T H E U N I V E R S I T Y O F M I C H I G A N COLLEGE OF ENGINEERING Department of Naval Architecture and Marine Engineering Final Report EVALUATION OF WAVE-RESISTANCE COEFFICIENTS FOR POLYNOMIAL CENTERPLANE SINGULARITY DISTRIBUTIONS Finn C. Michelsen Hun Chol Kim ORA Project 04534 under contract with, DEPARTMENT OF. THE NAVY.',: DAVID TAYLOR MODEL BASIN CONTRACT NO. Nonr-1224(40) WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1966

This research was sponsored by the Bureau of Ships Fundamental Hydromechanics Research Program administered by the David Taylor Model Basin, Contract Noo Nonr-1224(40).

ABSTRACT Wave-resistance coefficients for "thin ships" are obtained from the Michell integral by the application of the Birkoff-Kotik transformation, Mathematical expressions derived for the case of polynomial centerplane singularity distributions are shown to be of rather simple form and definable in terms of integral functions possessing nonsingular integrands. A computer program for the calculation of wave-resistance coefficients has been prepared and computed results for a simple mathematical hull form are given. These are similar to but differ noticeably from results obtained by means of asymptotic expansions. It is expected that the values of the waveresistance coefficients reported here are closer to being exact because no approximations are made in the derivation of expressions from which these are calculated beyond the approximations made in the linear wave-resistance theory. Furthermore, all functions involved in the computation are well behaved.

TABLE OF CONTENTS Page LIST OF FIGURES vii NOMENCLATURE ix INTRODUCTION 1 DERIVATION OF EXPRESSION FOR THE WAVE-RESISTANCE COEFFICIENTS Derivation of Expressions for Mo and II 6 Numerical Example 8 CONCLUSIONS AND RECOMMENDATIONS 20 ACKNOWLEDGMENTS 21 REFERENCES 22 APPENDIX A. COEFFICIENTS OF THE HULL FUNCTION POLYNOMIALS 23 APPENDIX B. COMPUTER PROGRAMS 27 Wave-Resistance Coefficient 27 Generalized Havelock P-Function 36 Hull Function 39 DISTRIBUTION LIST 43 v

LIST OF FIGURES Figure Page 1 Wave-resistance coefficient for h(u,w)=-8u, D/L=.10 10 2 M I x 10 versus F (D =.1; h(u,w) =-8u) 11 01 3 MI x 10 versus F =.1; h(u,w) =-8u) 12 3M11 L 4 M31 x 10 versus F (=.1; h(u,w) =-8u) 13 5 MII x 10 versus F (D.1 h(u,w) =-8u) 14 00 6 6 MM o x 10 versus F (D =.1; h(u,w) =-8u) 15 L II D 1D 9 MH0 x 10 versus F (D =.1; h(u,w) =-8u) 16 L 8 MII x 10 versus F =.1; h(u,w) =-8u) 17 11 9 M30 x 10 versus F (=.1; h(uw) =-8u) 18 10 M~I x 10 versus F (~ =.1; h(u,w) =-8u) 19 11 Flow Diagram for Wave-Resistance Calculation 28 vii

NOMENCLATURE f(x,z) Singularity distribution function g(x,z) Equation of the hull surface h( h,w) Nondimensional singularity distribution function u,w Nondimentional coordinates of center plan of hull x,y,z Coordinate system fixed in ship x,y Function variables A1iX Aii Coefficients of hull function polynomial B Half-breadth of ship C(x,y) Michell function C Wave-resistance coefficient [ cACSw ] I, Wave-resistance coefficient related to the a-: term of the hull l[ ACSC ] function II D Depth of singularity distribution F Inverse square of Frouide No ( = - ) H(S _,: ) Hull function L Length of ship M,N Maximum power of the variables of the hull function polynomial M( xpy), Special functions as defined in text MII(x,y) ix

NOMENCLATURE (Concluded) P2nl(X), Havelock P-function P2n(x) Generalized Havelock P-function R Wave-resistance, lb w V Speed of ship, fps z,5 Non-negative integer powers of the hull function polynomial (-cx)k Factorial function (=(-: )(-oc+l)...(-a+k-1)) 5(x) Dirac delta function i, 5 Variables of Birkhoff-Kotik transformation f (x,y) Special function as defined in text. aB~~~~~~~~~~

INTRODUCTION The analytical representation of the wave-resistance of a ship published by J. H. MichellI in 1898 forms in spite of many recent developments the basis for a major portion of the numerical and theoretical studies being conducted today in the field of wave-resistance theory. The work described in this report is in this respect no exception. What makes the "thin ship" linear theory of Michell particularly attractive is the simplicity of the surface on which we consider the singularities representing the ship's hull to be distributed, this surface being in the case of the Michell theory the centerplane of the ship. The authors believe that the "thin ship" theory provides the greatest flexibility for many studies of fundamental character, such as the problem of minimum wave-resistance, and therefore, that developments related to this theory, will be centered interest if they facilitate computational work, increase accuracy and, perhaps more significantly, provide us with additional analytic properties of Michell's integral. Within limitations of the linear theory the evaluation of Michell's integral may be expected to give pertinent information.about the wave-resistance of "thin ships" of general shape. It is a quintuple integral, however, and its numerical evaluation is a formidable task. Furthermore, the integral is improper, requiring by most methods of evaluation the application of numerical integration techniques. This will introduce numerical approximations which may prove to be of rather serious magnitude. Several methods of evaluation of Michell's integral have been proposed in the past. One of the most noteworthy is due to Weinblum2 who developed a set of functions suitable for the case of affine hull forms. Because of this geometric feature his method was severely limited in scope. Other researchers made use of assymptotic expressions for the Michell integral. Inui3 in particular has followed such a procedure. A fresh approach to the problem of formulating the wave-re istance of thin ships was taken by Birkhoff, Korvin-Kroukovsky, and Kotik in their joint papers presented before the Society of Naval Architects and Marine Engineers in 1954. Based directly on the Michell integral two transformations were proposed, which would lead to a suitable separation of the functional parameters. The first of these transformations revealed a separation of the functional integrand into two functions, one depending only upon hull form parameters and the other upon ship speed and draft. As a consequence it offered the prospect of providing a better mathematical definition of the relationships between hull form and resistance. Furthermore, it appeared that the Birkhoff-Kotik transformation would lead to simplifications in the numerical work. Especially the idea of being able to tabulate wave-resistance coefficients as functions of hull parameters was attractive at a time when the full impact of the high-speed computers had yet to be felt. 1

The wave-resistance integral resulting from the Birkhoff-Kotik transformation was left by its proponents in a form which required a double numerical integration over a finite rectangular domain. That operation was complicated by the existance of a singularity located on the boundary of this domain. In an attempt to circumvent the numerical difficulties posed by the singularity, Michelsen5 assumed the general singularity distribution to'be given by a double finite series. Combining this procedure with certain mathematical developments he was able to formally integrate the wave-resistance integral and write the wave-resistance coefficient for each'term. of the singularity distribution function in terms of an infinite series. These results led to a contract in 1961 with the Bureau of Ships administered by the DT.MB. for a research project with the objectives of making a more detailed study of functional properties, and to prepare a computer program needed for the calculation of wave-resistance coefficients which could subsequently be tabulated. In the following are the results of that project. Because a relatively long time has elapsed following commencement of the study. it may be appropriate to review its history, the reason being that this in itself offers a lesson to be learned. The equations requiring programming were far from simple and did indeed demand the services of an expert programmero A number of functions had to be calculated from subroutines not available in the computer library. Some of these functions, such as the Confluent Hypergeometric Functions of the second kind, presented very interesting studies in themselves. After about a year of efforts expanded on the writing of suitable subroutines the total program was ready for trial. Results indicated complete failure. Wave-resistance was for a representative case given as approximately 103 times the expected value, and for a different set of conditions the computer dismayingly recorded it as being negative. A careful analysis of each part of the terms entering into the expression for the wave-resistance calculations revealed that the formulation was absolutely impractical for numerical work. No matter what speed was considered one term. or another would become very large, and terms of the alternating series would first increase in magnitude making convergence extremely slowo The main problem was simply computer overflow, however. Some terms reached magnitudes of more than 1038o Several. changes took place at this time. Firstly the programmer left the project for the reason of more profitable employment. Secondly, and on the other side of the ledger, a faster and larger computer was being installed. The project was not of very high priority, however, working from the very beginning on a curtailed budget. Application for additional funding was not approved and research results obtained at this point could hardly warrant that any different action be taken. The nature of the project was such that partial success could not be achieved. Either the objectives were fully realized or nothing at all would be gained from our efforts. 2

