T HE UN IVERS I TY OF MI CHI GAN COLLEGE OF ENGINEERING epartment of Naval Architecture and Marine Engineering Technical Report A NOTE ON WAVE-MAKING RESISTANCE OF CATAMARANS Ryo Tasaki Project Director: R. B. Couch ORA Project 04886 under contract with: BUREAU OF SHIPS DEPARTMENT OF THE NAVY CONTRACT NO. NOBS 4485 WASHINGTON, Do Co administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1962

TABLE OF CONTENTS Page LIST OF FIGURES v Io INTRODUCTION 1 IIo WAVE-MAKING RESISTANCE OF CATAMARANS 3 1. Coordinate System, Notations, and Non-Dimensional Expressions 3 A. Coordinate System 3 B. Notations 3 C. Non-Dimensional Expressions 3 2o Formulas For Estimating Wave-Making Resistance 4 5. Formulas For Evaluating Wave-Making Resistance for Symmetric Ship Forms 4 IIIo EVALUATION OF FORMULAS 7 1o Resistance of Each Hull 9 2. Resistance Increase Due to Interference: Monotonically Increasing Term and Oscillating Term 10 IV. EFFECT OF DRAFT 15 V. CONCLUSIONS 17 APPENDIX: DERIVATION OF FORMULAS FOR ESTIMATING THE WAVE-MAKING RESISTANCE OF CATAMARANS 19 REFERENCES 23 iii

I

LIST OF FIGURES Figure Page 1. Coordinate system. 25 2. Monotonically increasing term, Il(n = 0; Ko, t = co, k) 26 3. Contribution of I,, 12, and 13 to the resistance increase due to interference for a given advance speed. 27 4. Oscillating terms, I2(n = 0; Ko, t = 0, k = 0.3) and I3(n = O; Ko, t = oo, k = 0.3). 28 5. Contribution of I,, I2, and 13 to the resistance increase due to interference for a given distance between two hulls. 29 6. Fundamental term, oscillating term, and total wave-making resistance coefficient of each hull. 30 7(a). Factor representing the draft effect: Z2(G) = (1-e-Kot sec G)2. 31 7(b). Integrand of one of the oscillating terms, i.e., I21. 31 8. Interference term plotted against the distance parameter 2k/Lo 32 9. Interference term plotted against Froude number for,T/L = 0. 33 10. Interference term against Froude number for T/L = O. 1 34 v

I. INTRODUCTION For years, many sailing catamarans have been built, and because of their good reputation their numbers are increasing. Few large sea-going catamarans have been built, however, and very few are actually in service,1-3 This does not mean that interest in sea-going catamarans has been meager. The catamaran has certain recognized advantages, such as a high degree of stability, large deck area, etc., and from time to time independent studies appear in the literature. For instance, W. H. Michel discusses the features and feasibility of ocean-going catamarans from a practical point of view, and offers a particular design of a catamaran for an oceanographic research vesselo A number of theoretical papers discussing the resistance of catamarans are also availableo In 1946, M. Kinoshita and S. Okata calculated the wave-making resistance of a catamaran of Weinblum's mathematical ship form,5 In 1950, K. Yokoo and R. Tasaki carried out theoretical calculations of simple ship forms, varying draft and distance between two hulls for speeds up to a Froude number of 0.5.'7 This study was followed by an experiment in which models corresponding to the theoretical ship forms were used, The study showed fairly good agreement between the theory and the experiment and it shed some light on the interference effect between two hulls. This experimental study also included hull forms which were asymmetrical, with respect to their centerlines, i.e., the waterlines were half-moon shaped, Asymmetrical hull forms of this type were found to be inferior to the symmetrical hull forms, howevero6 In 1955, K. von Eggers carried out theoretical and experimental studies on catamarans as well as on two ships in tandem, etc, His studies included the effect of shallow water and remain the only source of information available for such an effect. In previous papers by Yokoo and the present author,6,7 the general characteristics of the wave-making resistance of a catamaran have been presented. The present paper attempts to describe these characteristics in further detail. It will be shown that the interference effect between two hulls can be divided into two parts: (1) the monotonically increasing part, and (2) the part which oscillates with respect to the increasing speed or to the decreasing distance between the two hullso Numerical examples of these components as well as of total resistance are given for simple hull formso In previous papers, fairly good agreement between theory and experiment has been established for the models corresponding to the theoretical forms, 78 These forms may not be altogether suitable for a practical catamaran, but a theoretical representation of practical hull forms and a calculation based on such a representation are not yet at hand. At this stage, therefore, an inde1

