THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING WAVE RESISTANCE SOLUTION OF MICHELL'S INTEGRAL FOR POLYNOMIAL SHIP FORMS Finn C. Michelsen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 July, 1960 IP-447

Doctoral Committee: Professor Richard B. Couch, Co-Chairman Associate Professor Hadley J. Smith, Co-Chairman Professor Kenneth M. Case Professor Charles L. Dolph Associate Professor George L. West, Jr.

ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor Richard B. Couch and Professor Hadley J. Smith, who, along with other members of his doctoral committee, generously gave suggestions and encouragement during the course of the work presented herein. A debt of gratitude is owed to Professor Russell A. Dodge who made it possible for the author to continue his graduate studies. He is also indebted to Professor Georg Weinblum of the University of Hamburg who, during his visit to The University of Michigan during April of 1960, discussed with the author the many aspects of the theoretical wave resistance theory and reviewed this thesis. The author was assisted financially during the major part of his work by a National Science Foundation Faculty Scholarship. ii

TABLE OF CONTENTS Page ACKNOWEDGMENTS................................................... ii LIST OF FIGURES.................................. iv LIST OF SYMBOLS...................................... v CHAPTER Is INTRODUCTION...................................... 1 Statement of the Problem................... 1 Historical Background................... 6 II. THE MICHELL INTEGRAL.................................. 10 Boundary Conditions........................~......... 10 The Wave Resistance Integral......................... 15 III. SOLUTION OF MICHELL'S INTEGRAL FOR POLYNOMIAL SHIP FORMS... 24 Introduction..........*...................24 A Transformation of Michell's Integral................ 25 The Hull Function......**.....*...................... 27 The Michell Function........................ 34 Wave Resistance.................................37 IV. CONCLUSIONS..........................................4....45 APPENDICES APPENDIX I. THE HULL FUNCTION FOR A SIMPLE SHIP FORM.... 47 APPENDIX II. A BESSEL FUNCTION RELATIONSHIP..................50 APPENDIX III. NOTES'ON THE CONFLUENT HYPERGEOMETRIC FUNCTIONS...................................52 APPENDIX IV. CONVERGENCE OF SOLUTION..................... 54 BIBLIOGRAPHY, 4......................ii...........i *. 56 iii

LIST OF FIGURES Figure Page 1 Coordinate System Fixed in Stationary Ship in a Uniform Flow.................................. 12 2 The Hull Function for G(u,w) = (1-Lw)(l-4u2).......... 49 iv

LIST OF SYMBOLS anp' a Fourier expansion coefficients mnp' mn c constant velocity of the ship f Froude Number f(x,z) ship-surface slope function g acceleration due to gravitational field g(x,z) ship-surface equation h(uv), h(u,v) non-dimensional ship-surface slope function 5h(u,v) small change in h(u,v) m, n, p Eigenvalues p fluid pressure 5p fluid pressure on ship surface caused by waves po atmospheric surface pressure s, t variables in the Michell Function u, v, w perturbation velocity components of fluid u, v, w fluid velocity components with respect to a fixed coordinate system A A I u,u,w,,w non-dimensional variables of integration x, y, z moving coordinate system A A x, z variables of integration z(x,y) Free surface equation Arn, Brn Fourier expansion coefficients v

AI A" Hull Function polynomial coefficients B Half-breadth of ship C(s,t) Michell Function Cw wave resistance coefficient 5C_ differential wave resistance coefficient [AtCw]1i [6ACw]Ii contribution to wave resistance coefficient from terms of Hu11 Function polynomials of power a and P D Draft of ship F inverse square of Froude Number F l+v;x2(+l1 a generalized hypergeometric function o;Li+v> 4. G(u,v) non-dimensional ship surface equation H(, ),H(ty) Hull Function H (x) Struve Function V 5SH( ~, ~ ) small change in H(te) [H(, I,,[H(S, )]II Hull Function symmetric wavemaking integral ]il 1,1Y I3 auxiliary functions J unsymmetric wavemaking integral J (z) Bessel Function of the first kind Ko, K1 modified Bessel Functions of the second kind L Total length Of ship L' Half length of ship Vi

M, N, M, N Highest power of the variables in the Hull Function polynomials M(^) ( e) special function Rw Wave resistance I, T, Z Stationary coordinate system YO Bessel Function of the second kind a, a non-negative integers in Hull Function polynomials a, 3 Phase angles J Euler's constant e Phase angle,,, T non-dimensional variables e, X variables of integration P Density of water cp(x,y,z) Steady state perturbation velocity potential -q(X,Y,ff,t) Time dependent velocity potential r(a) Gamma Function A, A1, A2 Differential areas on hull surface ~(a, c, t) Confluent Hypergeometric Function Y(a,c,t) Confluent Hypergeometric Function vii

CHAPTER I INTRODUCTION Statement of the Problem The determination of wave resistance of ship hulls is one of the most important and interesting subjects in ship theory. Model testing facilities were primarily built for the purpose of investigations of this part of ship resistance. Since Froude's time, an. immense amount of experimental results have been published. It can, however, be safely stated that the aim to represent the resistance of a ship in-terms of its form has not been solved by experimental methods in a general manner. The appreciable differences in resistance observed at times between ship models of seemingly insignificant variations of shape can, for example, not be satisfactorily explained on the basis of our experimental experience. Considering the complexity of even the simpler cases of wave phenomena, such as a sphere moving at constant speed and fully submerge, this state of affairs is not surprising. TO predict ship wave resistance phenomena in a rational manner, an analytic formulation; should become available, and this formulation should preferably be simple enough so that basic deductions can easily be made from it. This investigation has primarily concerned itself with the problem of providing a simple.. method for the evaluation of the ship wave resistance. In doing so, it is believed that further insight into -1