Optimism again prevailed when it was found that the mathematical formulation could be modified to avoid some of the difficulties referred to above~ Details of these modifications were described by Michelsen in a paper read at the International Seminar on Theoretical Wave-Resistance in 1963o The second programmer did now leave the project, however, and with all funds expended it became a, matter of the authors completing the research in their spare timeo During the ensuing work it became apparent that the formulas given by Michelsen in 1963 were not fully satisfactory~ The expression for the waveresistance coefficients related to the individual terms of the singularity distribution function had an assymptotic series which appeared to lead to some erratic behavior of the total wave-resistance coefficient of a simple singularity distribution. To the researcher it is always amazing how painful it is to find the most logical path to the solution of his problem. The mathematical derivation'of',the next section is a case in point. As far as the authors can judge it avoids all the pitfalls of previous work and leads to a simple concise formula for the wave-resistance. Accuracy can easily be checked because the functions involved possess several recurrence relations. More important may be the fact that computer time needed is relatively short so that it should prove useful in the determination of optimum hull.forms. 3

DERIVATION OF EXPRESSIONS FOR THE WAVE-RESISTANCE COEFFICIENTS The Michell integral for the wave-resistance coefficient fgr a "rthin ship" can, upon the application of the Birkhoff-Kotik transformation, be written as 1 2 C 6F2 D )2 d j d: H(S,,) C(Ft, F ) (1) w Tr L L o o This coefficient has here been defined by Rw C, 1 2 2 p V B 2 The hull function H(~,~) is continuous. It is, however, defined by two separate expressions referred to two subregions as follows: 1 Region I: [H(L,)]I = du dw h(u,w)h(~+u,l-w) 1 o 2 O < < 1; O < 5 < 1 (2a)! 1 Region II: [H( I,)]II = du dw h(u,w)h(~+u,-w) 2 < < 1 < < 2 (2b) It should be noted that the hull function in this form does not depend upon the uniform depth D of the singularity distribution. For a nondimentional singularity distribution given by M N h(u,w) = Cr. u w () m=O n=O 4

integrations indicated in Equations (2a, 2b) are readily carried out and the results can be written 2M+l 2N+l ~[H(i:)] = E E A'g t (4a):=O'=1 2M+l 2N+l [H(it~)]T = E E A.g3. (4b) a=o B=o I II Derivation of the coefficients A and A"p are given in Appendix A. To the approximation of the Michell "thin ship" theory h(u,w) L d g(x,z) = L f(x,z) B dx B at corresponding points: u = L w = - where g(x,z) is the equation of the hull surface. The function f(x,z) may be taken to be the singularity distribution function of sources and sinks located on the longitudinal center plane of the hull. Equation (1) can be evaluated for each term of Equations (4a) and (4b). By designating the wave-resistance coefficient resulting from the terms [CCB]I,II as [A KCw]I,ii, respectively, Equation (1) thus becomes 2M+l 2N+l 2M+l 2N+l Cw = Z Z AI M'9?A (, F) + e=O =L o=O a =o 2M+l 2N+l 2M+l = I I [aCwc]I + E [+ w (5)C a=O iB= The summation on f in the first term in Equation (5) above starts at 4 = 1 because the hull function vanishes linearly as + 0. This is property of the Birkhoff-Kotik transformationo 5

DERIVATION OF EXPRESSIONS FOR M4, and M~p In order to show the details of Equation (5) consider any one of the terms of the summation. With the Michell function C(x,y) defined by 00 C(x,z) = -Y e cos xN4 1/ dt (6) 1/2 o t we obtain from Equation (1) 11 M (D) 16F2 2 M (DF)= _ () dt d: 0O 1 2 e L e (l+t) cos (F ) d (7) 2 L 1/2 dt (7) o t Integration by parts with respect to 5 a number of times equal to f Equation (7) becomes: 1 00 I D L6F +1 -(+D 1) Me())(dL) 2 () (l+t) L' it2 FD O O FD Z (-' ~ p ~L p+l -(p+l) - L (l+t) ( D)P (-1)p (FD) (l+t) e ] p=O.: cos (F J71) t dt (8) 1/2 dt t The integral 1 00 dt I = j d d cos (F4t ) 1/2 +1/2(9) oL0 t (l+t) 1 can be expressed in terms of Havelock P-functions. This is readily shown by integrating by parts with respect to ~ a number of times equal to c. 6

Thus I1 2(-1) kF FF()k F-k (-k 2) +k (F) k=O + o,. p2p+(0) 5(1 + (-1) a (10) where 6(x) is the Dirac delta function. The Havelock P-functions are defined as follows P2n-l(X) 2 (-1) cos ( ) 1/2 n+ () t (i+t) 0 1 n dt 1 o( n e FD (l+t) = f tl dt (12) ( -) _sin ( x12t+l ) 1n+lco 2 1/2 o o t ( l+t ) can similarly be shown to be given by I2 = 2(-1) F-(a+lL F( ()k2k FL23 FD (l+t) dt +0'. P23+C (O) ((1+(-l)(1) 1 n dt can similarly be shown to be given by O0 P2nl(xy) = 2(-1) e cos (x) t/2(l)n+l/2 a2n-l (14a) = (-)n -y tneQcos cos(x sec e)de 1a

00 -1 n yt dt P2n (Xy) 2 1 eY sin (xRt) d +l 2n 2 O t (t+l) 2 _e2 y tan G 2n - () 2 te cos 8 sin (x sec a) dG (14b) If 12 is written as (F, F) we note the integral I1 will be given by!B(F.O). With this notation the expression for the wave-resistance coefficient becomes very simple in form. Substitution into Equation (8) gives M (F) 8 (FD) Ff Y[ (F,o) UP L L L FD L (- () Y (F L) (15) p L p p=O From these results and Equation (1) it follows that for the region O< 1< 1, 1 < <: <2 we can immediately write MaII ( F) = 8(T (-) (-) ( x I=0 Fo FD - 2FFD e L E (F -) - (2) e L T (F ]} (16) Equations (15) and (16) are the complete expressions for the wave-resistance coefficients of a' "thin ship" based:.on the Birkhoff-Kotik transformation of the Michell integral. It should be pointed out that all functions involved are well behaved and easy to compute. Details of the computer programs are given in Appendix B. NUMERICAL EXAMPLE To check on calculations throughout the development of our mathematical formulations and the associated computer programs it was decided to consider a simple hull form for which the wave-resistance had been previously computed from the "thin ship" theory. An easy choice was the hull form of perabolic WoLo used by Inui and others. Results for this case are readily available in the literature.3 8

Choosing D =.10 and h(u,w) = - 8u (17) we obtain from Equations (2a) and (2b) Re gion I I 16 AII I 32 01 3 11 31 3 Region II AI _ 16; All I 32 5 A11=16 5 Ol A 31= -T I 32 II II 64 o~= A-3; = -32 A00 2' 10 30 3 With these values for the coefficients of the hull function polynomial as input, the computer provides us with the wave-resistance coefficient. The graph of this coefficient as a function of Froude number is shown in Figure lo It is noteworthy that results differ significantly from those obtained by means of Inui's asymptotic approximation. The contributions made to the wave-resistance coefficient by the individual terms of the hull function are given in Figures 2 through 10. A somewhat peculiar behavior was noted in the coefficients for region II in the range of Froude number _.30 whenever P = 1. Accuracy of calculations have been verified several times and it has been concluded that the values of the coefficients are indeed as shown in the figures. 9

2.2 2.0 1.8 Present Calculation 1.6 ----- Inui's s-201, Asymptotic Expansion 1.4 N1.2 r ~1.0 0 I.8.6.6.4/ I/ Note: Humps and Hollows.2!01 Occur at the same speeds..0 0.2 1 0.3 I 0.4 Froude No. 2220 18 16 1413 12 11 10 9 8 7 6 F Figure 1. Wave-resistance coefficient for h(u,w)=-8u, D/L=.10

