Mathematic Model of Green Function with Two-Dimensional Free Water Surface *

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1 Applied Mthemtics, 23, 4, Published Online August 23 ( Mthemtic Model of Green Function with Two-Dimensionl Free Wter Surfce * Sujing Jin, Xing Wng, Junjun Du, Shesheng Zhng, Shengping Jin # Deprtment of Sttistics, Wuhn University of Technology, Wuhn, Chin Emil: # spjin@whut.edu.cn Received My 7, 23; revised June 7, 23; ccepted June 5, 23 Copyright 23 Sujing Jin et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl wor is properly cited. ABSTRACT Adopting comple number theory, mthemtic model of Green function is built for two dimension free wter surfce, nd n nlytic epression of Green function is obtined by introducing two prmeters. The intrinsic properties of Green function re discussed on verticl line nd horizontl line. At lst, the derivtion epression of Green function is obtined from the formul of Green function. eywords: Green Function; Free Surfce; Ship Hydrodynmics. Introduction The nlysis of interction between wves nd ship by singulrity distribution method involves clcultion for Green function [,2]. After computer ws used to compute hydrodynmics, finding fst wy to clculte Green function becme science reserch wor [3,4]. Newmn [5] published theories nd numeric methods for computing the velocity potentil, nd its derivtives, for linerized wve motions due to unit source with hrmonic time dependence beneth free surfce. Shen [6] gve n pproimted lgorithm to estimte Green function nd its derivtives by using truncted series epnsion of the Green function were to void the conventionl timeconsuming numericl integrtion. Zhu [7] showed subdomin pproimte method to evlute frequency domin free surfce Green function with sufficient ccurcy. Yng [8] compred the clssicl Green function nd simpler Green function ssocited with the linerized free-surfce boundry condition for diffrction rdition by ship dvncing through regulr wves. Shen [9] proposed the ordinry differentil equtions bout depth Green function nd its derivtive, nd rpid Green function clcultion method combining solving ordinry differentil nd interpoltion between nodes. John [,] showed vriety of representtions for Green function with finite nd infinite wter depth. Other epressions for the free-surfce Green s function, in two dimensions nd for infinite wter depth, hve been improved by Liu [2], Thorne [3], im [4], Greenberg [5], Mcsill [6], nd Dutry nd Lions [7]. Some representtions for the Green s function re lso discussed in [8-2]. A more generl two-dimensionl wter-wve problem tht considers surfce tension is treted in [22-24]. This pper will discuss mthemtic model of Green function with two dimension free wter surfce. In Section, the Green function is represented by using two prmeters. In Section 2, intrinsic properties of Green function re discussed. In Section 3, specil vlue of Green function is given for verticl line nd horizontl line. In Section 4, the derivtion of Green function is obtined for two dimension free wter surfce. 2. Two Dimension Green Function Suppose velocity potentil φ stisfies Lplce eqution: P Q (2.) P is field point Z iy, y. Q is source point i,. The right of eqution is delt function. The boundry condition is:, y y (2.2) * The pper is finncilly supported by Chin ntionl nturl science i, foundtion (No. 5395), nd (No ) nd Chin ntionl eduction deprtment doctorl foundtion (No ). # Corresponding uthor. is comple potentil. On the free surfce y =,

2 76 velocity potentil stisfies liner condition. We esy find its solution [2]: Re G Z, Z, ln ep G Z I i i Z 2 Z (2.3) G z, is clled green function, I is principle vlue integrtion of below I,, ep ep iu Z u du iu i y u du where the integrl lower limit is, The integrl upper limit is, u is zero point of denomintor, upper subscribe (, ) shows tht it is principle vlue integrtion t point u. From the epression of integrtion, the vlue I is determined by the prmeters of,, y. Let unit trnsfer s X i Z R R Z (2.4) The principle vlue integrtion my be written s, ep urx I du H, u R (2.5) where H, is clled H-function with two prmeters:, ep vx i H, dv X e (2.6) v here prmeters te vlue s, Intrinsic Properties First, confirm tht you hve the correct templte for your pper size. This templte hs been tilored for output on the custom pper size (2 cm 28.5 cm). Let Re(H) nd Im(H) re rel prt nd imginry prt of H, respectively. We will discuss Re(H) nd Im(H) with prmeters nd. 3.. Prmeter = In the cse =, from epression of H, esy now, Im(H) =, nd ReH H,, or, epv H, d, we v (3.) v Above formul is principle vlue integrtion with one prmeter, nd my be rewritten s: We esy obtin d H,e e e d e e d d d e e d d d e e e e e d!! And by dopting subsection integrtion method, we hve: d e! d log 2! So tht: H,e log! constnt is: C At lst, we hve d e! 2 log d d e!! C log! (3.2) d e (3.3)! n (3.4) n nn!,e C log H The integrl vlue of H(,) is shown in Figure. From Figure, we hve below theorem: Theorem. ) H, s. 2) H, s. 3) The lowest vlue H,.7423 when = ) On the domin.347, the curve is down. 5) On the domin.347, the curve is rise Prmeter If the vlue of prmeters is not zero, we hve

