NAVY DEPARTMENT THE DAVID W. TAYLOR MODEL BASIN WASHINGTON 7, D.C. x ¥ oof To ees (Jr THE WAVE RESISTANCE OF BODIES | ¢ OF REVOLUTION by PON si MA he Georg P Weinblum, D. Eng. Clery nob with a Contribution by oe J. Blum, National Bureau of Standards May 195l Report 758 NS 715-084 INITIAL DISTRIBUTION Copies 15) Chief, Bureau of Ships, Project Records (Code 324), for dis- tribution: 5 Project Records 3 Research (Code 300) 2 Applied Science (Code 370) 1 Design (Code 410) 3 Preliminary Design (Code 420) 1 Technical Assistant to Chief of the Bureau (Code 106) 5 Chief, Bureau of Aeronautics, Aero and Hydrodynamics (DE-3) 5 Chief of Naval Research, for distribution: 3 Fluid Mechanics (N426) 2 Undersea Warfare (466) 4 Commander, U.S. Naval Ordnance Laboratory, Mechanics Division, White Oak, Silver Spring, Md. 1 Dr. J.H. McMillen, U.S. Naval Ordnance Laboratory, White Oak, Silver Spring, Md. 1 Commanding Officer and Director, U.S. Navy Underwater Sound Laboratory, Ft. Trumbull, New London, Conn. "1 Commanding Officer, U.S. Naval Torpedo Station, Design Section, Newport, R.1I. uy U.S. Maritime Administration, Washington, D.C. 1 Vice Admiral E.L. Cochrane, USN (Ret) 3 Director, Technical Division 4 British Joint Services Mission (Navy Staff) 2 British Joint Services Mission (Technical Services) ] Australian Scientific Research Liaison Office, Washington, D.C. 1 Director, Hydrodynamics Laboratory, National Research Council, Ottawa, Canada 6 Director of Aeronautical Research, National Advisory Committee for Aeronautics, 1724 F St., N.W., Washington, D.C. 1 Capt. F.X. Forest, USN, Portsmouth Naval Shipyard, Portsmouth,N.H. 1 Director, Woods Hole Oceanographic Institution, Woods Hole, Mass. Copies 2 DO Newport News Shipbuilding and Dry Dock Company, Newport News, Va., for distribution: : 1 Senior Naval Architect 1 Supervisor, Hydraulic Laboratory Director, Experimental Towing Tank, Stevens Institute of Tech- nology, 711 Hudson St., Hoboken, N.J. Attn: Miss Jones Dr. Hunter Rouse, Director, Iowa Institute of Hydraulic. Research, State University of Iowa, Iowa City, Iowa Dr. R.T. Knapp, Director, Hydrodynamic Laboratory, California Institute of Technology, Pasadena 4, Calif. Dr. L.G. Straub, Director, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minn. Director, Experimental Naval Tank, Department of Naval Architec- ture and Marine Engineering, University of Michigan, Ann Arbor, Mich. Dr. V.L. Streeter, Illinois Institute of Technology, 3300 Federal Street, Chicago 16, Ill. Head, Department of Naval Architecture and Marine Engineering, Massachusetts Institute of Technology, Cambridge 39, Mass. Director, Applied Physics Laboratory, Johns Hopkins University, Silver Spring, Md. Prof. W.S. Hamilton, Technological Institute, Northwestern Uni- versity, Evanston, Ill. Prof. G. Birkhoff, Harvard University, Cambridge, Mass. Prof. K.E. Schoenherr, Dean, School of Engineering, University of Notre Dame, Notre Dame, Ind. Prof. W. Spannhake, Armour Research Foundation, 35 West 33rd St., Chacagoni = lalalie Dr. M.S. Plesset, California Institute of Technology, Pasadena 4, Calif. Dr. Alexander Weinstein, Institute of Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md. Dr. J.V. Wehausen, Editor, Mathematical Review, Providence, R.I. Librarian, American Society of Mechanical Engineers, 29 West Thirty-ninth Street, New York 18, N.Y. Admiral S.M. Robinson, USN (Ret), Webb Institute of Naval Archi- tecture, Crescent Beach Road, Glen Cove, Long Island, N.Y. Copies 5 On Mr. Tingey, Bethlehem Steel Corp., Shipbuilding DAL, 5. Ohulaliaveryy Mass. Como. H.A. Schade, Director of Engineering Research, University of California, Berkeley 4, Calif. Prof. T.H. Havelock, 8 Westfield Drive, Gosforth, Newcastle-on- Tyne 3, England Commanding Officer, U.S. Naval Training Schools, Massachusetts Institute of Technology, Cambridge 39, Mass. Mr. C. Wigley, 6-9 Charterhouse Sauare, London EC-1, England Prof. G. Schnadel, Ferdinandstr 58, Hamburg, West-Germany Senior Naval Architect, Gibbs and Cox, Inc., 1 Broadway, New York 4, N.Y. Dr. J.F. Allen, Superintendent, Ship Division, National Physical Laboratory, Teddington, Middlesex, England Mr. R.W.].. Gawn, Superintendent, Admiralty Experiment Works, Haslar, Gosport, England Dr. A.G. Strandhagen, Head, Department of Engineering Mechanics, University of Notre Dame, Notre Dame, Ind. _ Dr. Max Schilhansl, Engineering Department, Brown University, Providence, R.1I. Dr. W. Prager, Chairman, Graduate Division of Applied Mathematics, Brown University, Providence, R.I. Supervisor of Shipbuilding, USN, and Naval Inspector of Ordnance, Electric Boat Company, Groton, Conn. Prof. §.P. Timoshenko, Stanford University, Stanford University, Calif. Dr. C.H. Lee, U.S. Naval Postgraduate School, Annapolis, Md. Librarian, Society of Naval Architects and Marine Engineers, 29 West 39th St., New York 18, N.Y. Mr. J. Blum, National Bureau of Standards, Washington 25) 5° 1D) Gi TABLE OF CONTENTS ABSTRACT . 1. INTRODUCTION . 2. THE REPRESENTATION OF SINGULARITY DISTRIBUTIONS AND SECTIONAL-AREA CURVES BY POLYNOMIALS . 2.1. Connection between Body Form and eure Hydrodynamic Singularities Sess 2.2. Representation by Polynomials . 2.2.1. General Remarks 2.2.2. The TMB (Landweber) Class of Bodies and Some Generalizations . Hey 2.3. Connection between Surenevh of ee aueg pos and Body Shape ao Biller’ Se Bante anes 3. EVALUATION OF HAVELOCK'S INTEGRAL 3.1. General Considerations 3.2. Tabulation of Resistance Integrals for a Five-Parameter Class of Bodies c 4. REPRESENTATION OF RESISTANCE CURVES 4.1. The Maen eion Factor Cy and Dimensionless Representations . aneclted cia ge ME REESE 4.2. Resistance Curves of Simple Symmetrical Bodies. 4.3. Resistance Curves of Asymmetrical Bodies 4.4. Limiting Depth of Immersion . 5. BODIES OF REVOLUTION OF LEAST WAVE RESISTANCE 5.1. Two-Parameter Forms . 