pendent experimental study of practical ship forms, guided by theoretical studies such as the present one, is a necessity for design purposes. Experiments on such conventional hull forms are currently underway at The University of Michigan and the result will be available shortly. The present study, University of Michigan ORA Project No. 04886, was sponsored by the Bureau of Ships, U.S. Navy, Contract NOBS 4485. 2

II. WAVE-MAKING RESISTANCE OF CATAMARANS 1. COORDINATE SYSTEM, NOTATIONS, AND NON-DIMENSIONAL EXPRESSIONS A. Coordinate System The coordinate system to be used is defined in Fig. 1. The origin of the coordinate system is at the midship, the still-water plane, and a center plane equidistant from both hulls. The positive x-axis is in the direction of advance and the positive z-axis is directed upward. B. Notations The notations to be used are defined as follows: 2. length of ship; 2. = L 2kA distance between the centerlines of the two hulls V advance speed of the catamaran V m(,Tq,t) source distribution, strength of source (out-flow per unit tim) at a point (~,T,5) T depth of source distribution R2w wave-making resistance of the catamaran Row wave-making resistance of each hull of the catamaran 2Rw increase in wave-making resistance due to the interference between the wave systems of both hulls G direction of propagation of elementary waves; zero angle is on the negative x-direction. C. Non-Dimensional Expressions = x/~, 5 = y/l, t = T/~ Ko= - = g — where F is Froude number V2 2F2,W - VR2w - Row, Rw C2w = V2L ow = 1 V2L2' w 1 PV22 3Z --

The first approximation to the wave-making resistance of a catamaran is obtained by superimposing the velocity potentials or free wave patterns of the two hulls, as shown in the Appendix, and assuming that the boundary condition is not disturbed by the presence of the other hull. This approximation shows fairly good agreement with experimental results.1'2 In fact, the presence of the other hull has only a secondary effect on the wave-making resistance, as is explained in the following section. One method of correcting the disturbance on one hull caused by another hull is to superimpose upon the original source distribution a suitable distribution of doublets oriented perpendicularly to the direction of advance. Since Lagally's theorem shows that no additional force in the direction of advance is generated by these doublets, their influence is necessarily of secondary nature. 2. FORMULAS FOR ESTIMATING WAVE-MAKING RESISTANCE The total wave-making resistance, R2w, is the sum of twice the resistance of each hull, Row, and the resistance increase due to interference between wave systems of both hulls, 2Rw (see the Appendix). Thus: R2w = 2Row + 2Rw (l) where Tr2 Row = Kp0V2- \ (P2 + Q2)sec3GdG, (2) ir/2 Rw - P cos(2Kok tan Q sec 9)(P2 + Q2)sec3GdG (5) P o Cos - / / (,(Ko0 sec G)ddt. (4) Q -t L1 sin 5. FORMULAS FOR EVALUATING WAVE-MAKING RESISTANCE FOR SYMMETRIC SHIP FORMS In the case where the sources are distributed on the center plane of each hull, antisymmetrically with respect to the midship section and uniformly in the direction of depth, the wave-making resistance is given by the following formulas. This distribution of sources represents the wall-sided ship form of Michelts theory; waterlines are symmetric with respect to the midship section. When the new expressions 4

=Z l-e-Kot sec29 Z = 1 - e (5) and 1 M = o m(t)sin(Ko sec2e)d (6) are introduced, Eqs. (1) through (4) become R = pV2L2 M2Z2cos GdG (7).22vL Tr/2 R = cos(2Kok tan 9 sec G)M2Z2cos dG. (8) 5