3 77 Figure. One prmeter principle vlue integrtion H(, )., v cos e H, cosvsinisinvsin d v v And:, ep v cos Re H, cosvsin dv v, ep v cos Im H, sin vsin dv v (3.4) From bove formul, we hve below theorem: Theorem 2. The rel prt of function H, is Symmetric function, or Re H, Re H,. The imginry prt of H, is ntisymmetric function, or Im H, Im H, ; By using epression of Green function, we hve ordinry eqution: H, ixh i So tht we my epress Green function s: i iv H, epe H,e i epe dv Using series epnsion, we get: i H, epe H,e i i epe C log i n n n n ni e nn! ep ni nn! (3.5) constnt C = It is esy to clculte H-function by using bove formul with given prmeters. The numeric results re shown in Figures 2 nd 3. Figure 2. Rel prt Re(H) vried with prmeters. Figure 3. Imginry prt Im(H) vried with prmeters Property of = /2 Consider = /2, then H-function my be written s: H, 2epiC logi 2 n if if f ep i i ep 2 f C log f 2, n2 2 R, n2 n n i nn! n n i nn! 2 2 f fr f f2 tn f2 4. Properties of Green Function n n i nn! (3.7) By using epression of H-function, we hve below theorem: Theorem 3. Consider field point Z iy nd source

4 78 point i re below free surfce, the Green function my be epressed s: i Re Z GZ, ln HR, 2 Z i Z iep i Z C i i, ep e log H where constnt C = Verticl Line n (4.) n ni e nn! (4.2) Consider field point Z iy nd source point ζ = ξ + iη te vlue t verticl line S :, y y. According to the define of X, we hve X =, or =. In this cse, the Green function is y 2 y, ln H y, G Z iep y (4.3) From bove formul, lst term is imginry prt of Green function, others t right is rel prt. Above formul lso show tht the Green function my represented by using H-function t = Horizontl Line Consider field point Z iy nd source point ζ = ξ + iη te vlue t horizontl line S: y y, 2. According to the define of X, we hve X 2 y i R. In this cse, the Green function is GZ, ln HR, 2 2y i iep 2y i R y tn 2 y (4.4) On the free surfce, y =, X = i. Consider X = i, or δ = π/2, the Green function cn be written s: ( i(- ) G ( Z, ). H( -, / 2) ie (4.5) On the free surfce, the Green function my represented by using H-function t Derivtion of Green Function We now, the derivtion of potentil re: dg dg Im Re y d Z d Z (5.) It is esy to obtin the derivtion of Green function s: GZ Z i H R 2 Z Z,, ep i Z (5.2) Consider field point Z iy nd source point ζ = ξ + iη te vlue t verticl line S:, y y, we hve formul of two prmeter y, : i y GZ Z, ln 2 y y y i H y y 6. Conclusion, ep In the pper, the Green function is simplified from integrl formul by using two prmeters. The intrinsic properties of Green function re discussed on verticl line nd horizontl line. The derivtion of Green function is obtined by using comple theory. REFERENCES [] P. Andersen nd W. Z. He, On the Clcultion of Two- Dimensionl Added Mss nd Dmping Coefficients by Simple Green s Function Technique, Ocen Engineering, Vol. 2, No. 5, 985, pp doi:.6/29-88(85)93-4 [2] J. V. Wehusen nd E. V. Ltoine, Surfce Wves, In: S. Flügge, Ed., Encyclopedi of Physics, Springer, Berlin, 96, pp [3] A. H. Clement, An Ordinry Differentil Eqution for the Green Function of Time-Domin Free-Surfce Hydrodynmics, Journl of Engineering Mthemtics, Vol. 33, No. 2, 998, pp doi:.23/a: [4] N. uznetsov, V. Mz y nd B. Vinberg, Liner Wter Wves: A Mthemticl Approch, Cmbridge University Press, Cmbridge, 22. [5] J. N. Newmn, Algorithm for the Free-Surfce Green Function, Journl of Engineering Mthemtics, Vol. 9, No., 985, pp [6] H. Shen, Computtionl Method of Surfce Green Function with No Numericl Integrtion, Journl of Dlin institute of Technology, Vol. 7, No., 988, pp [7] Q. B. Zhou, G. Zhng nd L. S. Zhu, The Fst Clcultion of Free Surfce Wve Green Function nd Its Deri-

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