5.2. Isoperimetric Problems, One-Parameter Forms . 6. RESISTANCE CURVES OF THE FAMILY (2, 4, 6; ¢; t) SUMMARY APPENDIX I - APPROXIMATE CALCULATION OF THE SURFACE S OF A CLASS OF ELONGATED BODIES OF REVOLUTION . APPENDIX II - EVALUATION OF THE AUXILIARIES INTEGRALS* . APPENDIX III - AUXILIARY INTEGRALS FOR VARIOUS FROUDE NUMBERS. REFERENCES , *By J. Blum, National Bureau of Standards Page itis ise rT —S ae oe Qa © © ~ eM FP ES Se Q ies] IS). (2) Ine) i] el i] A eae [eal It Sr = fey 3 Qs 3 ct 2a) - jeu} eo) Be) fey < = NOTATION With index, a coefficient Area Area of meridian section Sectional-area curve Dimensionless sectional-area curve Half length of distribution As index, antisymmetry Midship radius of body of revolution Form parameter coefficient (Reference 7) Constant Prismatic coefficient Wetted surface coefficient Midship diameter Froude number Depth Froude number Depth of immersion Wave amplityde Wave number Length of body Auxiliary integral Auxiliary integral Auxiliary integral Auxiliary integral Intermediate integral Intermediate integral Intermediate integral Intermediate integral Resistance, wave resistance Total Resistance Viscous resistance Wave resistance Resistance coefficient Resistance coefficient BB SS} tea ep) Wetted surface As index, symmetry Speed of advance Longitudinal coordinate Longitudinal distance of centroid Ordinate of the meridian contour Dimensionless ordinate of the sectional-area curve Dimensionless ordinate of the sectional-area curve fore and after body Dimensionless ordinate of the sectional-area curve even and odd part Variable of integration Doublet distribution Dimensionless longitudinal coordinate Dimensionless longitudinal distance of centroid Density Source-sink distribution Prismatic coefficient; afterbody Prismatic coefficient; forebody THE WAVE RESISTANCE OF BODIES OF REVOLUTION by Georg P. Weinblum, D.Eng. ABSTRACT Following a brief review of prior work on wave resistance of bodies of revo- lution carried out by Havelock and Weinblum a discussion is presented of the appro- ximate relations between the shape of sectional-area curves and of hydrodynamic irregularity distributions. The latter are expressed by polynomials, which lend them- selves to an evaluation of the basic resistance integrals by computing intermediate integrals. Values of the functions thus obtained are tabulated in an appendix. These functions are then used to calculate the resistance of some simple bodies of revolu- tion. Also investigated is how the resistance is influenced by asymmetry with respect to midship section. Distributions leading to bodies of least wave resistance are cal- culated, assuming rather severe restrictions. A rather complete set of resigtance curves is given for an important family of bodies. 1. INTRODUCTION When a body moves uniformly and rectilinearly in an unbounded liquid the only resistance experienced by it is the viscous drag. Our knowledge as to how this drag depends upon the body form is very limited, but it is well-established that for streamlined, elongated hulls—with which we are only concerned—the drag is roughly proportional to the wetted sur- face and is rather insensitive to reasonable changes in the shape.?* The well-known airship form with a rather blunt forebody and finer tail appears to be close to the minimum resistance attainable, although it must be empha- sized that earlier resistance data obtained in wind tunnels at low Reynolds numbers are utterly unreliable. But that there is a slight advantage in introducing some asymmetry with respect to the midship section appears to be unquestioned, at least when larger end-radii are used. Matters become different when a body moves close to the free surface; see Figure 1. A wave pattern is then produced and therefore a wave resistance arises. The laws governing the wave resistance Rh are quite different from those valid +References are listed on page 58. *Problems of cavitation are not considered here. for the viscous drag RL: Henceymeain this case forms of least total resis- tance R, must be derived from addi- tional considerations and may differ, phy, at least in principle, from the fa- miliar streamlined forms. In the present report it is intended to analyze the wave re- sistance of a rather wide class of Figure 1 - Scheme of Submerged Body elongated bodies of revolution, using an integral relation based on the work of Havelock.? The first classical solutions for the circular cylinder (Lamb)?7 and the sphere (Havelock)?® have contributed much to the general understanding of the subject, but these solutions must be applied with great caution to problems connected with elongated bodies. The reason herein is the extreme simplicity of the cylinder and sphere; the resistance curves of these bodies do not show the characteristic interference effects which are peculiar for prolate bodies of revolution. From physical reasoning we infer at once that in the latter case two similarity parameters are involved: the common Froude number F = U/VgL referred to the length L and a parameter characterising the depth of immersion f, say f/L or the depth Froude number Fp = U/Vgf, while the shape of the wave-resistance curves for the circular cylinder and the sphere depend only upon Fe» and the parameter f/L appears as a scaling factor only. Thus, for instance, the peak of the resistance curve is located at Fp = 1 for the cylinder and just below Fe = 1 for the sphere. It can be easily shown that this unity value of the depth Froude number has no special significance for the wave resistance of a very elon- gated body of revolution. Solutions for the spheroid and general ellipsoid due to Havelock®’ + lead to results which admit of qualitative and even of quantitative esti- mates of the resistance of "normal" bodies of revolution. The importance of the spheroid for general research on the subject cannot be overemphasized. Using Havelock's general expression valid for a plane source-sink distribution,® formulas were obtained which represent the wave resistance of a rather wide class of bodies of revolution.