IIIo EVALUATION OF FORMULAS The following expression of M2 in a power series of (Kosec 9) is obtained by successive partial integration of M, which has been applied to the evaluation of Eq. (7) by T. Inuio9 00 _M2 21 1Z y (-1) n+i mimj 2 Ksec2 n=o (2n=i+j) (Kosec 9)2 00 1 cos(2Kosec 9) n mim 2 Ko2sec2G n=o (2n=i+j) (KOsec 9)2 00 1 sin(2Kosec 9) 7 7 n+l mimj 2 Ko2sec2G L L (-1) 2n+l 2 0 K 02eC2n=o (2n+l=i+j) (Kosec ) 2n where mi = mi) (l); i.e, mi is the value of the ith derivative at the bow, t = 1o By inserting Eqo (9) into Eqs. (7) and (8), the wave-making resistance coefficient of the catamaran is given as follows: -C2w = 2Cow + 2Cw (10) Jt0/2 + (-1)i mimj Z2cos(2Kosec 9)cos2n+3Gde (2n=i+j) 1 n+r1/22Gn (1 K - mimj m Z2sin(2Kosec G)cos2n+4d (11) (2n+l=i+j)

oo c - 2 (_1) 2n Ko2 Ko2n n=o -r/2 x j mmimj Z2cos(2Kok tan 9 sec G) cos2n+3dG L_ 2n=i+j) ir/2 + (-l) i mimj J Z2cos(2Kok tan 6 sec ) cos(2KOsec ) cos2n+3GdG (2n=i+j) rt/2 1o 7 mimj o Z2cos(2Kok tan G sec ) sin(2Kosec G)cos2n+4GdG K0 L- m tan G sd (2n+i=i+j) (12) These expressions show that in this case the wave-making resistance is easily calculated, if the following integrals are known. st/2 2cos2n+3dG, n = 0,1,..., (13) 0 Jo Tt/2 Z2cos(2Kosec G)cos2n+3GdG, n = 0,1,..., (14) o /22 2n+4 Zsin(2Kosec G)cos2n Gd, n = 0,1,..., (15) o and j/2 I(n; Ko,t,k) = Zcos(2Kok tan 9 sec G)cos2n+35Gd, n = 0,1,..., o (16) (t/2 I2(n; Ko,t,k) = Z2cos(2Kok tan O sec )cos(2Kosec )cos2n+53d, n = 0,1,..., o (17)

7t/2 I3(n; K,t,k) = Z2cos(2Kok tan G sec G) x sin(2KoSec g)cos2n+4gd9, n = 0,1,... (18) 1. RESISTANCE OF EACH HULL The first part of the wave-making resistance, the sum of the resistance of each hull, has the same characteristics as the wave-making resistance of a single hull. The integral in Eq. (13) is the so-called fundamental term of the wave-making resistance and is evaluated by the following formulas.9 J/2 - 1 Z2cos 2n+3 Un+l(Kot) = / ZCos2n+3Gd Cn+l, = 1 - 2En(Kot) + E(2Kot) (19) where Cn+l - cos2n+3GdG = 2.4.6 (2n+2) (20) 2o 35.5.6., (2n+3) 1 <2 f + (V sec2dcosn GdG En+i(Ko,t) e (21) Cn+l o The integrals in Eqs. (14) and (15) are the interference terms.9 (In the following discussion they are called the oscillating terms to distinguish them from the resistance increase due to the interference between two hulls.) For 2Ko > 5 or F < 0.45, the method of stationary phase leads to good approximations as follows: r/2 Z2cos2n+39 cos(2KoSec G)dG - (i/4Ko)2 Zo2cos(2KO + I) (22) 0 casTo+4e sin(~%sec e~ae -J4%)~ Zo~sin~eK, + Z0 IT) r/2 z Z2os 2n sin(2Kosec 9)dG (Tr/4Ko)2 Zo2sin(2Ko + ) (23) 9