-2the more general problem of interpretation of results has been achieved. Michell's paper( ) on the wave resistance of ships moving at constant speed in smooth water of infinite depth was the first attempt to treat the wave resistance analytically. The basic.assumptions made by Michell in his investigation were as follows: 1. The wave heights are small compared to wave lengths. Thus particle velocities due to wave motion are so small compared to the ship's speed that second order terms in velocities can be neglected. 2. The effects of trim and sinkage are not sufficient to affect the wave motion appreciably. 3. The angles made by the hull surface with the center line plane (longitudinal plane of symmetry) are everywhere small. 4. The motion has persisted long enough so that a steady state has been reached. 5. The fluid is non-viscous and the motion has started from rest. Thus the flow is considered irrotational. 6. The free surface conditions are to be satisfied at the undisturbed water level (e = 0). 7. The boundary conditions to be satisfied on the hull surface are assumed to hold at the center line plane, and only the velocity component perpendicular to this plane is accounted

-3for. The vertical slope of the hull surface is neglected. To this list, one should add that it is explicitly assumed that the total resistance can be broken down into three major components: (i) Wave resistance (ii) Frictional resistance due to viscosity of the water (iii) Eddy-making or viscous form resistance and that there are no interaction effects between these components. It is thus clear that Michell's theory of wave resistance is a linear theory and is theoretically valid only for an infinitesimally thin ship. During the past thirty years, much effort has been exerted in determining the applicability of Michell'8 integral expression for wave resistance of ships. In principle, two major questions have been asked: (a) Does the Michell Integral represent the wave resistance of common ship forms with reasonable accuracy? (b) Can the integral be evaluated for real hull forms in a reasonable time? Question (a) can be said to have been answered in the affirmative by several investigators such as Havelock(3'4Y5'8'9) Wigley(10,1112,14 16), Weinblum(20), Lunde (18,24), Shearer(31). It may be argued that quantitatively the theory does not give sufficiently accurate results. A great part of the discrepancies between theory and experimental results has been shown to be due to neglect of viscous effects.

-4The effects of viscosity may be summarized into the following two items: 1. In a viscous fluid the wave amplitudes decrease as the waves propogate. 2. Due to the presence of the boundary layer the effective wave making form differs from the actual form of the ship. Furthermore, behind a ship, a portion of the regular free waves propogate in the ship's wake, and their speed of advance is slowed as much as the wake's speed. Their wave lengths, are consequently shortened to some degree. (45) Inui(3) has pointed out that if the Michell theory is extended to ships of finite beam, the slope of the hull in the integrand of the Michell Integral and the limits of integration must be changed or the integral will represent the wave resistance of a somewhat different hull form. Having obtained several such modified forms, he performed a series of tests and compared resulss with predictions from theory. The correlation between theory and experiments was extremely good. In his calculations he included corrections for viscous, finite wave amplitude and hull interference effects. Inui's results show that the mathematical theory is far better than had been anticipated on the basis of works by previous investigators. They also emphasize the shortcomings of the standard methods used in obtaining the wave resistance from experimental measurements, i.e., deducting the frictional

-5resistance of an equivalent flat plate from the total resistance of the model. To obtain a more realistic value of the frictional resistance Inui used data from submerged double model tests. As previously stated, the present work is concerned with the answers to question (b). It is safe to say that the complexity of the evaluation of Michell's Integral has been the main obstacle to the application of this theory to practical ship design problems. In making actual computations, investigators have in most cases been forced to consider simple mathematical shapes which often bear only a vague resemblance to usual ship forms. Even so, calculations have been lengthy, involving numerical integration. As soon as numerical calculations are initiated, further analytic evaluation of wave resistance is only possible through systematic variations of parameters and plots of numerical results. It should therefore be the aim of the theory to obtain expressions for the wave-resistance of a ship in terms of its hull form and known mathematical functions. As far as this author has been able to establish, the present work presents for the first time such functional relationships for the wave resistance coefficient for any ship form whose surface can be represented by a polynomial of integral powers of coordinates in the longitudinal plane of symmetry.

-6For each term of such a polynomial, these expressions could be evaluated by high speed computers and tabulated. The calculation of wave resistance has thus, once such tables become available, been reduced to a minimum of labor, involving only a few multiplications and additions. Indeed, the mathematical wave theory could become a powerful tool in ship design. The assumption that the ship's surface be represented in a polynomial form is not a serious restriction. In fact, several researchers in the field are strong advocates of such a representation which dates back to D. W. Taylor's work on his famous Standard Series Ships. The many problems in connection with the layout of ship dimensions in the yards also seem to favor the polynomial representation. Historical Background Several excellent reviews of the development of the mathematical theory of wave resistance can be found in the literature, some of which include extensive lists of references. Lunde(26) for instance, gives a total of 185 references published before and including 1953. The Transactions of the Institution of Naval Architects, vol. 100, 1958, gives a complete list of-papers published by T. H. Havelock on hydrodynamics during the years 1908-1958. A detailed account of the development of the wave resistance theory is beyond the scope of this work. If some investigators are not

-7mentioned here, it is not because their contributions are considered less important but rather that their works have no direct bearing on the results of this thesis. Lord Kelvin in 1887 was probably the first physicist to investigate three-dimensional waves. His picture of the pattern generated by a moving pressure point, revealing the existence of diverging and transverse waves, is well known. In presenting his classical paper, however, he made it a condition to the Council that no practical results were to be expected from it. Thus Michell's paper, published in 1898, marks the beginning of the theory of wave resistance of floating bodies. For many years, this paper was unfortunately overlooked and forgotten, In 1923, Havelock rediscovered Michell's work, and a few years later Wigley put the theory to test by initiating his series of papers on the comparison between theoretical and experimental results. Weinblum, in Germany, started his work on the Michell wave theory around 1930. His first concern was the determination of ship forms of minimum wave resistance. It may be of interest to mention that von Karman also has contributed to our knowledge on this facet of wave resistance theory. In 1936 Dr. Weinblum published a paper on the theory of bulbous bows and later became interested in the systematic evaluation of the Michell Integral. During his stay at the David Taylor