° By these formulas the resistance of various forms has been investigated;® especially, some endea- vors were made to find forms of least wave resistance.° These forms vary Obviously with the Froude number and to a lesser degree with the depth parameter f/L. The rather striking results found in this way were checked experimentally and good agreement between theory and measurements was eS- tablished as to the general trend.° As with surface vessels, theoretical forms of least wave resis- tance are symmetrical with respect to the midship section. Any departure from symmetry causes an increase in wave resistance, and this increase can become appreciable in some ranges of Froude numbers when the asymmetry is pronounced. The degree of asymmetry can be described in the usual way, though roughly, by the location of the center of buoyancy X52 or the dif- ference of the prismatic coefficients oa Py of the fore and afterbody. For instance, a difference Pp - ?, = 0.2 means a large deviation from sym- metry. Again, the resistance results are qualitatively supported by experi- ments .° An extensive hydrodynamic study of bodies of revolution is under- way at the Taylor Model Basin. It is based on a systematic variation of analytically defined forms.?»**+ As an extension of this work it was decided to make a more comprehensive theoretical investigation on the wave resistance of bodies of revolution. This is the subject of the present report. In Section 1 of this report polynomials are discussed which are Suitable for the representation of hydrodynamic singularity distributions (doublets, sources and sinks); to the first approximation the equation of the doublet distribution coincides with the equation of the sectional-area 48 A class of curves is selected which in- curve except for a scale factor. cludes the TMB Series® generalized by one additional arbitrary parameter. For this family a set of auxiliary integrals covering a large range of Froude numbers has been tabulated. The values of these integrals furnish immediately the variable part of the wave resistance of the simplest forms (parabolas of the type 1 - any. In the general case the wave resistance is given by a quadratic form of the parameters of the body in which the tabu- lated values appear as coefficients. Thus the computation of the wave re- sistance involves only some multiplications and an algebraic addition. The auxiliary integrals mentioned have been computed by the Bureau of Standards. A short description of the work involved, contributed by Mr. Blum of that Bureau, and tables of functions are found in Appendices II and TUT, As mentioned before, the resistance formula for a line distribution of singularities used throughout this report follows immediately from a more general expression due to Havelock®’> and therefore will be called Havelock's integral. Using the tables annexed, resistance curves are plotted for vari- ous basic forms of sectional-area curves (doublet distributions); they cover three depths of immersion ratios f/L except for the spheroid where a fourth f/L ratio has been added. Special investigations are made on the influence of asymmetry, and some examples of resistance curves refer to forms selected from the TMB Series. Following an earlier attempt distributions of least wave resistance are investigated.° Former results® are checked and refined. Particularly, the distributions obtained lead to rather peculiar "swan-neck" forms, for higher Froude numbers. Finally it is shown how systematic sets of resis- tance curves can be obtained for families of sectional-area curves (doublet distributions). 2. THE REPRESENTATION OF SINGULARITY DISTRIBUTIONS AND SECTIONAL-AREA CURVES BY POLYNOMIALS 2.1. CONNECTION BETWEEN BODY FORM AND GENERATING HYDRODYNAMIC SINGULARITIES In establishing a relationship between body form and generating hydrodynamic singularities two well-known problems can be formulated: a. Given a distribution, find the shape of the body (sectional-area curve A(x)). b. Given a body form (sectional-area curve A(x)), establish the appro- priate distribution. In the present report we disregard the complications connected with problem b and treat it in a very approximate way. The contemporary rudimentary state of knowledge on problems of wave resistance justifies this procedure to some extent; our investigation deals essentially with resistance properties of hydrodynamic distributions and merely some assumptions are made as to the probable shape of the bodies generated by these distributions. Thus two essential sources of error are involved when investigating the wave resistance of bodies of revolution: a. The approximate character of the wave-resistance theory, and bd. The generally admitted approximation that for a given body the deep-immersion distribution of singularities can be used instead of the actual distribution valid for near-surface conditions. The second assumption (b) appears to be a serious one when the body is close to the surface. It has been proved by Havelock® that it leads to inconsistent results with respect to added masses; however, by following numerous comparisons between theoretical and experimental results referring to surface ships it works reasonably well when applied to the resistance problem. In the present report the assumption will be made that the shape of the body generated by singularities moving close to the surface is identical with the shape of the corresponding body generated by the same singularities in an unbounded fluid. It is well known that in the latter case one can construct the contour of a body of revolution for any given singularity distribution along the axis; auxiliary tables for this work are available,?»