2. RESISTANCE INCREASE DUE TO INTERFERENCE: MONOTONICALLY INCREASING TERM AND OSCILLATING TERM In the author's previous paper6'7 the resistance increase due to the interference effect given by Eq. (8) was calculated graphically and was not separated into three integrals Il, I2, and I3, as it is in this paper. These integrals show the general characteristics of the interference effect in the wave-making resistance of catamarans, The integral Il corresponds to the integral (13) but is not represented in an explicit form such as Eq. (19). The integrals I2 and I3 correspond to (14) and (15), respectively, and are transformed as follows: 1 I2(n; Ko,t,k) = 2 (I21 + I22) where n/2 I21(n; Kot,k) = Z2cos(2Kohl)cos2n+3GdO, it/2 I22(n; Ko,t,k) = Z2cos(2Koh2)cos2n+3dG (24) and I3(n; Ko,t,k) = 2 (I31 + 132) where Zr 2 2n+4 Is2(n; Ko,t,k) = Z cos( Khl) cos2n+40dO, I32(n; Kotk) = / Z2cos(Koh2)cos GdG (25) with h, = (1 + k tan O) sec G h2 = (1 - k tan ) sec G (26) In these expressions h1(~) does not have any stationary point, but h2(G) has two stationary points between G = 0 and ir/2 when k f 0 and k < 1/2 P =03554, 10

Therefore, it seems possible to apply the method of stationary phase to the evaluation of the integrals 12 and I3. Unfortunately, however, the stationary phase method does not give good approximations for practical values of Ko. For example, 12 and 13 have large values for the practical values of Ko even when k > 1/2 42, as shown in Figo 35 In this paper, therefore, graphical integrations were carried out for several values of Ko, t, and k. Now, for simplicity, Eqso (11) and (12) are examined for the linear distribution of sources; that is, m( ) = al~ This expression corresponds to the wall-sided ship form, with parabolic water lines, of Michel's theory, In this case,!l = a1, and m2 = m3 = = 0 Equations (11) and (12) now become Cow = al2F [{C3U3 + (f F) C5UJ}-{)2 Zo2F(1 - (2 F)4 cos(2Ko + ) - (T)2 Z0F( F)sin(2K + 4 ] (26) Co = 4 al2F4 [I(n = 0) + (J F)4 Il(n 1)) + (I(n = 0) - (N F)4 I2(n = 1) - (2(1 F)2 I3(n = 0))] (27) Equation (27) shows that examination of the integrals Il(n = 0), I2(n = 0), and I3(n = 0) gives some general information about the resistance increase due to interference when the Froude number, F, is not large. The integral Il(n = 0; Ko,t,k) is a monotonic function which decreases with an increase in (Kok), The numerical value of the integral for infinite draft, that is, Il(n = 0; Ko,t = ook) is shown in Fig. 2. The integral I, corresponds to the fundamental term of the wave-making resistance of the singlehull ship and is same as the fundamental term of each hull when k is equal to zero. It should be noticed that this term increases rapidly when the distance between two hulls, k, decreases, or when the speed of the ship, F = l1/2~f, increaseso 11