-8Model Basin from 1948 to 1952, a partial computation program was sponsored under his supervision. This program has later been continued in Germany. In 1939 Guilloton published his thesis on wave resistance in France. Guilloton used the Michell potential to calculate the pressure disturbance of a simplified geometric body and by adding these disturbances was able to obtain the wave resistance of actual ship forms, and in addition he was able to trace the streamlines of fine hulls. After having published many papers on wave profiles and wave resistances for pressure points, sphere, etc., Havelock developed a wave resistance theory from a somewhat different approach. By considering a distribution of singularities (sources or doublets), he was able to simulate the presence of the ship, presenting formulae for the velocity potential and the wave resistance. In principle, Havelock's theory is capable of satisfying the boundary conditions on the surface of the ship exactly. In practice, however, the same assumption about form has been made as in Michell's theory, placing the singularities on the center line plane of symmetry. Wigley(3) demonstrated that, under these conditions, Havelock's and Michell's formulae for the wave resistance are identical when applied to the same hull form. In 1953, Timman and Vossers(33) were able to demonstrate the complete agreement between the two theories by means of Fourier Transform techniques, and a source of argument was removed. Birkhoff, Korvin-Kroukovsky, and Kotik presented an excellent

-9(34) study of the significance of the wave resistance theory() in 1954. In it, they also proposed two new transformations of the Michell Integral The first of these transformations form the basis for this thesis. A number of authors have extensively exploited the linearized theory of wave resistance during the last 35 years, notable Wigley, Weinblum, Lunde, and Guilloton. A considerable amount of work has also been done in Japan, notably by Inui(32'3'44). Reference(4) lists over seventy papers on the subject, many of which are unfortunately not translated from Japanese. No real basic modification of Michell's theory has been made, however, and our efforts are even today directed toward the application of his analysis.

CHAPTER II THE MICHELL INTEGRAL Boundary Conditions As a ship moves with constant velocity on the surface of an infinitely deep, incompressible, and inviscous fluid, it gives rise to a perturbation velocity field. If the fluid is assumed initially at rest, this field will be irrotational and it follows that a velocity potential $ exists. Using a rectangular coordinate system with the XY-plane located in the undisturbed water surface and Z-axis vertically downward, the velocity components u, v, w, which satisfy the equation of continuity x + ay as O are given by oX,.= — b Here p, is a solution of Laplace's equation 2 2 a, + 9X 4 + 0, ~ The Bernoulli's equation for unsteady irrotational flow is -^ i~~~u\^^^^^Z' ^~O~~

-11To reduce the fluid flow to a steady state case, a coordinate system fixed with respect to the ship is introduced. The origin is located amidships in the center line plane and the positive x-axis is in the direction of motion (Fig. 1). Furthermore, the ship is assumed stationary and a uniform flow of velocity c equal to that of the ship is superimposed, in the negative x-direction. It follows then that = -c + u; v = v; W = w where u, v and w are the components of the perturbation velocity caused by the presence of the ship in the uniform flow. These components are assumed small - i.e., u < c; v < c; w << c Introducing the perturbation potential qp defined by U cx; ay; one has that = cs + and 9= 0 (2.1) 2 where V is the Laplace operator. Neglecting small quantities of second order, the Bernoulli's equation now becomes P + 2(C, 22C-,-)- -,= C?0

12 f J ~ X Y z Figure 1. Coordinate System Fixed In Stationary Ship In A Uniform Flow

-13The constant C1 can be determined from the condition that far upstream the surface elevation must be equal to zero. Thus C 2 Po C1 = 2 PO where p is the atmospheric pressure. Letting the free surface be defined by S = - (x,y) the condition of constant pressure on the surface becomes xC n9 (2.3) The kinematical boundary condition to be satisfied by an inviscous fluid is that the velocity of a particle on a bounding surface must be tangential to it - i.e., DF cFF PF ) -D- at+ u x V i +w - 0 O (2.4) where F(x,y,z,t) = O is the equation of the surface. For the free surface F(x,y,z) = z + z(x,y) = 0 Applying Equation (2.4), the kinematical free surface condition becomes (-C+LU)) + V C + \ O ^x o^

-14Neglecting terms of higher order, this expression reduces to C 3-+ =; z= - (2.5) Eliminating z from (2.3) and (2.5) gives 2 oXZ C92 C. - ~ (2.6) Since perturbation velocities are small, Equation (2.6) is assumed to hold at & = 0. The kinematical boundary condition of (2.4) must also hold at the surface of the ship. If that surface is represented by y = g(x,z) (2.7) the boundary condition then is, by (2.4) o-(-C+ -ie t p o O If one introduces the restriction that the tangent plane of the ship surface makes a small angle with the xz-plane, i.e., ax < I. a< I then the boundary conditions on the surface of the ship simplifies to ~y = f(; Lj = 9(x Z) or i =- cf(x.; - (2.8) c3y

-15The boundary conditions given by (2.6) and (2.8) are necessary but not sufficient. To make the solution unique additional restrictions must be introduced. Since the ship is assumed to advance into still water it will be required that the waves are trailing aft. Furthermore, the perturbation velocities are zero at infinite depth. The velocity potential cp must therefore satisfy the following requirements: 2 1. VCp = 0; -oo < x <; 0 < y < oo; 0 < z < 0 2. -^ - O i o; z = 0; -ao < x < o; O < y < ax2 c2 az 3. - cf(x,z); y = 0; -0 < x < a; z > 0 ay 4. z -oo; 0 = 5. x -+oo; P = 0 6. y -+ oo; = 0; -oo < x < oo; z > 0 The Wave Resistance Integral The derivation of the wave resistance integral which follows is due to Michell(1) It is included in this thesis for the sake of continuity and completemess. It is at first assumed that depth of water is finite and equal to h. Hence the velocity potential cp must satisfy _ 0; z=h (2.9) Ay