° especially for eases in which the distribution is given by polynomials. Flat noses—as discussed by Weinstein*® — will not be dealt with in the present report, although it is possible that such forms are advantageous from a point of view of wave resistance at high Froude numbers. When dealing with "normal" shapes, the important approximation developed by Weinig’ and Munk® holds; i.e., for very elongated bodies the sectional-area curve of the generating body A(x) is affine to the doublet distribution w(x). This approximation will be used throughout the present report although its limitations should not be forgotten. Some explanation—if not definition—must be given as to.the concept of a "normal" shape of a doublet-distribution or a sectional-area curve. It means essentially a curve whose trend is similar to sectional-area curves of common ocean-going ships; these curves generally are monotonic with not more than one point of inflection in the fore and afterbody. Since for closed bodies the source-sink distribution o(x) is the derivative of the doublet distribution w(x) the latter is monotonic over the range of the forebody when o(x) consists only of sources in the same range. This condition (though not necessarily a required one) is sufficient to ob- tain bodies such that the circle of curvature at the nose lies inside of the meridian contour. We mention some conditions under which the affinity between the doublet and the sectional-area curve becomes strained: a. For larger values of the elongation D/L the divergence between the sectional-area curve A(x) and the doublet distribution u(x) becomes more pronounced even for "normal" shapes. This divergence can be roughly de- scribed. First, in the mutual relation of the prismatic (area) coefficients which are the decisive form parameters of the two curves—the one , $4, de- noting the prismatic or area coefficient of the distribution, and the other, Po» the corresponding one for the sectional-area curve—the following state- ment holds for a wide class of normal bodies:°> ?° for finite D/L ¢, > ¢3 When ¢, < 2/3 ¢,<¢g When 9, > 2/3 The equality %q = Pq is valid only for the ellipsoid; see Figure 2. Second, in the prismatics a differ- ence arises between the length of the body L and the distribution 2a, @ ——Ps—0 eel 2a being smaller than L. For the _— IE ‘spheroid the relative difference Figure 2 - Spheroid. Sectional-Area Curve A, Doublet and Source-Sink Distribution pa bn2a = De 2a oH BigP Zeke where ¢ depends on the shape of the distribution, especially at the ends. (Since this problem is being thoroughly investigated by L. Landweber of the Taylor Model Basin, we confine ourselves to these brief remarks. ) b. When complicated "abnormal" distributions like "swan necks" or curves with very steep ends are investigated (for instance, Rankine's ovoid) the divergence between these distributions and the sectional-area curve can become appreciable even for smaller D/L. 2.2. REPRESENTATION BY POLYNOMIALS 2.2.1. General Remarks In former reports polynomials have been used for the representation of the generating doublet (source and sink) distribution along the axis>’®! The doublet and source-sink distributions u(x), o(x) can be split up into dimensional factors Hy, % and variable dimensionless parts w*(é), o*(&); u(x) = wou*(€) a(x) = a,0%*(&) with € = x/a; see Figure 3b. The dimensional factors will be established later; in the succeed- ing discussion the functions w*(é) and o*(&) will be treated in the same way as Ship lines and their derivatives. Generally following Munk and Weinig the doublet distribution u*(é) is identified with the sectional-area curve A*(é) and the symbol 7 is used for both of them. Actually the resistance computations refer to given distributions for which the corresponding sectional-area curves can be easily calculated?’?° when Munk's approximation is not accurate enough—as for instance in cases dealt with in Section 5. The first adequate representation of ship lines by polynomials is due to Taylor;**»+ the equations obtained are, however, suitable for a separate description of the fore or afterbody only. Taylor locates the i] Figure 3a - Landweber's Axes Figure 3b - Present Axes Figure 3 - Systems of Axes origin at the bow or stern. The present writer has proposed!?:?* other sets of polynomials referred to a system of axes with an origin located midships. This approach has definite advantages when investigating the wave resistance. Landweber* has generalized Taylor's equation by adding one more term and by introducing appropriate boundary conditions; he uses the ex- pression obtained as the equation of the sectional-area curve of a four- parameter form.* The parameters are interpreted geometrically as the pris- matic coefficient, the location of the maximum section along the axis and the nose and tail radii of curvature. It will be immediately shown that Landweber's equation transferred to an origin at the midship section can be split up into a two-parameter symmetrical and a two-parameter skew part with respect to this section; thus expressions are obtained for which the wave resistance can be calculated in a simple way. 2.2.2. The TMB (Landweber) Class of Bodies and Some Generalizations The TMB (Landweber) class of bodies of revolution is given by the equation of the sectional-area curve We = BUR a alx® + a!x? + alx? + ax" + alx! [1] referred to axes, as shown in Figure 3. We transform the equation of the body by shifting the origin to the midship section x = 0.5, reversing the direction of the axes, and putting the length of the body equal to 2. Thus for as. 0 Rao | 2 re 0.5 €=0 XS a f=- 1 The transformation is given by f= TEM 0.5) or 2 dee 2 ees [2] The resulting equation is Vie ont Ae ee Geka A eo Ae ae [3] CUS IES 6 a! 6 na! 6 1 a = A = 2) $7 = nin = ams ee i ee Equation [3] can be split up into a