The integrals I2 and 13 are oscillating functions with respect to Ko and k, which correspond to the oscillating term of the single-hull ship. In Fig. 3 the contribution of these integrals I1, I2, and 13, to the resistance increase due to interference between two hulls is shown plotted against the distance between two hulls, k, for a given advance speed, Ko = 5(F = 0.316). When k = 0, from Eqso (16), (11), (22), and (23): I2(n = 0; Kot = oo,k = 0) nr/2 1 = 0 Z2cos(2KosecG)cos2n+3GdG = ( )21.cos(2K0 + ), n = 0,1,.. and I3(n = 0; Ko,t = oo,k = 0) p r/2 1 I Z sin(2KosecG)cos2n+4d ( )2..sin(2Ko +, n = 0,1,... The variation of the integrals I2 and 13 plotted against Ko are shown in Fig. 4 for infinite draft, t = o0, and for a given distance between two hulls, k = 0.35 That is, I2(n = 0; Ko,t = oo,k = 0.3) and I3(n = 0; Ko,t = o,k = 0.3) are shown. In this figure I21, I22, I31, and I32 are given by Eqs. (24) and (25); the variation of the integral I, is also shown for reference. When Ko = 0, from expressions (16), (17), and (21) the integrals I2 and 13 are: Dr/2 I2 = / lol.cos2n+3Gd = C3 = 2/3'Jo and Tr/2 Is 2 1+cos2n+3 d = 0 1.0.cos GdG = 0 ~o By comparing Fig. 3 with Fig. 4 it is noted that over the range of the practical values of k and Ko, the variation of the integrals I2 and 13 with respect to k (2k2 being the distance between two hulls) is much more gradual than the comparable variations with respect to Ko (Ko corresponding to inverse square of speed). Furthermore, it is seen in Fig. 3 that, for the small value of k, the monotonically increasing term, I1, is dominant. For example, in this case, k _ 0.27 this limit depends on the speed parameter, Ko (see Fig. 8). 12

In expression (27) the oscillating term is: I2(n = 0) - 2 (I F)2 I3(n = 0) This expression shows that the integral 12 is predominant in the oscillating term for small Froude numbers, Figure 4 shows that in this case the resistance increase due to the monotonically increasing term is more than offset by the negative resistance due to the oscillating term near the value of Ko = 4.8 and 6,9. In Fig. 5, the contribution of these integrals to the resistance increase is shown plotted against advance speed for the case of infinite draft and for a distance between two hulls, k, equal to 0.35 In this figure B = F4 I1 is the monotonically increasing term and C = F4(I2-2(1 F)2 I3) is the oscillating term with increasing Froude number. The maximum cancellation occurs when the amplitude of the oscillating term takes the maximum value. It is expected, therefore, from Figs. 3, 4, and 8 that the k = 0.3 would give the maximum cancellationo In reference to Fig. 5, however, it is noted that, compared with the oscillating term the monotonically increasing term takes so large a part in the resistance increase that it has a decisive effect on the wave-making resistance of the catamaran, In other words, the gain obtained by the effect of the interference is cancelled by the monotonically increasing term, In the case illustrated by Fig, 5, the range of negative increase of wave-making resistance due to the oscillating term covers Froude numbers from 0.30 to 0.36. This range is reduced to Froude numbers from 0.31 to 0355 by the monotonically increasing term, The amplitude of the oscillating term is reduced by half by the monotonically increasing term, In order to see the contribution of these terms to the wave-making resistance of catamarans, the wave-making resistance of each hull is shown in Fig. 6. (The scale is the same as that in Fig. 5). The monotonically increasing term, B = F4 I1 in Fig. 5, corresponds to the fundamental term, B = F4 C3U3 in Fig, 6. The oscillating term corresponding to C = F4{I2 - 2(,f F) 13 in Fig. 5 is the sum of 4 Tr 2 F4 () F cos(2Ko + ) and 15 F4() F(~ F)2sin(2o + 9), 13

i.e., C = D+E, in Fig. 6. Comparison of Figs. 5 and 6 shows that the magnitude of each term is comparable. In the present case, the resistance decrease is expected to reach 50% of the wave-making resistance of each hull. The ratios of the resistance increase to the wave-making resistance of each hull for several values of k are shown in Figs. 9 and 10, plotted against Froude number. It may be said, comparing Figs. 5 and 5 with Figs. 8, 9, and 10, that the results obtained by both methods show good agreement. 14