-16A typical term of the solution to Laplace's equation, satisfying (2.9) is a cos n(z - h) cos(mx + a) cos(py + B) mnp where m2 + n2 + p2 = 0, and m must be taken real since -oo < x < co. n and p may be either real or imaginary, but if p is imaginary,e.g. p=ip' the last term must take the form e,y since due to symmetry it is sufficient to consider positive values of y only. The free surface conditions (2,6) will be satisfied if c2m2 n tan nh = - cm. (2.10) g For each value of m, this equation has an infinite number of real roots and one pure imaginary root given by c2m2 n' tanh n'h = cm; (n = in') It will be shown that the imaginary root alone is responsible for the wave resistance. As for p, it is always imaginary for the real roots of n and so for the one imaginary root of n if m > n't In order to satisfy Equation (2.8), let f(x,z) be expanded in terms of a Fourier series. f(x,z) = a a cos n(z - h) cos(mx + a). m,n mn

-17If it is supposed that f(x,z) is periodic, f(x + L', z) = f(x - L', z) then f(x,z) can be written }j(x,) =- A Cos L + BZfv i T rrx }(c n(z -h( ) (2.11) r'L The orthogonality properties of the trigonometric functions now imply that f (xZ)co5 r dx = LiA, A c n(Z- h) J f(x Z) stn y dx - L B r, Cos Y(a-h) -L and also that h L h L +A (2.12) h L h L' - T cosX m(2-h)dxd: LAJcos (-d o -L L L BL ^- (2^h + Sin 2z0 4' (21 h V sL' 22v) 44 4

-18When n = in' Equation (2.12) becomes H L' (j -(- co. ULx cosh'(z-h) dx dz o -L =LA'Tn(2k'h + sh 2n'h) (2.13) L' Jf f(,z) 5sin v co^h' (za-h d) xd L Br,,, (,_'h + sivi 2'h) Substituting Equations (2.12) and (2.13) into Equation (2.11) one has ~ CL'(2ni+Csi S2i n) x Z)co 1 X- x) cos,( Z- h) d d -L 0 L h L h'cah 2+ Si ) )J J(, ))cQs) -( x)cosk n(2-h)dd r -L' 0 If now L' - oo with T- r; dTr the infinite summation on r is transformed into an integral of the form oo Qi h i, J' j ( ) v h'-) cCh1 r'( $- h) Tn 2'"m + -ink 2m'h -x) d cos m(x -x) d t d c

-19If.flirthermore, the depth of water is allowed to become infinite, the following relationships are obtained: vih W- Tr Cos 1(z-z )= (-. Cos -(-= 1m - (2. ) 4 2o v _oo'2a, f - J; B 1< ) lim tiap^, v' = \ W o&^ n (i-h)co% n (Z-hl) _L -n(Z+) Ltm 2m'h + s;c^ -2'h = -e ^ Substituting these relationships into Equation (2.14), one finally has i ~'iz'' <, ~':) co~ c 0! 0 0~ ~ Cosrr(rx-x)dx dwid8vi (2.15) ~ ~ m _ Cos ('2 -x) ddtd,,

-20The velocity potential satisfying all boundary conditions is therefore 00 00 0000 -0 2C x V) C|05 (V cosin P 2- ) x O o O Cos rn (X -'X) m cx xC) ^ 2c ^ r_________ 2c2 - s ri van M (x -xJ+m Xm) / } Ciodm T^ r r XIZ) -Z M (X _X m o o -co: -rn/ - 9^e- T t ^92 d^xdddm The term sin {m(x-x)+m m y} was introduced by Michell to make, in g2 his words, the waves trail aft. Timmon and Vossers have obtained Equation (2.16) by means of a Fourier Transform technique and have shown that the velocity potential is uniquely determined by the boundary conditions given on page 15. If one lets 5p be the hydrodynamic pressure due to the wave disturbance, the wave resistance will be given by Rw =-2f 5p f(x,z)dxdz where the integral is taken over the center line plane of the ship. Neglecting terms of higher order in the Bernouilli's equation the hydrodynamic

-21pressure due to waves becomes 5p = -pc-. Thus one has that Rw - 2pc f(x,z)dxdz. 6J ax The first and third integrals of Equation (2.16) make no contribution to the wave resistance since fffff(x,')f(x,z)sin m(x-x)ddzdxdz = 0. Hence it follows that - -, R., 4 9 ((f 9t 0oo o 0-0 (2.17) C2 I Let X = so that the wave resistance is given by 4 2 e ~^ (r a) (2.18) where oo o I J f J i(x,) e C dxd o -o0 22 09 e Ar n g 0 co 0 -00 IOA V.k c v2 )d

-22Equation (2.18) is called the Michell Wave Resistance Integral, It is observed that for a ship symmetrically fore and aft I = 0. Furthermore theory predicts the wave resistance to be the same for motion in either direction along the x-axis. For real ships, f(x,z) is defined as non-zero only on a domain 5sf x i X; o<ZC b} where L = Length of ship D,= Draft of ship It will be found convenient to express the multiple integral of Equation (2,17) in a non-dimensional form. For this reason, the following variables are introduced at this point: U = i-' -' "/=.........A L L L ) L Furthermore, it is convenient to let = F = f where f is the Froude c2 Number. Then from (2.17) ~00 R W 4?92Zl:j(Lu L,w,) (LO, LW) - TI cz -'' 5' 5' x co[ Lx F-)1J L2dwd d da 5= - _^2 J o-i;'~ L }

-23Defining the maximum ship beam by 2B, the hull surface may be described by = G( L ) = G(u, w) (2.19) From the dimensionless slope function defined by h(u, w) = (2.20) 8u one has that Bh(u,w) = Lf(x,z). Thus, if follows that 4 4 2L ((2 2 i ( f 2 x- CS [AF(u->)] dJA c wcUl Ad k The wave resistance coefficient defined by C = 1^Rw w PpcB2 is therefore given by A^^8 IF (0\ ( IF(WAt) C 8FL Tr I(uW)(L, w) _f -i-0 o o (2.21) x os ^F( U) Ud dwdw d