symmetrical and an antisymmetrical part = 2 4 6 Maran ALé + Als IMG [4a] a = y =Ye= Y, oF vt i 3 5 Wo ag th Nee Ag [4b] The obtained form [3] has definite advantages when calculating the wave re- sistance since the latter is the sum of the wave resistance corresponding to the symmetrical and antisymmetrical part computed independently. Going further, we derive from [3] the following simple properties of the Landweber bodies: 6 6 ix? = x ons 2 Avs The coefficient Ay can be factored out and merged into a dimensional constant which defines the midship section. Thus, the normal form of our polynomial is obtained ta s n n=1 2 ag [4e ] -Ay With a, =—— al Ay The symmetrical part of [4c] is a two-parameter family — ia Choi ll Gui a Git ye! 2_ 46 we 4_ 26 LENG tage CO eG an UN me NG) ay (Gra?) because from the boundary condition el) 10 Such families have been called "basic forms" by the present writer’? and designated by (2,4,6;¢;t) since the aubitrary parameters a, a, can be de- termined by the prismatic coefficient @ =|. ndé and by Taylor's tangent value t = -@n(1)/0E. It is thought that the Landweber Series [1] meets almost all reasonable requirements as to wave-resistance properties presented by prac- tice although only two arbitrary parameters ¢,t are at our disposal for the main symmetric part. The reason for this assumption is that from investi- gations on surface ships it is well known that area curves of fine ships, based on the basic family equation (2,4,6;¢;t) are advantageous in the range of high and medium Froude numbers. At low Froude numbers other polynomials are preferable but there the wave resistance of submerged podies becomes rather negligible. We have, however, introduced an additional term a,é* for which auxiliary wave-resistance functions are also tabulated in this report; thus more elaborate investigations can be performed using the polynomial a yy i Ne SJ ¢ a.é 2,4,6,8 The asymmetric (skew) part is the function UA Os CUS ae [4d ] factoring out as we write =a n* = ai(é ar BLES ce be) [Ye ] Obviously the resultant curve n = Ne + N, can have its maximum section out- side of & = 0 and the area of this section will generally differ from one. This slight complication does not involve any difficulties in actual work. Let us investigate oie aoe tube [4e] This trinomial has to comply with the conditions Palo) =. 0 WEG). ttl es © whence 10 thus aes Gy Ihe | (RA [4g] The only arbitrary parameter ds can be fixed by one additional condition; as such we choose the tangent value te at the bow (at the stern the correspond- ing value is -t¥) On*(1 enn nz(1) a aroha i Gs hence aos ¥ b, 2+ oa 2 [4n] the corresponding tangent value te of nes Equation [4e], is obviously a ¥ t. ata The table below shows some examples of skew forms. The parameter oe = Wes dé is an area coefficient referred to the unit square. Plots of é - 8 , €- &° and some other "skew" forms used in the TMB Series are shown on Figure 5. The actual skew part No contains additionally the "strength para- meter" a; see Equation [4e]. Tee rar Be 1/6. O21 G6s's¢ Our numeric evaluations are primarily based on Equations [4c] and [5]—which are stated below—but the theoretical treatment will be carried out along more general lines. J Extended investigations have been made by Landweber and Gertler@ on the influence of an additional term atx! on the form of the body when the geometric parameters are kept constant. Using our system of axes it is easy to perform similar investiga- tions for the symmetric and asymmetric part of [4c] +n, = 1 = Bae anil (Gap 9 Grice tye) 2,4,6 Curve ® (-e?)? @ i-1.56*+0564 (©) 2 3) © © fo) 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 09 1.0 é Figure 4 - Dimensionless Sectional-Area Curves A*(é)(Doublet Distributions u*(é)) of Some Simple Bodies Symmetric with Respect to the Midship Section t ie erie a Irg | ¢- 1.2184 63+ 0.2188 5 ~ 0.55 | a™| ¢-0688 63 - o.3i31¢5 ~ 0.61 Twa | ¢+ 1.9273 63-2.928265 ; 0.8 v* WI = 0.6 ere = on OS: ye fe) Figure 5 - Examples of the Antisymmetrical (Skew) Parts of Sectional-Area Curves A¥(€) (Doublet Distributions w*(é)) Belonging to the Family é+ Da = (1 a b= 2 By adding to No terms with arbitrary parameters a,é" and ag 68 , a manifold ec ecg Bog) ar aé° is obtained. The polynomial [4¢] is completely defined by the four geometrical parameters Wea) On, (1) 2) oat dé dn, (0) (5] 3) ya a on i on, (1) : ) Teer ar eh ca When 1, has to comply with the four equations [5] it can be ex- pressed by n= 1, + CiAn(é) + C4 n(6) [6] eG Ane) = @= 5 tye = 56° [6a] complies with the conditions 1 fo ajnlelag = 0; 0 3 64, n(0) i 04 on(1 ) 4,n(0) = 4on(1) = BE FY: = 0 and AVG) SE) = GI [6b] satisfies OA (0) OA n(1) A n(0) = 4 n(1) aa MSE = 0 Thus, an addition of the functions 4, A, to No does not influence the boundary conditions, [5]. The shape of the curves Ain(é) and A,n(é) is shown in Figure 13. The advantage of this representation is obvious. While in the equations ule ats or u¥(§) =mk(E) + uF (6) the symmetrical (even) terms Ne» MS are the main parts, obviously in On = Ons + Ona 0€ 0g 0€ or CME) Som (et one) 5 the odd terms 0n,/8€, 0% become the main part. 2.3. CONNECTION BETWEEN STRENGTH OF SINGULARITIES AND BODY SHAPE The next consideration is to establish the dimension factors H, and Fo: The flux through the midship section may be written as q = o(2, u*)mb2u [7] Here the coefficient C(b/a, u*) is, as indicated, a function of the elonga- tion ratio b/a = D/L and of the shape of the distributionw*. For very large elongations C(b/a, u*) +1, but for shapes and values b/a used in actual Op- eration C differs from one. A closer investigation of the coefficient C will be given elsewhere by L. Landweber; for the present purpose we introduce C as a correction fac- tor which improves the accuracy of Munk's or Weinig's approximate affinity theorem mentioned on page 5. The dependence of C