IV. EFFECT OF DRAFT The integrals in expressions (13) through (15) and (29) through (34) consist of the following factors: z2 ( l -Kot sec29) 2a) Z2 = (l - e (a) and cos2n+3C, cos(2Kosec2) cos2n+3G, sin(2Kosec2G) cos2n+4; cos(2Kok tan G sec G)cos2n+3G; cos(2K0hl)cos2n+39, cos(2Koh2)cos2n+30; sin 2Kohl) cos2n+3, sin(2Koh2) cos2n+4 (b) The first factor (a), represents the effect of the draft change and is shown plotted against G with respect to several values of Kot in Fig. 7(a). It is noted that Z2 is constant for small values of G and may be substituted by its value at G = 0, i.e., Zo2. The first of the factors in (b) is cos2n+30. The others oscillate rapidly for values of G close to T/2 and their envelopes are cos2n+3G or cos2n+4G, which become small for value of G close to jr/2 (see Fig. 7(b)). In other words, the factors in (b) contribute the most to the integral for the small value of G, for which Z2 is also very close to Zo2. The integrals are, therefore, almost proportional to the value of Z2 at 9 = 0, or Zo2. It is noted from Eqs. (25) and (26) that the ratio of the resistance increase due to the interference to the resistance of each hull, Rw/Row = Cw/Cow, is not affected appreciably by the draft change. In Refs. 6 and 7 these ratios for the length-draft ratio L/T = 20 and 10 are shown against advance speed and several distances between two hulls. These figures also confirm the finding noted above (see Figs. 9 and 10). 15

V. CONCLUSIONS lo The wave-making resistance of the catamaran can be divided into two parts, one the resistance of each hull and the other the resistance increase due to the interference between two hulls. 2. The resistance increase due to the interference can be further divided into two parts. The first part, the monotonic and positive term, is a monotonically decreasing function with respect to the increase of the product of speed parameter Ko and distance parameter between the two hulls, Kok or k/2F2. For a catamaran with small distance between the two hulls, the wave-resistance increase due to this term is dominant over the second part. The second part, the oscillating term, which oscillates with respect to the speed parameter K0o does not change rapidly with respect to the distance between the two hulls. 35. The ratio of the resistance increase due to the interference to the resistance of each hull is not appreciably affected by the changes in draft. 17

APPENDIX DERIVATION OF FORMULAS FOR ESTIMATING THE WAVE-MAKING RESISTANCE OF CATAMARANS T, H Havelock1 showed that when the free wave pattern behind a ship is given by Ir/2 Y(x,y) = (Flsin A cos B + F2cos A sin B + F3cos A cos B + F4sin A sin B)dG (1) where A = Ko sec 9 and (B = Ko Y sin G sec, the wave making resistance of the ship is r/2 R = pv2 (F12 + F22 + Fs2 + F42)cos3GdG. (2) 4 A 0 The wave pattern due to a point source at (Q~,kl,~) is MKo sec2 e -Ko> sec2 %ws(x,y; s~,ke,3~) = Ko sec e x cos[Ko((x - o)cos G + (Y - k)sin G)sec29]dgO If we put X1 = Ko0 sec 9 and K = Kok sin 9 sec29, then 19

ws(X,y; ~,k~, ) / mKo sec 2 ws(xY; ~Iekly) = MK _ sec3G e-KO~ sec2G xi' -/2 x [cos K[cos X1 cos A cos B + sin X1 sin A cos B - cos X1 sin A sin B + sin X1 cos A sin B) - sin K(cos X1 cos A sin B + sin X1 sin A sin B - cos X1 sin A cos B + sin X1 cos A cos B)]d0. Now we calculate the wave pattern due to two point sources which are symmetrical with respect to the x-z plane, that is, ws(xy) -- ws(Xy; tkQ) + ws(xy; +,-kl,) In this calculation the terms containing sin K cancel each other since cos K(k~) = cos K(-kR) and sin K(kQ) = -sin K(-kA). Then 2mKo I s3 KoS sec2G ws (x,y) = - secos K x (cos X1 cos A cos B + sin Xlsin A cos B - cos X1 sin A sin B + sin X1 cos A sin B)dG. For the source distribution over the center plane of each hull of the catamaran, the free wave pattern is 20

