Empirical formula for wave length of ocean wave in finite depth water

Chinese Journal of Oceanology and LimnologyVol. 23 No. 1,P. 17-22, 2005Empirical formula for wave length of ocean wave in finite depthwater*GUAN Changlong (管长龙), JU Hongmei (鞠红梅)(Physical Oceanography Laboratory & Institute of Plhysical Oceanography, Ocean University of China, Qingdao 266003, China)Received July 18, 2003; revision accepted July 10, 2004AbstractIn this paper, function characteristics of dispersion of ocean wave in finite depth water wereanalyzed systematically. The functional form of the fiting function is reasonably proposed, in which the parame-ters are optimally determined by the least square method (L SM). For infinitely deep and extremely shallow water,the fiting function fits strictly the dispersion to be fited. A new technique is presented in application of LSM.An empirical formula with maximum error of less than 0.5% for computing wavelength in finite depth water ispresented for practical applications.Key words: wavelength; ocean wave; finite depth water1 INTRODUCTIONlength is required to be recomputed at each timestep. The Newton method is not applicable to suchFor ocean wave computation in shallow and fi-models, especially for forecasting purpose. So anite depth water, it is usually required to computeconvenient empirical formula with acceptablewavelength (or wave number) on the premise thataccuracy is needed to replace the Newton method tothe wave period is known. It is well known thatcompute wavelength. Although Hunt (1979) pro-wavelength cannot be computed directly from theposed an analytical formula with high accuracy, itdispersion of ocean wave in finite depth water.has not been applied widely probably because of itsNumerical or graphic/tabular approximation meth-somewhat complex functional form.ods are needed to obtain the wavelength previous.In this paper, characteristics of dispersion ofUse of the tabular method adopted in the nationalspecifications of China for harbor hydrology (MOC,systematically. On the basis of the analyses the1999) usually leads to the error whose magnitude isfunctional form of the fitting function is reasonablydependent on the users. In most numerical oceanproposed, in which the parameters are optimallywave models (W AMDI, 1988; Wen et al, 1999), thedetermined by the least square method (LSM).Newton iterative method requiring much computa-Eventually a concise empirical formula with excel-tion time, is applied to compute wavelength itera-lent accuracy for computing wavelength in finitetively under to condition of arbitrary accuracy. Fordepth water is presented for practical applications.fixed computational region and stationary waterdepth, computing wavelength frequency in advance2 FITTING METHODand using the data repeatedly later on could avoidthe problem from using the Newton method.Wave-current interaction in coastal region is anThe dispersion of small amplitude surfaceimportant physical phenomenon. Common practicegravity wave in finite depth water is given asin the field is to compute hydrodynamic environ-∞2 = gkth(kd)(1)ment in coastal area with wave-current coupledmodels (Guan et al, 1999; Xie et al, 2001) where* This study was fwater depth is not stationary and therefore wave-Programme of Chin:中国煤化IlgiesR&DTYHCNMH G.18CHIN. J. OCEANOL. LIMNOL, 23(1), 2005VoL.23where g is the gravitational acceleration, d thewhere a>0 is a constant to be determined.water depth, 0,k are angular frequency and wavec2(x) should satisfy the following conditions,number respectively. Letc2(x|x >0)>0(12)x=∞“}(2glimc(x)=0(13)y= kd(3limc2(x)=1(14)x,y are called dimensionless angular frequencyand dimensionless wave number respectively. Eq.(1)After trial, the following functional form wasbecomeschosen among functions, which satisfy Eqs.(12)-(14), sayx2 = ythy(4)c2(x)= th(bx)(15)where x, y are positive. The strict functional rela-where b>0 is a constant to be determined. Substitut-tionship between y and x, which is regulated bying Eqs.(11) and (15) into Eq.(7) yieldsEq.(4), is referred to y=y(x), to which the functionalform cannot be determined strictly and analytically.y(x)= xexp( - ax2)+ x th(bx)(16)Next we search for an appropriate function, .y= y(x)，which could best fit y(x). Thus weObviously j)(x) has the following characteristic,firstly analyze the functional characteristics of y(x)y(x)- .2 >0(x2)(17)in order to determine the optimal form, which y(x)should have. From Eq.(4) consideration of thecharacteristics of hyperbolic tangential functiony(x)- x→0_ >O(x)(18)givesThis means that Eq.(16) fits Eq.(4) strictly fory(x)-→0→O(x2)(5cases of both infinitely deep and extremely shallowwater, whatever values a and b have. We nexty(x)- -0→O(x)(6)determine the values of a and b by applyingEqs.(5) and (6) apply to infinitely deep and ex-LSM. The logarithmic discretization of y in thetremely shallow water respectively. y(x) isinterval of [10-2,10*], yields a set of data, { y;}expressed below in termsof X(i= 1,2,-,2000，the same hereinafter) called theprecise value. Substituting { y; } into Eq.(4) yields ay(x)=cx+C2x2set of data { x; } which vary roughly in the intervalIt is obvious that CI, C2 are functions of x ratherof (10~,102). For a set of given values of a andb, substituting { x;} into Eq.(16) yields data { y;}than constants. Now the task is to determine thecalled the estimate value. We definefunctional forms of c(x) and C2(x). Next we presentthe conditions which c(x) and C2(x) should satisfy.Q=' Z(9,-y1)2(19)C(x) should satisfy the following conditions,q(x|x >0)>0(8)Now we search for the optimal values of a andb which minimize Q. The results indicated that thelimc(x)=1(9)global effect of fitting in such way was unsatisfac-tory, because y varies over 6 orders of magnitude,upon which the distribution of { y; } with respect to(10). { x;} is considerably uneven (Fig.1). For small x，even if the relative error of the estimate is large, itschosen among functions which satisfy Eqs.