30 1.0 -1.0 - -2.0 Figure 2. 1 x 10 versus F ( =.1; h(u,w) =-8u) L

3.0 2.0 1.0 N) u 0.0 10. \ / 20.0.0 F -1.0 -2.0 I.1; h( u,) =-8u) Figure 3. M11 x 10 versus F ( =.1; h(,) =-8u)

3.0 2.0 1.0 H 0.0 10.0I\ 20.0 30.01 -1.0 - 2.0 Figure 4. 1 x 10 versus F ( =.1; h(u,w) =-8u) TL

.80.60.40.20 O.00'p- 10.0 20.0 \ 30. 0 -.20 -.40 -.60 -.80 Figure 5. I0 x 10 versus F (- =.1; h(uw) =-8u) 0 L

.80.60.40.20.00 \,Y 10 0 \ 20.0 F F 3. -.20 -.40 -.60 -.80 Figure 6. Ix 10 versus F (2 =.1; h(uyw) =-8u) 10 xL

.80.60.40.20 20.0.00 0.0 30.0 \ -.20 -.40 -.60 -.80 II.1; hu =8u Figure 7. MO1 x 10 versus F (.1; h(u,w) =-8u)

.80.60.40.20 < ~.00 — 4 000 0 0 20'.0 30.0 F -.20 -.40 -.60 -.80 Figure 8. I x 10 versus F (E =.1; h(u,w) =-8u)

.80.60.40.20 ~H.00,.0|\1.0 / \ / 20.0 30.0 F -.20 -.40 -.60 -.80 Figure 9. Io x 10 versus F (L=.1; h(u,w) =-8u) ~~O

.80.60.40.20 20.0.00,,A -.20 -.40.60 -.80 Figure 10. II x 10 versus F (D =.1; h(u,w) =-8u)

CONCLUSIONS AND RECOMMENDATIONS It is believed that the method of evaluation of the Michell integral presented in this report may well prove to be the most general, fastest, and most accurate in existance. Low and high speeds pose no special problems. In fact the middle speed range is where computation time is the longest. Even so it takes only a few seconds to calculate the wave-resistance for the sample hull form considered in the report. At this time we have not taken full advantage of all recurrence relationships of the generalized Havelock P-functions and other features which can further reduce computing time. It is therefore recommended that the computer program now available be modified to take advantage of all possible short cuts that can be incorporated. Of equal importance to the prepared computer programs in the mathematical formulation of the wave-resistance coefficients on which they are based. It is noted that the finite sums of the expression for Cw form a linear-system I II in term of the coefficients AI and Add of the hull function polynomial. These functions are in turn linearly dependent upon the coefficients Cmn of the singularity distribution function. The formulation of a minimum wave-resistance problem with suitable constraints is therefore fairly Straightforward and should be pursued as soon as possible. Another extension to the present work worthy of consideration would be the inclusion in the theory of concentrated sources and sinks, as well as singularities of higher orders. Such developments would be necessary to make the formulation of the wave-resistance of the "thin ship" as described in this report as flexible and complete as possible. 20

ACKNOWLEDGMENTS The authors would like to thank Messrs. James W. Thatcher and Charles M. Greene, research assistants and programmers, for their aid at the beginning stages of this research project. Although their programs are not now being used, preliminary computer results obtained with these programs where extremely helpful in arriving at a workable solution. 21

REFERENCES l o Michell, J. H. "The Wave Resistance of a Ship," Philosophical Magazine, 45 (1898), pp. 106-122. 2. Weinblum, G. P. "A Systematic Evaluation of Michell's Integral," Taylor Model Basin Report 886, June, 1955. 3. Inui, T. "Study on Wave-Making Resistance of Ships," The Society of Naval Architects of Japan, 60th Anniversary Series, II (1957), pp. 173-355. 4. Birkhoff, G., Korvin-Kroukovsky, B. V. and Kotik, J. "Theory of the WaveResistance of Ships," Transactions of the Society of Naval Architects and Marine Engineers, 62 (1954), pp. 359-385. 5. Michelsen, F. C. Wave Resistance Solution of Michell's Integral for Polynomial Ship Forms, Doctoral Dissertation, The University of Michigan, 1960. 6. Michelsen, F. C. "Expressions for the Evaluation of Wave-Resistance for Polynomial Certerplane Singularity Distributions," Proceedings of International Seminar on qTheoritical. Wave Resistance,. Ann Arbor, Michigan August 1963, pp. 859-892. 22

APPENDIX A COEFFICIENTS OF THE HULL FUNCTION POLYNOMIALS From Equations (2), (3), and (4) it follows that the hull function can be written 2M+1 2N+l [H(i,i)]I = C A~ t (19) c~=0 18=0 a=O P=O!-M 1 2 =- du dw lCiju iw 2 0 m,n( +) ) mO n=O M N M N 1 = C Cmn ui(u+)m du w (~-w) dw i j m n 1 0 Considering the integral [Il]I = Ui (u +5)m du (20).1 By the binomial theorem we have that m - m-k 2 lm i+k [I = - (k) Z 1 u+k du k=O 23

i+k+l -1 m-k (-1)i k(k)( i+k+l)l m-k k=O i+k+l i+k+l ~ 1 2 i+k+l- 1 i+k+ ~=0 m m'. where (k k. ( (k) = k'. (m-k)'. Similarly we find that 1 [I2I =; wj (-w)n dw 0 n Z p) P (-1) (j+p+l)-1 (22) p=O Substitution of (21) and (22) into (19) then gives M N M N m n i+k+l+p(m n H(~,~)]i k _ i j m n k=O p=O i+k+l (i+k+l)(j+p+l) O 2 =0 i+k+l-ik M N M N m n i+k+l i+k+p+i+ 1 () (i+k+l)( j+p+l) i j m n k p''~ 24

m (n) (i+k+l) m+i+l- a n-p C C k p ~ ij mn i+k+l + I I I I (-1) i+k+p (1 ) (23) (i+k+l)(j+p+l) i jCm n k. p m n)" m-k n-p k p CijC mn Following an identical procedure it can be shown that i+k+l+p+i n H(,)]I = Cij n(-l) n i,n k P i,j,m,n,k,p, ~ i+k+l (1I 1 m+i+l-~ n-p 2 (i+k+l)(j+p+l) j+p+l 1 13 ru n i+k+p+q+l n i+k+l!+ Cij Cmn(-l) ( () ( ~ ) i,j,m,n,k,p,~ q=O ( j+p+l (j1) Q 1 m+i+l-~ n+j+l-q (j+p+l) (-) q 2 (i+k+l)(j+p+l) i+k+p m n 1 i+k+l +Cij Cm (-1) () () ( + E mn( (k) (p) P ) (i+k+l)( j+p+l) i,j,m,n,k,p m-k n-p 25

+ C (1) i+k+p+q+l m1 n) j+P+j i,j,m,n,k,p,q 1 i+k+l 1 m-k n+j+l-q (~) ~ ~ (24) 2 (i+k+l)(j+p+l) The method used in determining the coefficients AI and AII is to calculate and collect the coefficients of equal. powers in ~and. U A computer program written in MAD language that will perform this task is shown in Appendix "B." 26

APPENDIX B COMPUTER PROGRAMS Because programmers at The University of Michigan are generally more familiar with MAD, this language has been used throughout. The programs can readily be translated into other languages, however. WAVE -RESISTANCE COEFFICIENT Wave-resistance is calculated from Equations (5), (15), (16) and from the coefficients of Equations (23) and (24). The flow diagram is shown in Figure 11 below followedby the associated program. 27