CHAPTER III SOLUTION OF MICHELL INTEGRAL FOR POLYNOMIAL SHIP FORMS Introduction The transformation of the Michell Integral proposed by G. Birkhoff and J, Kotik in 1954(3' led to a separation of the parameters contributing to the wave resistance of ships. By introducing auxiliary functions, they were able to divide the integrand in the Michell Integral into one part containing all the properties of the ship form and another describing the influence of ship speed. In making actual use of the method, it was suggested that numerical integration be employed. This led to two basic difficulties. First, the integrand possessed an irregular singularity on the boundary of the domain over which it had to be integrated. Second, the part of the integrand which is a function of ship speed was given in an integral form only. This integral, called the Michell Function, has been investigated in parts by Birkhoff, Kotik and Parikh(363738) This thesis presents for the first time a series solution of the Michell Function. A solution of the Michell Integral for wave resistance has also been obtained under the sole assumption that the function containing the properties of the ship's surface is of a polynomial form. -24

- 25A Transformation of Michell's Integral The following transformation is the first of two proposed by Birkhoff(34) In Equation (2.21), two new variables 5 and, are introduced by =u - u; w + w. (3.1) For u and w constant, it follows that dt = du; d = dw and (2.21) becomes CrJ T LA C2W).^ 2 Cw0, d w C+,'~ x| cs u, $'-) x F COS (AF If the order of integration is interchanged, Cw can be written Cw fd d Jd dw ( w ) +,-w To |e cos L F I cd or in a more compact form (.2) CW= 8 d H(,0)C(F) (3.2 -I

where L((,LA a d d-k (3 3) fdi fcBw (^w \k(U,- )w) C(s~tJ =Jec ~t;L -5(3) (3.4) Here H(t,5) is called the Hull Function and C(s,t) the Michell Function. In order to show that Equations (2.21) and (3.2) are equivalent, it is necessary to prove the following: Theorem I. 1 Suppose j h(u,w) du < M for some finite M. -2 Then (2.21) is an absolutely convergent multiple integral. Proof. Since the integrand of 2 W) jlljl= e li-W) x lI I d(, id \ bdwdadu dA is positive, measurable, and larger in absolute value than the original (41) integrand, it is, by the Fubini theorem, sufficient to show that one iterated integral is finite. By hypothesis, one has AtL The eqiaec of (221 an (3*2)Ui ao F 2 FZ The equivalence of (2.21) and (3.2) is now a corollary to Theorem I.

-27The Hull Function From the form of Equation (2.12), it follows that H(-Sjt) = H(S, ) Obviously, the Michell Function is symmetric with respect to s. It suffices therefore to consider positive values of t only. Thus (3.2) can be replaced by 1cw= Fw 2jdyj8> H(>J) CCFy, Fr) (3.5) o o Since h(u,w) = 0 everywhere outside the region S', the limits of integration of Equation (3.3) can be reduced somewhat. If = 0, then h(u,w) = O for u < -, and the lower limit on u in (3.3) becomes (-). In regard to the limits on w, two cases must be considered. (i). =; the limit G - can be replaced by zero, since L L h(u,w) = 0, w < 0. D < < 2D (ii) D-= = D; the upper limit on w can be replaced by L L D D w =, since h(u,w) = 0, w > L' The Hull Function may therefore be defined as follows: (a,>) Cn(, p~7) = dAu { dw h(,v) hl(>+u,~-w),$~20; o - (3.6) (^ t y =[ H,].Ji = d>-L dw - (uW w)L(W LA, 7 -w) 2o; I-Z 2

-28These expression show immediately that H(t,,) = 0 along the boundaries of S". Furthermore, it is noted that for small values of ~, H(t,) can be approximated by 1(, ) > m jf - ^(> ( u^,0) Thus H(t,~) vanishes at least as fast as a linear function of e on 0 = 0. By means of similar approximations, it can be shown that the Hull Function vanishes like a linear function (or faster) on the complete boundary of the region S". 5$ "1 _; 0- Ji Now by Leibniz rule = fdsw Jh( 4KU,-w) - h(u,W)duL s sn dw -t (,"-w)k (4 ), ) ( W A i d Also since -+ ff h(u.^+^ w) -^ "cP, -w) it follows that t o' J - [-(-, -w,( w)+f,,w) (u, -w)du] (. 7) If one lets w' = - w, (3.7) may be written y k(^ ^ +)dwk(-h\(t /) -w)*)k(i, -/V' k( l )w'^d j] (3.8) PI" o -/

-29Dropping the prime, one obtains from (3.7) and (3.8) 2 = 1 jdwL-2L(+,w)h(i.-w) UJj-WI (4) A/)du + \A/)JW) dkA _ -Z -1 f o z -- -f dw {hC, w) (i r-w) + h (-,AW)l(-),o If the waterline angles at the bow and stern are zero, then - = 0 at = + However if the slope is different from zero, the slope of the Hull Function has a jump' along the line e = O since H(.,~) = H(-g,). The Hull Function for a simple mathematical ship surface is derived in Appendix I. To gain insight into the behavior of the Hull Function, it is useful to consider the effects of variation in h(u,w) doe to changes in the ship's form. Let h(u,w) = h(u,w) + 8h(u,w) (3.9) be the new slope function after alterations. The corresponding Hull Function n then becomes

-30H(S,) = c(iu/cw [h(u,w)+bh(u,w)][h(e+u,-W)+h(+u,-W)] Defining the change in the Hull Function by 5H(E,)), it follows that &H(U,) = H(,S) - H(l,) = du dw [h(u,w)5h(t+u,t-w) + h(t+u,t-w)5h(u,w) (3.10) + du dw [6h(u,w)Sh(t+u,^-w)] The first integral of (3.10) represents interaction effects between original hull form and the change bh(u,w), whereas the second integral represents the Hull Function due to bh(u,w) when this is derived from the change as a separate body. The latter integral will always lead to a positive contribution to the wave resistance. Thus, for a change in hull form to be beneficial, the contribution of the first integral must be negative and of greater magnitude than that of the second integral. Equation (3.10) should prove to be valuable in the analysis of bulbous bow designs. So far, the general functional behavior of the Hull Function has been studied. To get the physical significance of the function, however, it is necessary to proceed in a somewhat different manner. Consider two elementary areas A and A2 (Al = A2 = A) located on the hull surface at (ul,w) and (u2,w2) respectively, and let h(ulwl) = hi, h(u2,w2) = h2.