/2 -K 1sec ^w(xy) = --- t m cos K sec3O e x (cos X1 cos A cos B + sin X1 sin A cos B - cos X1 sin A sin B + sin X1 cos A sin B)ddfd9 o -Ko~ sec2G Now, cos K sec30 e, cos X1, sin X1, cos A, sin A, and cos B are even functions of 9, and sin B is an odd function of 9, so that the above expression is equivalent to /2r 0o 1 (xy) 4KO t m cos K sec39 e K sec2 x (cos X1 cos A cos B + sin X1 sin A cos B)dgd~dO. Therefore, F1,...,F4 in Eq. (1) are: o 1 F1 = - cos K sec3g m e-Ko~ sec29in Xdd IT *-t,-1 ~- cos K sec3 * Q F2 = 0 o 1 4Ko~ 0 sec2 F3 = cos K sec3G m e K e os X dds 4KoQ - cos K sec3 *~ P F4 = 0 Then, Eq. (2) leads to 21

1 r/2 R2w = TpV2 (F12 + F32) cos3dG fo 4pK02V22 cos2K(p 2 + Q2)sec3GdG pK 2V2L2 o /2.p.K02V2L2 - (1 + cos2K) (P2 + Q2)sec3 dG 7r 0 2Row + 2Rw where PK02V2L2 if t/2 Row K(L2 + Q2)sec3GdG and pKo2v2L~ /2, Rw = p 2V2L2 / cos2K(P2 + Q2)secGdG. 4t 022 22

REFERENCES 1o Ley, W, "The Unchangeable Shipo" MIT Technology Review, Jan. 1951, pol49o ("Castalia" built in England in 1874 for channel crossing to France, 300 fto catamaran; "Veturi"-Gar Wood's 188-fto target catamarano) 2o Saller, "Russisches Expussgleitboato" Werft Reederei Hafeu, Sept, 1, 1940, po 224, (Russian high-speed passenger boat for crossing the Black Sea from Kurorteu Sotschi to Suchumn) 35 Nihon-kokan Co. "The Pleasure Boat Kurakake-Maru " Fune-no-kagaku, Volo 14, Noo 12, Dec. 1961, p. 62. 4o Michel, W, Ho "The Sea-Going Catamaran Ship and Its Feasibility." International Shipbuilding Progress, Vol, 8, No, 85, Sept, 1961, p. 391. 5. Kinoshita, Mo, and Okata, So "Several Examples of Application of Theory of Wave-Making Resistance." Journal of Zosen Kyokai, Vol. 77, 1949o 60 Yokoo, K,, and Tasaki, R. "On the Twin-Hull Ship: Report No, 1," Report of Transportation Technical Research Institute of Japan, Volo 1, Noo 1, 1951. (Translated by Ho Co Kim, Univo of Micho Department of Naval Architecture, Oct. 1962o) 7. Yokoo, K., and Tasaki, R. "On the Twin-Hull Ship: Report No. 2o" Report of Transportation Technical Research Institute of Japan, Volo 3, No, 3, 19535 (Translated by Ho Co Kim, Univ, of Micho Department of Naval Architecture, Oct. 1962,) 80 Eggers, Ko vono "Uber Widerstandsverhaltnisse von Zweikbrper - schiffeno" Jahrbuch der Schiffbautechnischen Gesellschaft, 1955o 9, Inui, To "Study on Wave-Making Resistance of Ships," 60th Anniversary Series of the Society of Naval Architects of Japan, Volo 2, April 1957, po 197o 10o Havelock, To Ho "Wave Pattern and Wave Resistance," TINA, Volo 76, pp. 430-32, 23

z,e JI - 2k _ Xk) Fig. 1syt V Vm(C) T Fig. 1. Coordinate system. 25

1.0 C3 = 2/3 9.5 II 0 a \* c ~ I' 0 5 0 Kok Fig. 2. Monotonically increasing term, Il(n = 0; KQ, t = oo, k). 26

A= B + (D+E) 1.0 A= TOTAL B= MONOTONICALLY INCREASING TERM (D+E) = OSCILLATING TERM B = I(n=O Ko= 5 t Iok) A= Cw 7'r/a2 F4.2.4.5.6 -.5 - D= 12(n=0; Ko=5, t= O, k) -1.0 Fig. 3. Contribution of II, I2, and I3 to the resistance increase due to interference for a given advance speed. 27

12 (n=0; Ko,t= o, k=.3) 13 (n=O; Ko, t=ao k=0.3) 1.Or I 1.0'2= (l1i+2+22) 12 =2 (1,+l12).8.8 2/3.6 = =.6. ro 0 2 82k 8 Ko -.2\ /-.2 -,4 I ~~~~~1 -,~-2-.4 -.6 -.6.8 -.8 - I.OL -.0 Fig. 4. Oscillating terms, I2(n = O; Ko, t = o, k = 0.3) and I3(n = 0; Ko, t = oo, k = 0.3).