READ DATA +" T START ~~N, KL(N), D/L [c 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NKQ),D/ >N F F=KL(N) j - cl.. #, A I pA -U +d,8 +1 FRNO=F -1/2 > cl max Cl maxSmox iti max F Iici~~~~~~~~l max k~~~ T nn=2 a PR7= PCOS.( nn,,o, ) ji o a= Odd F TERM 9 = c! * PR7/F c+ 1 [n TEM8OI j > T PR7 = 0 ". F >I1TERM 10 =16/7r (F D/L) Si (-I. )! jt!1' 1 1(TERM 8 + TERM 9 ) DELCWI (,,6 A)= Ae x TERM 10 TERM 3=0 = ~~~ ~~PIR6=|.PCOS~. (nntoF 9) | 1IM 8+-)*P6Fai jl Co T T PR I=PIN nPSIN. (, F, F,.U) k] ~"'-' 0ff nn=2* ki +j1 5 F t ven l v /'" > ( 3; TERM I= TR" 6 ) FT IPRI IIPCOS.( nn, F D/L, F TklcO~~~~~~~~~~~~~~ I~~~~~~~~~~~ II~~~~~~~~~~~ IIjio ~~~~~ 0 P2 PI.(nn FD/,0,U)TEM =C2PR/FC1 =kl > ~ > TERMi=TERM i33+ jI =- evnL IF B (- 1 ),a+ k I ( F D/L ).4+kl PR2= =0T7` IPRi2|PCOS.( nn, F D/L, F, (- TERM I + TERM 2/F jl+I Figure 11. Flow Diagram for Wave-Resistance Calculation ~' —ki~ ~~T PR 7~~~~~~~~~~~~~i!+1=od ~~~PR2=0EMI ER PR=PCOS.( nF D/Lo.. (EM,+TR2) Figure 11. Flow Dia~gram for Wa~ve-Resista~nce Calcula~tion

NABTMP=16/7r( F D/L)19 e F D/L PRINT RESULTS FOR REGION I TERM 3 F, c,,, AC, DELCWI (a,&, C))=CW 6 DELCW2,(a8 )=AdrS NABTMP H DELCW2 (qC,,B ), DCW (a,,) + DCW (0, DCW (,,a) DELCWI (,) DCW (ca,8)/ A a + DELCW2 (Q,By) I I I SIMILAR I CALCULATIONS CW ( 2 )= CW ( 2 ) CWT= CW ( I ) PRINT RESULTS I I + DCW ( d,) +CW (2) F, FRNO, D/L, CWT I FOR REGION U II Figure 11 (Concluded) ro

$COMPILE MAD, EXECUTEDtJMPPRIAT OBJECTI/O DUMPPUNCH OBJECT CWK 018659 08/19/65 6 56 59.0 PM MAD (09 AUG 1955 VER1,SION) PROGRAM1 LISTINIG.I n MICHELL INTEGRAL F.C.; MIOHELSEN -. H. KIM FACT. (XvN) PSIN.(tN,FDL F, MU) PCOS.(NIFDLF, MU) INTERNAL FUNCTION Z.(KK)=FACT.(1IKK) *001 EXECUTE FTRAP. - *002 PRINT COMMENT $I$ *003 PRINT COMMENIT $4 MICHEL INTPGRAL$ *004' PRINT COMME'NT $4 *005 1 FINN C. MICHELSEN AND HUN CHOL KIM$ *005 PRI_ T COMMET___ _______ ______0______ *006 START READ AND PRIN4T DATI N *007 READ FORMAT KLFRMI KL1)...KL(N) *008 PRINT COMMENT $U REGION I INPUT DATA$ *009 START. i. READ AND PRINT _DATA.MAXA_. MAX BDOVERL _ A. *010 READ FORMAT ABIN, ALPHA(1)...ALPHA(MAXA) *011.YaEC& VALUES ABIN=S16I5*5 _012___ ___ _ ___ ___ *012 RFAD FORMAT ABIN, 3ETA(l)...3ETA(MAX6) *013 MDI.M( 1 ).=- AXR+22 -0.. -----—.14 MOIM(2)=MAX B *015 0 D) IM I ( I ) =5f3.TA,(MAX i' )+2 *016 OIMI (2)=BETA( MAXB) *017 READ AND PRINT QA LA AA-(4ALPHA, BETA) *Qj8 PRIINT COMMENT R E REtGGION II IN PUT DAT TAs$ *019 READ AND PRINT DATA MAXAI, MAXRl, ALP MAX *020 READ FnRMAT A8IN, ALPHAL(1)...ALPHAI(MAXA) *021 READ FOR- \MA!T AOI NPE:TA1( i)... OEFTAI(MAXB) *022 MDIMI( 1)=MAXD~.1+ *023 MDIMI(')=MAX ___ *024 DIM2(1j)=3-CTAI(MAXBfl.+2 *025 D I M 2 ( 2 ) = B F T A 1 ( M A1X Q i ) *026 READ AND PRINT UOTA AABI(ALPHAI, BETAl) *027 PRINT COIMMENT $3 SPEED INPUTt *028 PRINT RESULTS KL()...KL(N) *029 THRD Li, FOR__I=14,l.N - - ______ __ *030 PRINT COMMENTT$1$ *031 0 F=KL(I) *032 01 FnL=F*n OVER L *033 01 F' NO=F *P.-O.S5 *034 01 CW ( 1 ) =) #035 01 Nr=ALPH A(AX AL! -*RETA X_____, ___AX__*036 01 WHENEVr2? NN7:.L. ALPHIAI(MAXAI)+2*FITAI(MAXBI) *337 01 " J Nl ALPHA1(NI A l XAI)+2*RFTAII (MAXB1) *038 01 01 rEtn OF Cr njDITID4"'L *039 01 01 WHERNJ'VEVR SW( 2 )i PRIN IT RFSULTS AAD(1,1),A I( 1,1) *040 01 Tt~liJ2H: I L(,OPF FO J=OFO, 1,J NN *041 01 PSJv(J) p ___ ______ *042 02 PCNSW~J Ii ~J ( J ) LI: i30*043 02 dS1lSw- (J) *044 02 PC~Sb I J) ~ *j45 02

PiSNiSW6IJI=0&8 ____ _ — *046 0 PCNSW6 (J )=08 *047 0 PCNSW7( J) =0B *048 0 PSNSW7( J)=OB *049 0 R SX5W 4 (JI~f *050 0 PCNSW4(J)=DB *051 0 -_ _ _ _ S S 5 f 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * 0 5 2 0 LOOP PCNSW5(J)=OB *053 0 WHENEVER SW(l.) *054 0 PRINT COMMENT$0 RESULTS FOR REGION 1$ *055 0 0 END OF. CONDITIONAL o05 *057 0 THO~ Zi FORJ=1 sl J. Go MAX A __ _ *05~ 01 ALP=ALPHA (J) *059 0 THROUJGH L2.ERK=191,K.G. MAX B *060 0 BtT=BETA (K) *061 0 - - __- 4- *~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 6 2 _ __ _ *063 0 _EECUTfE 7ERQ_..Tu1EN 81 __ _-*64 -Q THROUGH L89 FOR Jl-0,1'.Jl.G.ALP *065 0 NN=2*BET+Jl *066 0 NNH=NN/ 2 *067 0 W HLE N ERYEAiN~ZfNB4_____________________________ *068 0 WHENEVER PSNSW6CNN) *06901 4 PR6=PSNS6 (NN) ~- *07Q02_4 OTHERW ISE *07102 4 PR6=PSIN. (NNtO.,tFMU) *072 02 0 WHENEVER SW(2), PRINT COMMENT$ PSINS *07302 4 WHENEVER SW(2), PRINT RESULTS NNt. BET, Jlt PR6 - *-0.74 - 2 0 PSNSW6(NN)=1B *07502 4 H ~~~~~PSNS6(NN)=PR6 --- - *702 4 END OF CONDITIONAL *07702 4 OTHERWISE *078 01 0 WHENEVER PCNSW6(NN) *079 01 0 P R -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _* 0 8 0 - 0 OTHERWI SE *081 2 0 --- -PR 6=PCOS AN.4PQ. i,EAKUL _ - ____ 22 0 WHENEVER Sw(2), PRINT COMMENT$ pCOS$ *083 02 0 WHENEVER SW(2), PRINT RESULTS NN, BET9 Jit PR6 *084 02 0 PCNSW6(NN )=1B *085 02 0 PTNS6tNLi{ -- 06 2 0 END OF CONDITIONAL *087 02 0 -END.OF —CDNDI.TIQONA4Lx- --— ____ - __ __*088 01 0 LB TERM R=TERM 8+FACT.(-ALPJiL)*PR6/F.P.(J1lKL) *089 0 ALPH=ALP/2 *090 0 WHENEVEP ALPH*2.NE.ALP *091 0 NN=2*bET+ALP - __ _ - _ *092 01 0 NNH=NN/2 *093 01 0 WHENEVER PCSINV__ ___ — *094 01 0 PR7=PCNS7( NN) *095 02 0 OTHERWISE - *096 02 0 PR7=PCOS. CNNO.,O.,MU) *097 02 0 -WHENEVER. SW(2)t PRINT COMMENT$ Pcos$ _ -*_098 02 0 WHENEVER Sw(2)v PRINT RESULTS NN, BETvALP, PR7 *099 02 0 WH11EN EVER SW(2)i -PR-INT RESULTS PR7, _NN~ - _-*0 2 0 PCNSW7( NN )=IB *101 02 0 N S 7 ( N N )? & r _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _* 1 0 20 23 ENO OF CONDITIONAL *103 02 0 U T H ERW I, SEl_ *104 01 0 PR7=u. - *105 01 0