upon u*, although apparent- ly negligible within the range of presently used submarine hull forms, shows some interesting features. Earlier brief investigations lead to the follow- ing table for C(b/a, w*) (Reference 5). b/a = D/L ee eS a eee areal 05 8 a ceo eho. fos From these results we gather that C(b/a, u*) values for normal submarine shapes apparently can be estimated from the spheroid; an empirical - 3.0825€° + 0.165@2° + 1.9175822|0.820 14 formula C(b/a) = 1 + 3b7/a® may fit the facts reasonably well. For fuller bodies lower values seem to be suitable. The constant aN shalt HO, = are is therefore obtained as Hy = yg w*)b7U [7a] The flux [7] or the strength of the doublet distribution at the midship section must be somewhat higher than the product of the cross section times the speed of advance. For the source, we have o,=% c(z, L*) ay [8] 3. EVALUATION OF HAVELOCK'S INTEGRAL 3.1. GENERAL CONSIDERATIONS The wave resistance experienced by a continuous doublet sheet wu, distributed over a vertical plane and moving uniformly on a straight hori- zontal path, has been calculated by Havelock.® Concentrating the distribu- tion w(x) along a horizontal straight line we obtain immediately “i PY yet a ae Sanju a oee R VémpKs | jee + oe} sec?ed0; K, = [9] with we P = exp(-K-f seca) | * a(x) cos (K_x sec 6)dx = exp(-K_f sec*6)p [9a ] 1 oO a6 {@) [@) 1 + Q = exp(-K_f sec79) [ * u(x) sin (K_x sec 6)dx = exp(-K_f sec“@)q [9b] 1 0 Bee fo) 0) il hence R= 16mpK* ies ~ a) exp(-2K f sec*@) sec?@d0 [9c ] ‘ Using a source-sink distribution we obtain similarly T/ R = 16mp Ke | * (p2 + a?) exp(-2K,f sec?@) sec°@de [10] 0 +a p= | o(x) sin (K,x sec 6)dx [10a ] ak CL q aa a(x) cos (K,x sec @)d6 [10b] -a VB) Introducing dimensionless coordinates x = aé and the expressions u(x) = uou*(é) o(x) = o,0*(&) various forms of the integral for R can be derived for purposes of numerical evaluation. We confine ourselves to the source-sink integral.*® Splitting up o*(&) into a main antisymmetrical and a symmetrical part o*(§) = o%(6) + o%(E) and remembering that an integral taken over an odd integrand between limits of equal absolute value but opposite sign vanishes, we obtain with the desig- nation = nn b* Ze it 2 3 R = 4C“mpg ih i, expl-4 + ¥, sec“@] sec°@. [11] | froxe sin(y, Esec ayachs {forte cos (a6 sec ayats|ac = Ve f 2 3 2 2 const exp [-4 = % Sec @| sec°@[p*= + q*=]d@ [12] 0 We introduce further polynomials for ~ * wr i} 1 2 aa6 3] hence GAG) = Dyin eb [14] or ONG) = wile) a Gall) oy aie (amet je = (ia = 3 with k, m as integers. For the main antisymmetrical part the intermediate integral p* becomes 1 1 = a 2k-1las p¥ = i o*(£) sin (y,€ sec 6 )dé = ees a, Jie sin (y,é secé)dé = 2, a, ul 1 lig Ee O) [15] with ° i Re pe al Se aeon 1(% sec 6) =H (7) = [e* sin(y ¢ sec @)dg = | e**”tsin(ye)as [16] 16 Here for brevity the designation y = % sec @ has been introduced, [16]. For the symmetrical (even) part il q* = j o*(&) cos(y,§ sec @ )dg = - >" (2mt1 Ja La MU ye Se che) [17] ™m™ with = = 2m 2m Meg (om sec 6) My (7) Me = | é cos(y, é sec 6 )dé =i é cos y dé [18] inserting [15] and [16] into [12] one obtains ing be ee f = F R = 40? mpeg A % | exp [-4 Tite eae a] (2% ai Mes) + + (Ps (anal M! )=i sec°oda [19] 2am This formula is suitable for numerical computations above in special cases, since tables of the functions My, a> M! (y) are available and will be m published in a TMB Report. 3.2. TABULATION OF RESISTANCE INTEGRALS FOR A FIVE-PARAMETER CLASS OF BODIES As mentioned before, auxiliary integrals have been prepared for the three-parameter symmetric distributions of Equation [5], ux(é) (asymmetric in o3(§)). = n st -1 MelG) Ss » a, é beth Gy e(G)) SS » na, §" 2,4,6,8 2,4,6,8 and the one-parameter skew distribution uX(é) (symmetric in o%(é)) WE(E) i= 6 DESY (HDs ylang image) eal ptt lO) Ga The computations are based on a slightly different form of R (see Appendix II). Substituting 7 =) ynse cue sec @ = me es = V (7/%,)= -1 one obtains (7/¥,)*dy Vays -1 3 (y/ = 4c2 met | em[-4 ro] wees (2 ia (y))? + + (2° (2m ase in) a} dy [20] Lie dys sec> @sin 6d6; ee sec? 9d@ = hence V7 putting for abbreviation 4 ; 4p all : A= 6 I exp |- re] fly) Paes 2) iggy kaa + & (ret) (2st Jay, 89M), ms bar [21 ] with i, j, s, r integers. R can be built up of terms of the type 2g Til 2j71 36 Sil. aq So = Uf y® fe exp|- 7 2|f(y)M,._, (vIM, _, (v)ay =m [22] Y ie) for the symmetrical part of the sectional-area curve and 2s em [- oF |ey My (IMs, (v)dy = my [23] for the skew part. The final result is therefore obtained as a quadratic form in the parameters a or, better, na, By : F : a 61 ee 2J ies aly Bq ma ig & (ert (asi NE cea 2s Mra with the tabulated integral TERE UD ss pet a. as main parts of the coefficients 2i Zi Moen oy ond IES We mention again the fortunate circumstance that the contributions to the wave resistance due to the symmetrical and antisymmetrical parts can be calculated independently and added. Returning now to a family of distribution curves given by Equation [4c] but generalized by one additional term a, é,: & n n-1 1 npaid Ie DY abs be tae cee a - Dn af + 8a,§ [44] i 1 The wave resistance can be calculated by the functions My, M5 ™ 5 ™ ; Moo Moe Moa 33 ™ 35 ™ 37 Nee ™ 24 ™ vu 55 ™ 57 a ™ 18 tabulated in the Appendix III. The integral R and the functions ™M and ™m depend upon the two parameters Ue 1/2F2 and f/L. The tables have been pre- pared for a range 0.5< Mes <€ 10 and f/L = 0.125, 0.25, 0.50. Additionally, for ms an intermediate depth of immersion ratio f/L = 0.1875 has been intro- duced. From the wave resistance integral it follows immediately that the ratio depth of immersion over length f/L is theoretically preferable to the more commonly used ratio f/D, since f/L appears explicitly as factor of the exponent of the e-function under the integral. With elongated bodies the ratio b/a or D/L influences primarily the constant