-31Since H(i,) =/du dw h(u,w)h(i+u,2-w) the Hull Function will be different from zero only for specific values of t and ~. For example, H(O, wl) = A hl2 H(O, w2) = A h22 H(ul-u2, w1+w2) = A h2hi H(u2-ul, wl+w2) = A hlh2 From these expressions, one concludes that H(O, ) represent the wave resistance of all the elements of area considered separately and that H(t,), t > O, represents interaction effects. The contribution to wave resistance by the two elementary areas, considering interference effect only, is given by (3.2). Thus it follows that 8F'2 2 a F2 f'h h, rE =., C = 8 F(I-U2) Co This expression is the same as that obtained by MF(u,-ichell from This expression is the same as that obtained by Michell from the original form of the Michell Integral.

-32If the two elementary areas are located on the same vertical line, ul = u2 or 0 = 0, the interference effect becomes -C^. T-' IT Now Michell(l) has shown that at -t 4d e [8 K,,(J K ( -z (3.11) where Ko and K ) are the modified Bessel functions of the second kind. From this it follows that = 4 F2 2z -F _ _A+ Now if hi and h2 have the same sign, 61 Cw(S = 0) is always positive. The favorable effect of a bulbous bow in the area immediately above the bulb is therefore due, to a large extent, to the reduction of hull slope over the portion where the bulb is located. The contribution to wave resistance by the two elementary areas considered separately is given by 6 C, 8 CF z (2AZ? _____ t ( h.-24Rw2 / \ I i Fz,2 { 2 -^W F W M A +> e [ Vo(w(\) ( + W, ( )] ^ ^ e" ^ [vJ o^) ^ K (^wJ] }

-33This expression shows that 52Cw is always positive, and its magnitude depends upon how far the elementary areas are located below the undisturbed surface and upon the absolute value of the hull slopes. From what has been said, it becomes apparent that the wave resistance of a ship is made up of two parts, namely: a) The sum of the effects of all elementary areas taken separately; and b) The sum of interference effects between any two elementary areas taken in pairs. It has been shown that the interference effects depend upon the horizontal distance between the areas and the sum of vertical distances to the undisturbed surface. The horizontal distance is represented by the variable t and the sum of vertical distances by the variable ~. It follows then that for a given value of i and ~, the Hull Function H(t, ) represents the total interference effects of all areas, taken in pairs, having a horizontal spacing equal to t and for which the sum of vertical coordinates is equal to ~. The wave resistance is determined by the products of the functions H( ^) and C(Ft, Ft). Over any region in the (3, )-plane where these functions are of equal sign, one may say that interaction effects are detrimental, whereas opposite signs indicate favorable conditions, Since

C(FS, FC) is completely determined for a specified F (a given Froude number), the Hull Function represents all the wave resistance characteristics of the ship at any given speed of advance. This function should therefore prove' within the limits of the theory, to be an invaluable analytic tool in determining ship forms of minimum wave resistance. The Michell Function The Michell Function is defined by (3.4) as Cslt) Je co5 A 5A Differentiating under the integral sign, one observes that C(st) satisfies the homogeneous heat equation~ 2u u; t > 0. (3.12) 6s2 6t It is known, by reference 40 that C O's x Tr TT!(3o13) J Co0 x.- - - jY(x) (5 13) where Yo(x) is the Bessel function of the second kind. Furthermore, it can be verified that 90 60 s i so^J SL0 l C e Cos s Y Cos _t a Hence it follows that li^m cc^~t) -^ ^ y~o js) (3.14)

-35An expression for C(0, t) is obtained by making use of the relationship (3.11), namely C(O, t) = e [Ko() + K14)] (15) An asymptotic expression for large values of s and fixed t, obtained by J. Kotic and rederived in reference 37, is given by Ctst) = S e "^Co5( )- ^(9-t)5 okJ (3.16) - 12 ( ItCo Et f 40 - s 0) I It follows from this formula that C(s, t) tends to zero for large values of s and t. Birkhoff(37) outlines methods and procedures for obtaining numerical values of the Michell Function. If a formal integration of the product of Hull Function and Michell Function over the region S" is contemplated, however, where S"{O < 1; 0 < < -} numerical values are not sufficient. A general expression of the Michell Function must be found. One method of obtaining such an expression is as follows: In Equation (3.4) let -2 X'I so that the Michell Function becomes C -i) e ~efJ Xo sVTK dX (5,17) i -t rf~~; ~ ri = _ v L oo s \/ -,

where L{ } is the Laplace Transform operator. By reference 42, the transform of (3.17) exists. From the well known relationship j 2 )ir1 one has that CoS \A + I ( S5 +I (S Furthermore, it can be shown that (see Appendix II) (-)'( A#, ~', )..(...- ~ = (- _ ) Thus -_ (5 Tr / >) A_ (3.18) Cos ^ ST \ -l ^^'2^j),^ a' Y / Substituting into Equation (3.17) o00 rC(st) -#( (2 -e (#) d —l?- -d (3.19) By referee 39, p. 2 By reference 39, p. 269 f ^& (1+L) -0Lg - Vlt) (520) o where T(n+~, n+2; t) is the Confluent Hypergeometric Function, Since (n+2) is always a positive integer, (n+4, n+2; t) is logarithmic near origin. (See Appendix III,) Using the relationship r(n+2) - ('

