= Eno% sinh[ nk(h+n)]sin(n0),is the maximum relative error defined as le,|m = max {e|} . Thecalculations are performed with N2=32, while N is set to be nN27=7.+ 27, cos(n0).(33) (n=1, 2, 3, 4). It is shown that the modified method cannot onlysignificantly reduce the mean errors of the free surface, but alsoThe following N; +N2+2 equations can be obtained by satisfy-the maximum errors, especially for the waves in the relativelydeep water. The phenomenon that the errors of the kinetic freeEq. (29) at N +1 points and Eq. (30) at N2 +1 points equallysurface boundary condition decrease with N, increasing can bespaced in half wavelength.readily explained owing to the fact that, in the modified system,the kinetic free surface boundary condition is satisfied at N+1f,= +......o.)=0(34) points. However,中国煤化工table decreaseof the errors of thry condition is+.w+.*...... ,k,z)=0, (35)somewhat surprisi|YHC N M H Gare performedwith the same N. This may be due to the increasingly accurateZHAO Hongjun et al. Acta Oceanol. Sin, 2014, Vol. 33, No. 5, P37-4715Table 3. Relative errors for the free surface kinetic and dynamic boundary conditions under different Fourier series truncations:N andN2Caseexp(-kh)kH/2N:=nNze|..1).10.425 9413.7x10-31.4x10-41.4x10-28.7x10-4n=24.0x10-49.9x10-61.7x10-31.5x10-45.1x10-53.3x10-62.3x10- 41.2x10- 4n=46.7x10-63.0x10-62.9x10-51.2x10-420.90.042n=11.8x10-23.3x10-31.4x10-13.1x10-28.2x10-45.7x10-21.5x10-2n=32.8x10-34.5x10-42.7x10-21.3x10-34.8x10-41.3x10-2computations for the free surface (Fig. 8a). As aforesaid in Sec- 5 Conclusionstion 3, the iterations for waves in deep water might encounterA computational method for steady water waves is presentedthe overflow of hyperbolic functions due to the higher trunca- on the basis of potential theory in the physical plane with spatialtion for the Fourier series of the potential function. Although variables as independent quantities. It involves expanding thethis phenomenon cannot be avoided in the modified system，free surface and potential function into the finite Fourier series,it at least provides us a clue for the Fourier approximation obtaining the nonlinear algebraic equations of the Fourier coef-method on how to increase its accuracy under finite Fourier ficients from the free surface boundary conditions, and solvingseries truncations. For the wave with the steepness kH/2: =0.042 these algebraic equations through Newton's method with a re-in a depth of exp(-kh)=0.9, Figure 8 respectively plots the non- laxation technology. Numerical computations are performed fordimensional wave surface kn, the kinetic boundary error e1 and waves with a wave height up to 98% of the highest in the finitethe dynamic boundary error e2 in the half wavelength. It is seen depth. It is seen that the convergence of the Fourier series is well,that although the free surface boundary errors decrease with n and the present results are as accurate as those of Cokelet (1977).increasing, the results of the free surface for n=2, 3 and 4 are The set-up and the set-down are numerically analyzed, showingalmost indistinguishable everywhere. Therefore, if the Fourier that they are both non-monotonic variables with the steepness.series of the free surface is truncated as twice as that of the po- Owing to the simultaneous Fourier expansions for the free sur-tential function, good results will generally be obtained.face and the potential function, the present method can directly0.200.06-.101人s 0.03心0.000.00--0.10-一0.03-103.02.0.047]exp(