-ND OF CO',OITIONAL *106 01 03 TRN 9=:Z.(ALP)*PR7/F.P.(ALP+i) *107 TERM 10:C=,./(3.14159*FDL.P.(BET-1))*Z..(,'T.ET)*(-!).P.(BET)* *108 I (TER{M 0+TERM 9) *108 OELCWI( ~LP,I3ET)=AAB(ALP, BET)*TERM 12 -109 EXECUTET ZFRO. (TERM 3) 0110 THROUGH L3, FOR K1=.),1,K1.G. 3ET'111 EXECUTE- ZiRO. (TERM1,TERM2) *112 04 THROUGH L6 _.._FOR_ jl=-! J..G..LP. *113 04 NN=2*Ki+Ji *114 05 WHENEVER (NN/2)*2.E. NO *115 05 WHENEVER PSNSW(NN) *116 01 05 PR:=PSNSI (NCN) *117 02 05 nTHERWI SE *118 02 05 PRi=PS!N.(NI, FDLF M.U).' *119 02 05 WHENEVER SW(2), PRINT COMMENTS PSINS *120 02 05 WHENEVER SW(2), PRINT RESULTS NN,KI,J!,PR1 *121 02 05 PSuSW(NN)=1B *122 02 05 PSlSi(t)N) =PRJ *123 02 05 END OF CONDITIONAL *124 02 05... OT HS7,_[S$F:/,. *125 01 05 WHENEVER PCNSW(NN) *126 01 05 PRI=PCNS! (NN) *127 02 05 nTitERWISE *128 02 05 PR I=PCnS.(NN, FL,FLFMU) *129 02 05 WHENEVER SW(2), PRINT COMMENT$ PCOS$ *130 02 WHENEVER SW(2), PRJT RE SULTSNNK 1J 1, PRI *131 02 05 PCNSW(NN)=iB *132 02 05 PCNSIUJN ) =PR1 *133 02 05 END OF CONDITIONAL *134 02 END OF Cnq\,DITIONAL *135 01 5 L6 TERNi=TERiM1+FACT.(-ALI),J1)*PR!/F.P. (Jl+l) *136 05 ALPH=ALP/2 *137 04 WtHEN'VEP ALPH*2.'IE. ALP *138 NN=2*K 1+ \L P *139 01 04 vHEI F V ~71 (\)A/-,)*2 N~' *140 01 04. -EVER PSN'SW,(NN,) *141 02 04'p5( A S J) *142 03 04 { 7 Ti-: w I SJ -1'IlL TJ 5 __ __ *_143 03 04 )o=P S I?14 (N11, FOL, ),Ii) *144 03 04 RH IEV R SW(2)2 PR[INT COMMENTS PSIN$. *145 03 wH-r.V' SW(2), PRINT RESULTS NN,KlALPPR2 *146 03 04 P \4 SI:( JN\ R2 *148 03 04 F'F) OF CON[)ITIONAL.................. *149 03:"1T iH n —"'< W I S -*150 02 04 WI-:EVER PCNSWO(NN) *151 02 04 )"1'=PCNSO ( xl"I ) *152 03 04 JTtI-(WI SE *153 03 P'.72 =,CnS. C>IN, FDL,'.,MU) -154 *154 03 04 WJHENEVERSW_(2), PRI [NT COMMENT$ PC$S$ *155 03 WHENEVER SwC2), PRINT RESULTS NN,K1,ALP,PR2 *156 03 04 (C-UW IN ) )=l *157 03 04 PCNSQ (IN) =P B2 *158 03 04 ErO OF CONnITIO:JAL *159 03 F%'O OF CONDITIONAL *160 02 OITHIERWI SE *161 01 04 ~~?::'-'- -162 01 04 NO nF rm'j[)ITIOnllAL *163 01 04 "=7.('LP)*P;?/F.P.(tLP+l) *164

L3 ~~~~TVRM 3=TElR. 3+FACT.(-BET i1)*(-l.).P.(BET+K1)*FDL.P.(BET-Kl) *165 0 I *(TEM 1+ TERM 2) *165 NJABTMP=-16.*TERM 3/(3.14159*FDL.P.(BET-1)*NAPE.P..FDL) *166 0 DELCW2(ALPRET)=AAB(ALPBRET) *NABTMP *167 0 DWIALPBT)=E C1AA P EJJDELCW2 (ALPBEET1 ____ *16803 WHENEVER SW(l) *169 0 - - aRINIG4~~COMENIIO __ AL HAIIFTA _.AAB F IRS __ — *17001 3 1 T SECOND sum COEFFS *170 PRINT FORMAT-OUTPUT,F,ALP,BET,AAB(ALP,BET)2DELCW1(ALP-,-ET), *17101 3 1 DELCW2(ALP,BET), DCW(ALPRET),DCW(ALPBET)/AAB(ALPI3ET) *171 0 F-1ECON-DITA1-04AL. -------- 17320 WHENEVER SW(5) *7 PU-NC~HFORMAT__M~jil~VEKRL1 FKKDW(LBT/BA LPBET) - _ —_ —*17401 3 VECTOR VALUES MK=$2Fl0.5,15,55, E25.9*$ *17501 3 END OF CONDITIONAL *17601 3 L2 CW(l) = CW(l) + DCW(ALPHA(J), BETACK) *177 0 WHENEVER.SWLIK -___ __ - ~~~~~~~~~~*178 0 PRINT COMMENT $j RESULTS FOR REGION 2$ *17901 0'INDQ iN DITOAL -*18001 Q CW(2)=0. *181 0 THROUGH L4, FOR J=1l,1J.G. MAX Ai ___ *18 2___ 1 ALP= ALPHA 1( J * 183.0 LHRQUGJJ L4 FQR K1,1, KG. MAXIU - - ~~~~~~*184 0 BET=ETA1(K) *185 0 KKK=KKK(+I ~___ *186 0 AAB1 (ALP, BET) =1 * *187 0 EXECUTE ZERO. (TEIRM3,TERM6) *188 ___0 THROUGH. L5, FOR K13,Il,K1.G. BET *189.0 EXECUTE: ZERD-.A-LE-RMI,-2TE.RM2,TER?,M4,TE.RM5) --- *190 Q THROUGH L7, FOR Jl=O,lJ1.G. ALP *191 0 NN2*KI+JI ~~~~~~~~~~~~~~~~~~~~*192 0 N NH = NN / 2 *193 0 WHlENLERkNNH*2 4. MN *194 0 WHENEVER PSNSW(NN) *19501 5 PR' =?SN51 (NM) *19602 5 O T HE R WI S E *19702 0 PRIl=PSIN.(NNFDLvFtMU) *19802 0 WHENEVER SW(2), PRINT CnMMENTS PSIN$ *19902 5 __ _______ ~~~_WliLNEVR RiLLFSULTS NN,Kl9JI,PR1 *200 2 0 P S'lS W( N N) I10 *20102 5 PSN-S1 (NM,) =PRl *20202 5 ENO OF CON4DITIONAL *20302 5 W H,'- ENEV -R. P SNSW 4 (NN *20401 5 PR4=PSNS4(N'l) *20502 5 -DT LIB'J S~_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * 2 0 60 25'IR4=P SI N1. (Ni, 2. *FDL, FvMU) *20702 5 WHENEVER SW(2),i PRINT RESULTS NN,K1,JI1,PR4- *20802 5 PSASW4 (NIA) =1 *20902 5 PS Nl$4(N N) =PR4 *21002 5 END OF CONDITIONAL *21102 5 0 T H1 FR W I SE -— _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _- - - *212015 WHFENEVER PCNSW(Y1) *21301 5 PRI=PCN S2 (Ml) *21402 5 OTH R WISE *21502 5 PR>.pcDlS.(INN,FDLFIMU) *21602 5 WHcENEVER SW(2), PPINT RESULTS INN,K1,J1,P RI *21702 5 P'ciSW(NNll)=1D - *21802 5 PCMS ~~~~~~. (NM) =PR j ~~~~~~~~~*21902 5 9~ 0 F C oN'D I T I nlA L *22002 5 ~I'VZ- PC"IS',44 (NN) *22101 5