-37where (2)n = ~(~ + 1)( + 2) -* *+n -. ) and substituting (3.20) in (3.19), the expression for the Michell Function becomes C(ts) (.) e, X = h f V(+- Y+2: tj (3.21) n'o Wave Resistance The wave resistance of a surface ship moving at constant speed on a straight course in water of infinite depth is given by Equation (3.5) with the Hull Function as defined by (3.6) and the Michell Function by (3o21) It will now be assumed that the Hull Function can in general be expressed as polynomials —i.e., (3.22) LH(:,$)j: 4 L I II o where A and A are constants and a and P are non-negative integers. 0Y3 03 Since H(, ) vanishes at least as fast as a linear function as - 0, one concludes that in [H(S, )]- >= 1. In the following, expressions for the contribution to wave resistance due to a general term of the polynomials of (3.22) will be presented. Depending upon whether the term belongs to the first or

-38second polynomial, this contribution will be defined by [FACOCw]I and [AaC] II respectively. From Equations (3.5), (3.21) and (3.22), it follows that o J (5.25) I O~ 2 Introducing new variables defined by F = I and F2 = 4, Equation (3.23) becomes C L - ttLC]: = 4'p,, 4^d $ fdF ^/3e (35.24) O O 0 \ 1[J-L t ( r) (0 i0 0; 2 ] T.-po F - e ( "+o2 Now from reference 59 If (5.25) is integrated by parts, one has ED LJ $ e-^(^^,^i;2)6.S;^k~r3

-39So that by iteration, 4/k-k i 1,1 f1-e -'/ ),(' "nl-k;)I (3.27) C - k-o e Substituting (3.27) in (3.24), it follows that LA (, - -- 2 F F. -.~0o n=o' 0 03+ _~ /s -k kso If the integral representation of the Confluent Hypergeometric Function and relationship (a) of Appendix II is used, the second integral in (3.28) becomes F k- k 12 = livvl2 ( Tr) Jc 5e la (-t) S Je cos.... dN^ In the limit as e - O, I2 reduces to 0 ( Tir) 1 |COS (3.29) o o W'

-40where the M(~)(-) depends upon the value of 3. Now from (3.13), YoC) = I ffcos;" d7 0 so that ^d ip[M 5)] = (-^1)!iYod) ^(3 30) Moreover, from (3.29) one has 00 This may be transformed by ^^ i^^QZ8; d'- ( 2! so that r M(P(o) = (w*) ~!o C^ ^ede T k0 d () - (f f) dZ ( P 2 )" ezl coS ~ ~h~+l drt "f Z ("R)'" Also, by iteration it can be shown that the derivatives of M(~) at = O are d ~=((-1)P( f )1lf 2(,,>{,t( I) ( d"1WI'M ( -- -0 (1,3,5-..2 /(- 0 ( 2. () T= 0,,4 ~- ({z-2)

-41Now the Bessel Function of the second kind and of zero order is given by Y.0)=Xt8~ (X#3 ) + log, ylx i )}, = ib s~ ^ 10 t) 1-(-t (+{ + 1 + t* 1 where. -) n-o so that integrating Yo() repeatedly and using (3.32) and (3.31), it can be shown that Tr / (t..+ */ 5) If this is substitute into (39) If this is substii tuted into (3.29), + then 7.- CZ VI F ^ __________ t OL ( J +y( /) F________(____ (+, + +( -lLk). _^^ ^?,,( (t... +2\{ ) ) ] i1=-0 k-r

-42Returning to Equation (3.28), one observes that the first integral is of the type I3.f L 2+ (y)di (3.35) o Let u=j; d^-^I d^ dv- i -) =~' " ~..(Ref. 40, p. 192) so that integrating by parts gives 15 ~ ++J a 0)-2 _-c Continuing by iteration, one obtains I: 2 k7? +(- (3.36) k=o This Equation is valid, however, only for a an even integer. For ca an odd integer the last integral of (3.35), after integration by parts -] times, is of the form 2 + C (-+ C where kl = i — the greatest integer in +l^ Thus by reference 40, p. 194, for a an odd integer, 13 must be replaced by

5 -43k-o (3.37) f { (i F) H4 C() - JI2~ CF) ++ *k- 2( FJc'u'4 (4-) Z),,..bXk)+ I z++ ~ where Hv(x), the Struve Function, is defined by H,) (x) = (-1) r(<+ l) From Equation (3,23), (3.34), (3.36) or (3.37), one finally obtains r A^ - A L-]r {-2n n! L (3.38) k-o By inspection, the corresponding expression for [AbOCw]jI becomes -r I 3 - 4r~X ^ ( )-i r n(o k o ((3.39) () L) (, k; )/} 1ko

-44 The total wave resistance coefficient is obtained by adding the contributions from the individual terms of the Hull Function polynomials as given by (3.38) and (3.39) —i.e., Cw _ Wcl f ~(^ Id cl [ ew (3.40) The expressions for the wave resistance as given by (3.38) and (3.39) are convergent. (See Appendix IV.) It should be emphasized that Equation (3.40) is not restricted to ship forms symmetrical fore and aft. The expression is however, only valid for ships with rectangular longitudinal plane of symmetry.