Cia 4c? mpg b*/a only, though in a very decisive way. Although the lower speed limit Le: 10 (F = 0.224)—up to which the auxiliary integrals have been computed—is rather high, it is thought that for normal hulls with g< 2/3 moving at greater depths than D, the wave resistance becomes unimportant when F <~ 0.224. The low-speed range may, however, be interesting in connection with other research problems. In principle the wave-resistance equation, [24], solves the problem for any sets of a, within the family following [4i]. Actually since the relative error of the tabulated functions is approximately 0.0001, a loss of accuracy may occur—when the coefficients an reach high absolute values with alternating signs. It is not probable that difficulties of this kind will be important in connection with submarine work; besides, they can be overcome to some extent by plotting suitable simpler resistance curves and by inter- polating. 4, REPRESENTATION OF RESISTANCE CURVES 4.1. THE DIMENSION FACTOR Co AND DIMENSIONLESS REPRESENTATIONS The dimension factor in Equation [20], Cie 4c? mpg b*/a, has a rather unusual form, but it will be widely used throughout this report because of its theoretical merits and the comparative ease with which it can be connected with more familiar expressions. We rewrite, in terms of the displacement A, 2 2 Gl =) grcbe apie ce Ae ce [25] a 2 2n2 aU. YO =i ga or t= Po eb C fe) 2bacs gaz 19 Hence we can immediately derive the resistance per unit displacement for a given b/a and shape when r, is known. The introduction of the displacement A in [20] is open to objection Since so far we have not distinguished between the length of the body and the distribution. We repeat the definitions: 2a is the length of the distribution along the axis L is the length of the generated body 2b = D is the diameter of the generated body Obviously for the displacement of the body we must use L = 2/. Then 2C*b@a garl Further, the ratio b/J = D/L is technically more important than b/a; hence Car wimr pieNC= bs aed [25a] noo ea ; ils een pel [25d] ON: OCA be al or 2 Fee a % [25e] r =g-G4($)2 [26] One should not, however, overestimate the influence of the length correction. For the spheroid aril 1 2 2 2 poe nee A iccalray i.e., influences the C* correction by less than 10 percent. Further, even the introduction of the more important C factor does not lead to an exhaustive correction since we know that not only the midship section but the whole trend of the curves changes with increasing b/a. Thus within the limited accuracy of the present wave-resistance theory we generally can put //a=1. It is of course important to use all approximations in a consistent and clearly defined way, so that fair comparisons can be made. We note particularly, that for the spheroid R = = Ino Cs Da /ee 20 For comparison with experiments the coefficients Cy referred to the wetted surface S is advantageous. We write 2,2 2 a. = lines Raa ry Ant“v"(2) [27] p/2 UPS SF? or introducing a surface coefficient S (Reference 17) Cy = S/nDL CIN CB NNT ab ee a C2 ab Cy = To C (a) Chee agitic ¢, (a) % [28] with », = 1/2F2. For elongated spheroids Cy = 0.79. The importance of the resistance coefficient Cy referred to the wetted surface S justifies a short digression on the calculation of S for bodies of revolution. Solutions of the exact expression (Equation [29]) can be obtained in a closed form in exceptional cases only, as for the spheroid. Of course it presents no difficulties to evaluate the integral numerically, but a simple approximate formula can be derived at least for the surface area of a restricted class of very elongated bodies of revolutions complying with the condition that the end tangents of their meridianal contour do not become vertical; it is similar to the well-known expression for the length of a slightly curved arc, see Appendix I. 4.2 RESISTANCE CURVES OF SIMPLE SYMMETRICAL BODIES Since the presentation and the discussion of resistance curves is the main subject of the present report, various sets of such curves have been computed. Essentially, the resistance properties of the following three groups of body forms (distributions) have been investigated: (a) A set embracing a wide range of prismatic coefficients, which fur- nishes a general review of the resistance as function of the form (IV,2). (b) A set dealing with four TMB models. This raises the problem of the influence of asymmetry with respect to the midship section (IV,3). (c) A group consisting of systematically chosen forms belonging to the two-parameter family (2, 4, 6; ¢; t) (VI); for the same family some calcula- tions of shapes of least resistance are presented (V). The procedure adopted leads to repetitions which, having in view the importance of the subject, have been thought to be advisable. Because of the complicated dependencies involved the interested reader can more 0.8 0.7 0.6 0.5 R 4cen Pg b*/a 0.4 fr, 0.3 0.1 2| - 32,3 + 16%, = 12M 3 + 433 Curve wa" represents the additional resistance function r) due a an asymmetric function é - e3, see Figure 5. -— 6M + 2M2o ia Zz 10 | | eo 0.707 0.500 0.408 0.354 0.316 0.288 0.267 0.250 0.236 0.224 aed PS Tae R Figure 6 - Wave-Resistance Coefficients r easily WENT 6 and at stance, Figures 6, 7 and 8. _ of Symmetrical Unc? og b*/a Bodies as Defined on Figure 4, f/L = 0.125 draw conclusions from the rather comprehensive plots than from any We are mainly interested in the range of Froude numbers F below the maximum of the large hump in the resistance curve; see, for in- Above the maximum the absolute value of wave resistance decreases comparatively slowly with growing F, but the ratio wave resistance to frictional resistance drops quickly. Therefore, at high speeds 22 Prismatic No. form Coefficient Resistonce ENN (U=% oe (ony maori + 167, 1- 1.567 + 0.564 9m,, - 12M, + 4am. 4m, (f/L = 0.1875) Curve 7 