PR4=PCNS4(NN _ ___ *222 02 05 OTHERWISE *223 02 05 PR4=PCOS. (NN,2.*FDL,F,MU) *224 02 05 WHENEVER SW(2), PRINT RESULTS NN,K1,J1,PR4 *225 02 05 PCNSW4(NN)=1B *226 02 05 PCNS4(NN)=PR4 *227 02 05 END OF CONDITIONAL *228 02 05 END OF CONDITIONAL *. 229 01 05 WHENEVER SW(3), PRINT RESULTS K1.JlPRL PR4 *230 05 TERM1=TERM1+FACT.(-ALP,Jl)*PR1/(F.P.Jl*NAPE.P.FDL) *231 05 L7 TFRM 4=TERM 4+FACT.(-ALPJl)*PR4/(F.P.JI*NAPE.P.(2.*FDL)) *232 05 ALPH=ALP/2 *233 04 wHENEVER ALPH*2.NE. ALP *234 04 NN=i*K1+ALP *235 01 04 WHENEVER PCNSWO(NN) *2.36 01 04 PR2=PCNSO(NN) *237 - 02 04 OTHERWI SE *238 02 04 PR2=PCOS.(NNtFDL,O.,MU) *239 02 04 WHENEVER SW(2)o PRINT RESULTS NNKlALP,PR2 *240 02 04 PCNSWO(NN )=lB *241 02 04 PCNSO(NN)=PR2 *242 02 04 END OF CONDITIONAL *243 02 04 WHENEVER PCNSW5(NN) *244 01 04 PR5=PCNS5 (NN) *245 02 04 OTHERWISE *246 02 04 PR5=PCOS.(NN,2.*FDLO. tMU) *247 02 04 WHENEVER SW(2). PRINT RESULTS NN,K1,ALP.PR5 *248 02 04 PCNSW5(NN)=lB *249 02 04 PCNS5(NN)=PR5 *250 02 04 END OF CONDITIONAL *251 02 04.~1 OTHERWISE *2'52 01 04 PR5=0. *253 01 04 PR2=0. *254 01 04 END OF CONDITIONAL *255 01 04 WHENEVER SW(3), PRINT RESULTS K1,JlPR2,PR R5 *256 04 TERM 2=Z.(ALP)*PR2/(NAPE.P.FDL*F.P.ALP) *257 04 TERM 5=Z.(ALP)*PR5/(NAPE.P.(2.*FDL)*F.P.ALP) *258 04 WHENEVER SW(3),PRINT RESULTS TERM1,TERM2,TERML+TERM2 *259 04 WHENEVER SW(3), PRINT RESULTS TERM4, TERMS, TERM4+TERM5 *260 04 TERM 3=TERM 3+FACT.(-BBET-Kl)*(TERM1+TERM 2)* *261 04 1 (-1.).P.(BET+Kl) *261 L5 TERM 6=TERM 6+FACT.(-BET,Kl)*(2.*FDL).P.(BET-Kl)*(TERM4+ *262 04 1 TERM 5)*(-1.).P.(BET+Kl) *262 TMP1=16,*TERM3/(3.14159*FDL.P.(BET-l))/F *263 03 TMP2=16.*TERM6/(3.14159*FDL.P.(BET-1))/F *264 03 DCWL(ALP,BET)=AAB1(ALP,BET)*TMP1 *265 03 DCW2(ALP,BET)=AAB1(ALPBET)*TMP2*(-l.) _ 266 03 DCW3(ALP,BET)=DCW1(ALP,BET)+DCW2(ALP,BET) *267 03 WHENEVER SW(1) *268 03' PRINT COMMENTSO F ALPHA BETA AAB1 THIR *269 01 03 1 D FOURTH SUM COEFF$ *269 PRINT FORMAT OUTPUT,F,ALPBETAABALP,BET),DCW1(ALP,BET), *270 01 03 1 DCW2(ALP,BET),DCW3(ALP,RET), DCW3(ALPBET)/AAB1iALP,BET) *270 END OF CONDITIONAL *271 01 03 *272 03 *273 O0 03 *274 O0 03 L4 CW(2) = CW(2) + DCW3(ALP,BET) *275 03 CWT=CW(1) + CW(2) _ *276 _ __ PRINT COMMENT $8$ *277 01

L1 PRINT RESULTS F,FRN]QDOVERL, CWT,CW(1),CW(2). *278 01 PRINT COMMENT$I$ *279 TRANSFER TO START 7__ _ _ _ _ *280 VEC T OR VALUS E KLLF.RM=$16bF5.*$.......281 VECTOR VALUES MDIM=2,1,iO *282 VECTOR VALUES MDIM1=2,1,1, *283 VECTOR VALUES OUTPUT=$SOF5. 1,214,15E15.7*$ *284 VECTOR VALUES SW=-'B,,BtO R...._ *285 VECTOR VALUES NAPE=2.73218421828 *286 INTEGER KKNNH, KKK *287 INITEGErR, M XA,MAXA:, MAX,MAXP. I,I,J,K,J1,Kl,ALPHA,ALPHAl, *288 1 RETA,RETAI,NN,ALP, BET, ALPH *..._..288 VFCTIR VALUES DIM1=2,1, 0 *289 _ VECTrR _VALUES DIM2=211I, *290 DIMENSI O AAB(1)'OJ,MDIMh,AF Al (iZO3:,MDI.M1), I2AB(1000,DIM1), *291 1 D-ELCW1I(:0',OIM1)., OELCW2( JOO,DIMi)l, DCW(lOGO_1DIMi1),.*291 2 DCW1(lO3,nIIMM2), CW(2), ALPHA(50), AL-PHA1(50), fPETA(50), *291 3 BETA1(5')), KL(i1iC-), SW(5).. *291 DI.MNSION! DCW2(:i0-0,DIH'2 ),DCW3(1'..vODIM2) *292. I _ ES LON PSL LSI O ), PC.N S1...___O { 10' } PCNSO 100 ). *293 I PSN!S4(1?,), PCNS4( lu ),P S;1S5( iD), PCNS5(1-0) *293 2 PSNSW(1C.'), PPCNSW(i:),PSNSW;)(l- ) l,PCNSW'A4(l]ul.)PSNS.W4(t10J), *293 3 PCNISW4( 1 C ),PSNSW5(lO);PC^-ISW5(1CG:) *293 LIME)ISI.O" PS'JSSW6 (100 ), PCNSWS( 100), PSNSW7(-lJCiQ)__PPCNSW7(O1300), *294' i PSNS,6( 1-0';), PCfIS6( Cr)),PSNS7(1iO), PCNS7(100) *294 P-n rL E S^?P SL wPSNS iP,, PCNSWI P _S N SJ.W, PCNSWLPSNSW4, PCNSW4, PSNSW5_ _ _ _ _ *295 k.i PC'tSW5 *295 P3 OLE A N PS'ISW6, PCNSW6, PSNSW7, PCNSW7 *296 END DOF PRO)GRAM *297 THE FnLLOWING C'AN'ES HAVE OCCUORED q9LY ONCE Il THIS PROGRAM. COMPI L A T IO W I L L C T.J......__ ___ ___ ___..._,........._ _.......... S rATi A J1L;

GENERALIZED HAVELOCK P-FUNCTION Generalized Havelock P-function in Equations (14a) and (14b) are called PCOSo(KK,FDL,FF, MUM-) and PSIN.(KK,FDL,FF, MUMU) respectively, where KK: integer order of P-function i.e., 2n-1 or 2n. FDL: argument y FF: argument x MUMU: a number corresponding to an allowable maxirmum error in numerical integration. As e is increased toward 2, the oscillating frequency of the integrand increases but the amplitude decreases making the contribution to the integrand smaller The method of integration is by Gauss-Legendre 9-point quadrature formulae applied in the intervals between two consecutive zero values of the integrand. Zero values of integrands are found from Cos sin(x sec e) = O Whenever the area under a half-cycle is smaller than a specified number (MUMU), the integration is automatically terminated. The error is always smaller than that due to the oscillation. 36