CHAPTER IV CONCLUSION A general formula for the wave resistance of polynomial ship forms has been obtained. The result is based on the Michell Integral, and all assumptions made in the linearized wave resistance theory of thin ships apply. The series form solution is not too complicated in form. It does, however, involve several transcendental functions, some of which are not readily available. Also, one notes that even for the simplest ship form, at least about 20 series expressions will have to be evaluated at each Froude cumber investigated, a prohibitively long task if it was to be repeated for each individual design. The real significance of the results obtained lies in the fact that a systematic program of evaluations of the formula can be undertaken with the aid of high speed digital computers. The functions so computed depend upon the Froude Number and the power of the term of the hull polynomial, but not on the individual ship form directly. For each value of a and 3 of the Hull Function the wave resistance can therefore be tabulated for various values of the Froude Number. Once such tables become available, the wave resistance calculation will have been reduced to a few elementary operations. Model experiments, in conjunction with -45

-46theoretical calculations, have repeatedly shown that the linearized wave resistance theory is capable of predicting effects of even small variations in hull form. Indeed, it is the opinion of several researchers in the field that the linear theory will determine hull forms of minimum wave resistance with sufficient accuracy. As a result of the formula presented, it should become possible to investigate families of ship forms systematically and to determine such forms. This has always been a principle aim of the ship hydrodynamicist. The complexity of the formulae given makes it difficult to study the behavior of the functions involved analytically at this time. This task must be left to the computer. As an extension to the present work, it is hoped that it will become possible to obtain expressions for the complete velocity potential in terms of ship form and speed. It would also be interesting to consider perturbation potentials in order to improve the assumption made on both the ship surface and the free surface of the waves. These are but two of the problems of ship wave resistance still left open.

APPENDIX I THE HULL FUNCTION FOR A SIMPLE SHIP FORM The term Simple Ship Forms refers to ships' surfaces whose equations are given by g(x,z) = gl(x)g2(z). For these shapes, the Hull Functions can be written H(S,~) = H1(~)H2()' Consider the particular case g(x,z) = B(1 - (1- 4 which implies that G(u,w) = (1 -w )(1 -u2) and h(u,w) a = -8u(l - w ). 6u D By Equation (3.6) t9(S )],T =j j (u +})8u [- D(-u)l' DWJ dzdW 2 L. 2 =4 + + + where L 4- o D -47

-48Similarly - \,,)T)s -2: L^ _^ A plot of H1(s) and H 2() is shown in figure 2. 2

49 8 8 -~ —rn-rn — - - 4t \ 0.2.4.6.8 1.0 fr~~~ ~C.2.2 0 0.4.8 1.2 1.6 2.0 L Figure 2. The Hull Function For G(u.w.) = (-Lw)(l-u2) D

APPENDIX II A BESSEL FUNCTION RELATIONSHIP Consider the generalized hypergeometric function; - 4') [, v (I "2' ) Wl=O \"0 ko Z f (-.1)" 2"u -k!/'k)' where the last step follows from the Binomial Theorem, It can be shown that Ocg h 00 00 T >,7 A(k 2 (2,k2 A (i, k<) n-o k —o n-o0 k-0 from which 0^ L ^.;- _ _ j - ^~k Z~ ~1 k 2 oo oo,q ok^ 2ki*1 i x -jyx ((-Ox _(-() x oF, I +4; 4 3 ntO t= (I+@)M+k 2 k! "- ao 0 k k+ t Xkt = n o k=o However T( I4.) r( I+) t t+ k) T"(- (IJ~ ) - so that (+)\ ) ) r(('^+4+.L) -50

-51-. ) I-l) j 4 Hen/ce o. L \; 4+ J so that from these two relationships, one obtains - o ( k Zk^ U^ X^ [.. z-. -(a) ( 4-') (~X V — -) = T v () (- ^_.C)

APPENDIX III NOTES ON THE CONFLUENT HYPERGEOMETRIC FUNCTIONS In the notation of reference 39, the Confluent Hypergeometric Function is defined by K O In the notation of generalized hypergeometric series, this is 1F(a; c; x). Now e(a, c, x) is a solution of the differential equation X d'- (-x) - - (III.2) and a second solution of (111.2) is given by (,ck x) (111.2) 4 rr^ K-) X C (a-c+I, 2-c;X ) for c not an integer, For c an integer Y(a, c; x) becomes -52

-53where *p(z) is the logarithmic derivation of r(z). With a =n +; p = n -k, this becomes + {)(k)(T'(k) X 4 k Equation ((IIIz) is valid for k When k > n —i p < 0W an expression for +(a, pl; z) is obtained by (cIII.) from the transformation c; zC(a -c; z) ( (a, c;z) = z (a-c+l, 2-c; z) (III.5)

APPENDIX IV CONVERGENCE OF SOLUTION From the definition of M(J)(t) by Equation (.30) it is readily shown that a > 0 and. > 0 is a sufficient condition for I2 given by Equation (3.29) to be finite, This condition on a and a is satisfied by polynomial ship forms, One need therefore consider only the infinite summation on n. Equations (3.36) and (3.37) define I for a a n.even and a an odd integer respectively. It can be argued that for given values of a and [AcCw3 < [L+', Cw ] < [AAC+2, Cw]< or [A4, cCW]z > [Ai+ Cw ]I > [Ao+2, cw ]i Hence only the cases where a is an even integer need be considqred. Furthermore, to prove convertence of Equation (3.38), it is sufficient to consider the terms in the summation on n obtained from the product of a general term of the finite summation of Equation (.3536) and a general term of the finite summation on k of Equation (3.38). The resulting infinite series on n becomes 2n o where k (- L)k - f Oo Since W-)-2 Z/ 00 it follows that

-55and then ( t t{ ~ L [T- 4 ()]d (III.2) But from Appendix II o -),: (-0,F ^(Ft) =' (- F, t~) /-z /~. ^_ ~^ ^',~) k(tr^/4~ (j VI^) This relationship was obtained by a rearrangement of the terms of a oF1(-;a;x). The infinite series is therefore uniformly convergent and it follows that Equations (III.1) and (III.2) are identical. Substituting Equation (1113) into Equation (111.2)..,. (e tt Z(L+t),-+Jk -) ct' \ecause (Frt \ \;,<o: F t T o E P Cf tk( (III.4) < i C e t (, ) dt The upper bound for S given by (III.4) is finite by the convergence theorem for the Laplace Transform42) theorem for the Laplace Transform.

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