(8)-(10),contribution to Q is still small because of the smallvalue of y itself, and is much smaller than thesaycontribution to Q for large x even with very smallq(x)=exp(-ax2)(11)relative error. In other _wnrds, the minimization of Qonly makes中国煤化工rlarge x,YHCNMHG.No.1GUAN et al: Empirical formula for wave length of ocean wave in finite depth water19namely deep- water case, while it does not work wellrespect to { x; } is comparatively even. Letglobally. In order to solve this problem we considerthe distributive characteristics of Eq.(4) and thenQ'=Z(;-y)(22)defineThe optimal values of a and b which minimizeY;=yx7/2(20)Q’are9;=9x312(21)(a=1.115[b=1.325Fig.2 shows that the distribution of { Y; } with10* r10'0}o2> 10'>.10°|10~'theory-- fitting1002102101Fig.1 Comparison of dimensionless wave number with theFig.2 Comparison of reduced dimensionless wave numberfitting onewith the fitting one70厂,200m0tfitting4, 100m50 t50m40 t之2-20m20 t10m5m-2m10 1:20 2510~11(T(S)Fig.3 The variation of the ftting error of dimensionless waveFig.4 Comparison of wavelength with the ftting one undernumber with dimensionless frequencyvarious water depthsIt is shown that the effect of ftting in this wayin terms of y(x) in order to avoid confusion.is very satisfying. This technique is worth using inEq.(24) is compared with Eq.(4) in Fig.1, in whichthe LSM applications. Substituting Eq.(23) intothe solid line refers to Eq.(24) with legend“theory"Eq.(16) yields the ultimate ftting function to Eq.(4),while the dotted line refers to Eq.(4) with legend“fitting”". It is hard to distinguish the two lines iny(x)= xexp(-1.115x2)+ x2th(1.325x)(24)Fig.1 intuitiv中国煤化工d dimen-where the ultimate fitting function is expressedsionless waveTYHCNMH G.20CHIN. J. OCEANOL. LIMNOL.. 23(1), 2005Vol.23[Y(x)= y(x)x312日21(25)\y(x)= j(x)x-312..60m |100mFig.2 shows the comparisonof Y(x) with Y(x),日- 200min which the legend is the same as that Fig.l. Thedifferences between the two lines in Fig.2 are hard toidentify as well. In order to show the effect of fttingquantitatively, we defineR=||(x)-y(x)|,x 1000(26)otid|y(x)0152Fig.3 shows the variation of R, with x. TheFig.5 The variation of the fiting. error of wavelength withmaximum relative error of the fitting is shown to beperiod under various water depthsless than 0.5%. As x tends to zero or infinity, theerror tends to zero, which is inevitable from thefitting function should have is presented. Theintrinsic characteristics of Eq.(16).distribution of the ftting error is given quantitatively.The conclusion that the fiting error is less than4 RESULTS AND DISCUSSION0.5% is valid for various water depths and periods.This is the global estimate of the maximum error ofTo use the empirical formula proposed in thethe empirical formula proposed in the present paper.present paper to compute wavelength of ocean waveFor the cases of infinitely deep and extremelyin finite water depth for a given period, the proce-shallow water, the fitting function fits strictly thefunction to be fitted. It is known that the function todure is as follows: (I) Compute the angular fre-be fitted is the dispersion of water wave. This inquency∞=2π/T for a given period T，and thencertain sense is the new form of the function ofcompute the dimensionless angular frequency xdispersion of water wave for both infinitely deepby using Eq.(2). (I) Obtain y by substituting.into Eq.(24) and then compute k by using Eq.(3). .and extremely shallow water. If this point of view is(II) Compute wavelength L= 2π/k. Fig.4 showsacceptable, the present study is likely to break a paththe comparison of wavelength L, computed withfor new investigate on wave motion in infinitelythe empirical formula, with the precise wavelengthdeep and extremely shallow water according to theL, under various water depths. The legend in Fig.4 isroutine treatment of constructing wave equation bthe same as that in Fig.1. Fig.4 shows that thedispersion. The formula proposed by Hunt (1979),empirical results are very consistent with the precisewhich corresponds to Eq.(4), is as follows:values and it is hard to differentiate the empiricalresults from the precise results intuitively. In order(28)y=x|x2to indicate the effect of the fiting under various+0.0864.x8 +0.0675xI0water depths and wave periods, we defineThe maximum relative error of Eq.(28) is 0.1%.R=|L(T)- L(T)|(27)The accuracy of Hunt' s formula is higher than thatL(T)of the present result. However, the formula proposedin the present paper is simpler in functional formFig.5 shows the variation of R, with wave pe-than Hunt's result. And the 0.5% error is smallriod T under various water depths. It is shown again inenough for practical application in wave modeling.Fig.5 that the maximum error of the empirical formulaSo Eq.(24) is an alternative formula with excellentproposed in the present paper is less than 0.5%.accuracy different from Eq.(28) for computing waveThe ftting method proposed in the present pa-length in finite depth water.per is quite different from the usual polynomialAll the computations and plots were accom-fitting. On the basis of characteristics analyses ofplished by MA中国煤化工1age in thethe function to be fitted, the form of which thepresent paper.HCNMH G ..No.1GUAN et al.: Empirical formula for wave length of ocean wave in finite depth water21WAMDI Group, 1988. The WAM model - A third generationReferencesocean wave prediction model. J. Plrys. Oceanogr: 18:1775- 1810. .Guan, C.. V. Rey, Ph. Forget, 1999. The improvement of theWen, s.. C. Qian, A. Ye et al, 1999. 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