ICOMPILE MAO, PX'rINT UDJLCT, DUMP, PUNCH OBJECT PBAR C 75875 12/1K/64 1 51 8.9 AM MAD (24 SEP 1964 VER~SION) PROGRAM LISTING. EXIORNAL FUNCTI ON (KKFDL L.F,MUMU) *)i CALCULATION Of GENERALIZED HAVELOCK P FUNCTION BY NUMERICAL INTEGRATION ENTRY TU PSVIN. *C02 NA P = 2. -i.1P2 81 2 B * 3 LOW3f<=. *CC 4 A = F /3 * 415 927 MA=A+C.9999999 *Q0 7 THROUGH LOOP A2, FOR M=MA4-lM.G.6Vrr' *(^ UPPEP=ARCCOS. (A/M) 9 AREA *1 THROUGH LOOP Al, FOR K=L,1,K.G.9 *011 X = ( (iA+.HG(1 -K) )*I U P PER + ( 1. - 1 QG -K)) L OWE R). 5' *12 COSINE=CSLiS. ( * 13 S I4 =SI1 N. IX) *014 T ANOGTS= IWEICUS IE *) 1 5 c L T TA, G, T * T AN Gr T *16 WHENEVER C.G.3Y. *017 AREA=' *018 TRANSFER TO El *;19 tND OF CroNOiITIO'NAL 2 Z'* EPC=NAPE.P. C *-21 COSPR=COSINE.P. KK *.j 22 WHWYNE-Vt:R KK.L.ki COSPR=1.1' *223 IITGRO=COSPR/LPC:*S I ~.(FF-/COSINE) *024 LOOP Al AR E A= AR-E A + I ('(K)*INTG RD*(UPPER-LOWER)* 93.5 * 25 El LIMIT=A REA *26 WqH&'EV ADS (IHMIT).LE. MUMO, TRANSFER' TO LOOP A3 *27 LO i LR -UP PLR *28 LOO 0 P A?2 SUM= SUIA+ARr A *029 LOOP A3 ShUM-iJM*(-l ).P.(KK/2 ) *; 35 FUAC I(:;~I R- FUrR:J SUM *031 L ATRY TU DC,EOS. *032 NAAE=2.171828192 *0j33 LOwE- = ". *.' 3 4 SUO A= *J 3 5 AVF'3. 141Y)21F *036.1AA=A+'.4)9q9 *9f 37 THROU'lhi LOOP AU-, FOR" M=M4,.l l,M.G.6,,: *038 TH -tJGiI LO: A4, FUOR K=Ill,,K. G.9 U' 41 < = ( (l1. ~ i;f- ( I'-K) )*UIPPEW+(1.-HG(1 -K))*LOWE R)*.5 * 42 "h I' E=CO iS. (X) * 43 I *"44 r~4. TF=S 1 ~4:/rf) S I A F 4 5 C1 ~LVLR,3').47 a~~~~~~~_2=~, +~~~~~~~~~~~~~~~~~~~~,48 I' F'J i T!I E 4 49 4 1 2JHD[IIYJAL *. TI = I 51

COSPRtCOS(NE.P. KK *052 WHENEVER KK.E.O, COSPR*1.0 *053 INTGRD=COSPR/EPC*COS. FF/COSINE) *054 INTGRD:COSPR*COS (:FF/COS. (X)) *055 tOOP A4 AREA-AREA+IG(K)*INTSRD*(UPPER-LOWER)*0.5 *056 E4 LIMIT=AREA *057 WHENEVER LABS. (LIMT).LE. MUMU, TRANSFER TP LOOP A6 *058 LOWER=UPPER *059 LOOP A5 SUMbSUM+AREA *060 COOP A6 SUM=SUM*(-1.).P..((KK+1)/2 ) *061 FUNCTION RETURN SUM *062 INTEGER KK,MA*M,K *063 DIMENSION IG(lO), HG(10) *064 VECTOR VALUES HG(1) =.96816024,.83603111.,.61337143,.32425342`065 I,.U, -.32425342,-.61337143,, *065 2 -. 8360311 1,-. 9681 6024 *065 VECTOR VALUES IG(1=.o08127439,.18064816,.26C61070,. *066 1.31234708,..33023936,.31234708,. *066 2.2606107G0,.18064816,.08127439.066 END OF FUNCTION.067 OD

HULL FUNCTION The program below describes the calculation of the coefficients of the polynomials in the variables I and: given by Equations (23) and (24). The form of these expressions are such that a flow diagram is superfluous. 39

SCOMPILE MAD, EXECUTE,OUMPPRINT OBJECT,PUNCH OBJECT HULL 0130C3 12/09/65 5 01 4e.3 Pt MAD (09 AUG 1965 VERSION) PROGRAM LISTING. THE HULL FUNCTION POLYNOMIAL MICKELSEN AND KIM EXTERNAL FUNCTION FACT.(N,P) INTERNAL FUNCTION F.(KKK)=FACT.(1,KKKK) Ccl INTERNAL FUNCTION FAT. (I IJJ)=F. (II )/(F. (JJF.(II-JJ)) _ _ __ 02 PRrNT COMMENT $15 _ _. _ ____ 3CC3 PRINT COMMENT $8 HULL FUNCTICN$ *C4 PRINT COMMENT $8 *CC 1 FINN C. MICHELSEN AND HUN CHOL KIMS *CC5 PRINT COMMENT $1$ __*CC_ 006 PRINT COMMENT $4 GENERAL INPUTS'0C7 START READ AND PRINT DATA, MM_tNN_NSWSW(__1 ) ~ *C08 EXECUTE FTRAP. CC 9 FV.=NN+1 *C'1C AM=2'MM+1'C11 BM=2 NN+1 *012 AM1=MM *013 ____ BM1=NN ________ *C14 DIM(1)=NN+2 *C15 o OIM(2)=NN +1 _ C1 DIM1( I )=AM+2 iC17 *C19 *C2C'021 C1'C22 C2 *C23 G2 PRINT COMMENJI $S _ L. *C24 C2 PRIN-L COMMENT $8 POLYNOMIAL CCEFFICIENTS ARES C25 C2... C26 C2 PRINT RESULTS C(O,O)...C(MM,NN) C27 C2 THROUGH LOOP 1,_.FOR' A=C,t,A.G.AM *C28 C2 THROUGH LOOP 1, FOR B=C,1, B.G. BM C2S C3 AAB2(AlB.)=O., _ ___ __ _C3C 04 LCCP 1 AAB(A,8)=O. C31 04 THROUGH LOOP 2, FOR 1=0,1, I.G. AM1 C32 C2 THROUGH LOOP 2, FOR J=C,I, J.G. BM1 *C33 C3 THROUGH LOOP 2, FOR M=-0,1, M.G. AM1'C34 04 THROUGH LOOP 2, FOR N=C,1, N.G. BM1 C35 05 WHENEVER C(IJ).E.O..OR.C(M,N).E.0., TRANSFER TGENTR2 _ 036 C6 THROUGH OOP 2, FOR L=C,1, L.G. (1+1) C37 06 THROUGH OOP 2, FOR K=C,1, K.G. M C38 C7 TEMP= C(l,J)*F. (I+1)*(-1.).P.L*'C3 ce 1.5.P.(I+L-L)/(F.(L)*F.( [+1-L))*C(MN)*F.(M)*F.(N)/(FACT.(I+l, C3S 2 M+1)*FACT.(J+1,N+L))FAACT.(I+1,K)*.5.P.K/F.(K) C03 WHENEVER SW(1).040 0. PRINT FORMAT CHK, F.(I+1), (-1.).P.L,.5.P.(I+1-L), F.(L), 41 C 1 F.(I+1-L), F.(M), F.(N), FACT.(I+1,M+1), FACT.(J+1, N+1), *041 2 FACT.(I+1,K),.5.P.K, F.(K)'041