.03 B = F4I,(n =OKo, )= a, k.3).02 A C F= 2 2-(f2F)2 13} D =FI2 (n=O; Ko,t =:, k=.3) E = -F42(2F). 12 (n=O; Ko, t= oD, k=.3).01 I 0W~~~~~~~~~ I.1.2.3 5 I \ lI A =B+C - C= D+E A = TOTAL B = MONOTONICALLY INCREASING TERM C = OSCILLATING TERM.01 Fig. 5. Contribution of Il, I2, and I3 to the resistance increase due to interference for a given distance between two hulls. 29

.03 I.02 B=F4CU 3 I. \ A = Cow r/a D = F!(7/2)'/g F Cos(2Ko+-7/4) C= FE(7r/2) F Cos(Ko+r/) 2 I + J2 F)2 Sin(2Ko + 7r/4)} j | / = F(7rN/2)D FM L F) TSin(2Ko+ER /.01 30 0 "^ ^.2 \ / 4.5 A =B + C \ C = D + E A = TOTAL B = FUNDAMENTAL TERM C = OSCILLATING TERM.01 _ Fig. 6. Fundamental term, oscillating term, and total wave-making resistance coefficient of each hull. 50

Kot =:) 1.0 Kt.8. 0 30~ 60~ 90~ e (a) Fig. 7(a). Factor representing the draft effect: Z2(9) = (1-e-Kot sec ) 2 I.0.8.4 - X \. Z2Cos(2Koh,) Cos S.2 / FOR t= oKo=5, a k=O.3 0 I30$ 1190~ -.2 / -.4 -.6 -.8 -1.o (b) Fig. 7(b). Integrand of one of the oscillating terms, i.e., I21. 31

6 in i^^' oJ N - - -r 3O0 -~ O c+~C o ) 0 0 o) O 0 0 0 C) 0 C) C) o 0 o o ~ H - 0......... -- -- | -- --- --- -- ----- | ----------- C Oq O~ 0 O 0 (D' I'O!)I —-do d-,. OH~ P. D 0 op 0 CD' CD c+CD ~~d. H~. c'l- F., c+,0 0 ~<~ g ~.~^~ ^ 7 I-'b ('3 I-D. ( (D C (' 0') t~D ~d H' 0) 4=- c+,^-.^ —^^ ^ ^ ^.^^^ /C /A /^~~~~~~~~~,,, CD C N. (D cD % * (D<^^ " < /^/'d HCl D~i PiC Cl D ) I'd) ~~~j^ -X / (D. 0~~~~~~~ c-: pi C ^^z1 c - K ____________ -:.. ____\. ^ \ _ ____ / / _ ____________ s(s C>/ 7^ ~ H - \(D\ \ - CD Fl^ \ W o (D Ft c - CO (D (D ~ 0 ---------------- I --------— ______________

100 (Reproduced from Fig. 5, Ref. 6. 2k/L and Ko in the figure corre-.2-,3.4,5 FRO DOE_ N UM&EP~ Fig. 9. Interference term plotted against Froude number for T/L = 0. (Reproduced from Fig. 5, Ref. 6, 2k/L and Ko in the figure corresponds to k and 2Ko in the present paper, respectively.) 33

100 40 -20 -60 -80t -100. -.....,, Fig. 10. Interference term against Froude number for T/L = 0. (Reproduced from Fig. 35, Ref. 7. 2k/L and Ko in the figure corresponds to k and 2Ko in the present paper, respectively.) 34