VECTOR VALUES CHK=$S2,12E(.3*$ *C42 Cl CS PRINT RESULTS AAB(L-K+MN+J+1), C(IJ), C(MN) *C43 Cl CS END CF CONCITIONAL *+'. Cl C8 GOP 2 AAB( L-K+MN+J+1 )=AAB(L-K+MN+J+1 )+TEPP *C4.5 Ca ENTR 2 CONTINUE *C47 C6 LOOP 2 CONTINUE *C4e 06 THROUGH LOOP 3, FOR I=C,1,I.G. AMl *048 02 THROUGH LOOP 3, FOR J=0,1,J.G. BM1 *C49 03 ____RQUGH LOOP 3 -FR M=GClt M.G. A*l *C5C C'. THROUGH LOOP 3, FOR N=Ctl, N.G. 3M1 *051 C5 WHENEVER C(IJ).E.O..OR.CCW,,N).E.C.,TRANSFERTCENTR3 *052 C6 THROUGH OOP 3, FOR K=0,1, K.G. P *C53 C6 THROUGH COP 3, FOR P=Cl, P.G. K *C5' 07 TEMP= (-1.).P.I*.5.P. (I+1)*(II,,J) *055 C CCMNr)*F.C (M) *F. (N) /(FACT.( I+1,M+1)*FACT. (J+1,IN1) ).(-1. )P.P* *C 5 2 FACT.(I+1,K)*.5.P.P/.(F.(P)*F.(K-P)) *C55 00P3.AAB(M-PN+J+)=AAB(M-PN+J+1)+TEMP *0C56 C ENTR 3 CONTINUE *C57 06 LOOP 3 CONTINUE *C5e C6 THROUGH LOOP 4i FOR I=CI. IG.-AML C59 C2 THROUGH LOOP 4, FOR J=O,1,J.G,.BM *C6C C3 THROUGH LOOP 4, FOR M=C91vM.G. AMIl *061 C' THROUGH LOOP 4, FOR N=C,1,N.G. BMI *C62 05 WHENEVER C(IJ).E.0..OR.C(MN).E.0.,TRANSFER TCENTR4 *063 06 THROUGH COP 4, FOR.K=Ot.G M *C6' 06 THROUGH OOP 4, FOR P=C,1,P.G.N *065 07 THROUGH COP 4, FOR L=C1,L. G.II+K+1) *C66 CS TEMP= -C(ItJ)*C(MtN)*-1.).P. *C67 H-i 1 (I+K+P+L)*FA T.(NP)*FA T.(MK)*FA T.CI+K+iL)*.5.P.L/ *C67 2 CCI+K+1)*(J+P+l)) *C67 WHENEVER SWCI) *065 CS PRINT RESULTS -C(IJ),C(M,N),(-1J..P.CI.K+P+L) FA T.CNP),FA *06s 01 CS 1 T.CMK),FA T.(I+K+1,L),.5.P.L/ CI+K+l)CJ+P+1) ) *069 END OF CONDITIONAL 0C7C Cl CS COP 4 AAB2CM+I-L+1,N-P)=AAB2(M+I-L+1,N-P)+TEMP *C71 O1 ENTR 4 CONTINUE *072 06 LOCP_ 4 ONTINUE *C73 C6 THROUGH LOOP 5, FOR I=C,,Iv.G.AM1 *C74 02 THROUGH LOOP 5, FOR J=O,1rJ.G.BM1 *C75 03 THROUGH LOOP 5, FOR M=ClM.G. AL *0C76 C' THROUGH LOOP 5, FOR N=C,1,N.G. BM1 *077 CS WHENEVER CC IJ).E.O..OR.CCMt),NE.O.TRANSFERTCENTR5 *C78 C6 THROUGH 0OP 5, FOR K=C,1,K.G. M *C79 06 THROUGH GOP 5,'FOR P=C91,P.G.N *080 07 THROUGH COP 5, FOR L=OC1,L.G.(I+K+1) l0S1 08 THROUGH OOP 5, FOR Q=C1,Q.G.(J+P+l) *C82 C2 TEMP= CCIJ)*C(tMN)* *053 10 1 (-l.).P.(I+K4P+Q+L)*FA T.CN,P)*FA T.(MK)*FA T.CI+K,1,L)* *c03 2 FA T.(J+P+1,Q)*.5.P.L/CCI+K+l)*CJ+P+1)) *083 COP AAHC+I+L+1-,N+J+1-Q)=AAB2CM+I+1-L,+J+l-Q)+TEP *C84 10 ENTR 5 CCNTINUE *506 LCCP 5 CONTINUE *CEC 06 THROUGH LOOP 6, FOR I=C,1,IIG.AM1 *087 02 THROUGH LOOP b, FOR J=CvlJ.G.BM1 *CE8 03 THROUGH LOOP 6, FUR M=C,1,M.G. AM1 *08 C'. THROUGH LOOP 6, FOR N=C,1,N.G. BMI *CS0 C5 WHENEVER CC IJ).E.O..UR.C(W,N)EO.,TRARSFERTCENTR6 *CS1 06 THROUGH LOP 6, FOR K=C,1,K-.G. M *02 06

THROUGH OOP 6, FOR P=0,tl,P.G.N *Cq3 C7 TEMP= C ( I, J)*C(M,N)*(-1.).P. (I+K+P)* *C4 C8 1 FAT.(N,P)*FAT.(M,K)*.5.P.( I+K+1)/( ( I+K+I)*(J+P+I)) *C4 LOP N-P)A2AB2((MM-K,N-P)=AAB2M-KN-P)+TEMP *C95 Ce ENTR 6 CONTINUE CS C6 LOOP 6 CCNTINUE *C7 C6 THROUGH LOOP 7, FOR I=C,1,I.G.AM1 *CS8 C2 THROUGH LOOP 7, FOR J=C,1,J.G.BM1 *CS 03 THROUGH LOOP 7, FOR M=0, 1,M.G. AMI *lCC C4 THROUGH LOOP 7, FOR N=C,i1,N.G. BM1 *101 C5 WHENEVER C{ I,J).E.O..OUR.C(IM,N).E.O.,TRANSFERTCENTR7 *1C2 C6 THROUGH UOP 7, FOR K=C,1,K.G.M *1C3 CC THROUGH OOP 7, FOR P=Q,1,P.G.N *104 07 THROUGH OOP 7, FOR Q=C, 1,Q.G.(J+P+1) *105 C8 ~_. ___ ~~_.....TEMP= C(I,J)*C(m,N)*(-1.).P. *1C6 C 1 (I+K+P+1+Q)*FA T.(N,P)*FA T.(M,K)*FA T.(J+P+1,C)*.5.P. * 1C6 2 (I+K+1)/((I+K+1)*(J+P+1)) *1C6 COP7 AAB2(M-K,N+J-Q+1)=AAB2(M-K,N+J-Q+I)+TEMP *1C7 C ENTR 7 CONTINUE *1C8 C LCOP' CCNTINUE *lC9 C PRINT COMMENT $1$ *11C C2 PRINT COMMENT$O RESULT FOR REGION 1$ *111 C2 *112 C2 *113 02 *114 C3 *115 C4 WHENEVER SW.AND. AAB(A,B).NE.O. *116 C4 PUNCH FORMAT UUTAtB,AAB(AB),LLK *117 C1 C4 END OF CONDITIONAL *118 Cl C4 -I:"t~ ~ PRINT RESULTS LL,K,AAB(A,B) *11S C4 Po *12C C4 PRINT COMMENT$1$ *121 C2 PRINT COMMENT $0 RESULTS FOR REGICN 2$ *122 C2 * 123 C2 *124 C3 *125 C4 WHENEVER SW.AND. AAB2(A,B).NE.C. *126 C4 PUNCH FURMAT OUTl,A,8,AAt2(A,B),LL,K *127 Cl C4 END OF CONDITIONAL *128 Cl C4 PRINT RESULTS LL,K,AAB2(A,B) *129 C4 *13C C4 *131 C2 TRANSFER TO START *132 ECOLEAN SW' *133 INTEGER MM,NN,FV,K,I,J,M,N,P,Q,L,A,B,IM,BF,AM1,BM1,KK *134 INTEGER I I,JJ,LL,PP,QQ *135 FORMAT VARIABLE FV *136 VECTOR VALUES OUT=$S10,4HAAB(, I2,1H,,I2,2H)=,E25.8,215*$ *137 VECTOR VALUES OUT1=$SIC, 5HAAB1(,12,1H,,I2,2H)=E24.8,2I5*$ *138 VECTOR VALUES DIM=2,0,10 *13S VECTOR VALUES DIM1=2,1,10 *14C VECTGR VALUES SW=OH,0,O0B *141 LIMENSION C(10CC,DIM),AAb(CO,DIM1), AAB2(5CC, CIMiVl) *142 CIMENSION Sw(10) *143 END CF P;:UGRAM *144

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