represents the resistance coefficient lo for a distribution 1-€2 (spheroid) at f/L = 0.1875. ee a ed ee eee ee ee ee ee EE EEE Eee oo 0.707 0.500 0.408 0.354 0.316 0.288 0.267 0.250 0.236 0.224 Figure 7 - Wave-Resistance Coefficients of Symmetrical Bodies as Defined on Figure 4, f/L = 0.25 the wave resistance of elongated bodies such as torpedoes represents only a small part of the total drag. It has been shown in References 4 and 5 that in the limit of very large Froude numbers the wave resistance becomes pro- portional to the square of the displacement or eo Par In general, throughout the present report calculations have been extended to F=1 (y, = 0. ae and te F=1.58 (y, = 0.2) for the parabolic distributions 1 - hea - é4 - B® only. From an approximate investigation 23 0.07 oneeece (pire el Pep Bree AL eA iat os pepede gaia Resistance Coefficient ro - l2™;3 + 4735 4 5 6 v 8 9 10 2 | Mom ior? a) 0.707 0.500 C.408 0.354 0.316 0.288 0.267 0.250 0.236 0.224 = u FO Yat Figure 8 - Wave-Resistance Coefficients of Symmetrical Bodies as Defined on Figure 4, f/L = 0.5 it appears that the resistance curves R (y,) plotted over Me have a vertical tangent at en 0, but no attempt has been made to draw accurately the range of curves below 7, = 0.5. To obtain a general idea of the wave resistance for various symme- tric distributions w(é) (sectional-area curves A*(é)) graphs have been plot- ted for following simple cases:* *As before, by symmetry we mean symmetry with respect to the midsection. 2 0.533 2) 0.6 1 3) 2/3 2 4) 0.8 yb 5) 6/7 = 0.857 6 6) 8/9 8 Figure 4 shows these sectional-area curves and Figures 6, 7 and 8 the corresponding resistance coefficients as functions of Vo = 1/2F?, with an additional non-equidistant scale for F. The choice of es as independent variable yields an appropriate picture of the wave-resistance values at high speeds. From the Figures 6, 7 and 8 a rather complete understanding of the wave-resistance properties of various symmetrical forms can be derived. hef- erence is also made to Figure 12 and the pertaining discussions in the text. The influence of the depths of immersion follows immediately from a comparison of Figures 6 through 8; also, cross curves can be plotted over f/L as the independent variable. Figure 11 shows this dependency for 4M? which is the resistance function of a spheroid A¥(€) = 1 - €*, with y, = 1/2F° as parameter. We note that with increasing depth the resistance drops more quickly at small than at large Froude numbers F. This is rather obvious; it will be discussed later more thoroughly that the most indicative parameter is the ratio f/A , where A the length of the free wave is A = 2nF*L. In Figures 9 and 10 the resistance curves for three depths of im- mersion have been reduced to approximately the same maximum ordinates. This rather artificial approach yields a clear idea about the shift of the last hump (of its steep rise as well as of the position of its maximum) to higher Froude numbers with increasing depth of immersion; it further emphasizes again that the rate of decay of the wave resistance with ‘increasing depth is much higher for low Froude numbers than for high ones. Figure 12 represents a coefficient r= RAV a= /2Cbe)— r/o. For approximately constant C® (very elongated bodies) and given a*/b® ratio, a R/A, i.e., the figure yields a comparison of the resistance per unit displacement for various forms. The discussion of the various graphs leads to the following summary results: 0.7 0.6 0.5 0.125) 0.25) 0.50) so} iB 4M, (F/L 2.95 x 4M, ,(F/L 15.10 x 47m, (f/L ° a Resistance Functions 0.2 in| 25 0.707 0.500 0.408 0.354 0. Sich 0.288 0.267 0.250 0.236 0.224 F = ee Figure 9 - Comparison of the Shape of Wave- Resistance Curves for the Spheroid n(é) = 1 - &é*; the Curves are Reduced to Approximately Equal Maxima A. Small depths of immersion 1) Within reasonable limits, the peak value of the R/A curve does not depend too much on the shape of the body,* especially upon the prismatic co- efficient. 2) The merits of full forms, over a wide and possibly important range of Froude numbers 0.35 < F €< 0.50, are clearly emphasized, as well as 3) The heavy penalty which has to be paid for high prismatics at lower F. *If more elaborate results are desired they can be derived from Figures 28 through 35. 26 Resistance Functions .- 16[m, + ™,,- 2%,| for f/L = 0.125 and ~3.48 x 16[m, + ™,,- 2%M,,] for f/L= 0.25 PRA A Rete AN A 0 0.707 0.500 0.408 0.354 0.316 0.288 0.267 0.250 0.236 0.224 Fhe Figure 10 - Comparison of the Shape of Wave-Resistance Curves for n = (1 - &*)* Reduced as by Figure 9 B. For larger depths of immersion the dependence of the peak values of R/A upon ¢ becomes more pronounced; the advantage of high prismatics in the range mentioned in A(2) is, on the average, reduced. 4.3. RESISTANCE CURVES OF ASYMMETRICAL BODIES Further curves representing the wave-resistance coefficients of the four TMB models represented in Figures 14 and 15 are shown in Figures 16, 17 and 18. Before discussing these particular asymmetric models, however, an investigation must be made of the influence of asymmetry on the resistance. Figure 5 represents examples of asymmetrical lines belonging to the family n¥ = ¢ + bie me (iliact bs) €°, Equation [Ug]. arf [A EE eee NE eS A | ee) ee eee fe} 0.125 0.250 0375 0.500 f/L Figure 11 - Wave-Resistance Coefficients of the Spheroid as Functions of f/L with Ue = 1/2F° as Parameter (Re A ee eee ee ee eee co 0.707 0.500 0.408 ssa 0.316 0.288 0.267 0.250 Figure 12 - Wave-Resistance Coefficients Sere phar = of Symmetrical Bodies Shown on Figure 4, Ph = Ost25 28 0.10 0.08 0.06 Figure 13 - Distribution Functions Following Equations [6a] and [6b] The curves I, to IV, have been derived from the TMB models (Figures 14 and 15) by reducing the coefficient of & to unity. The procedure of ob- taining the symmetric and the skew part from graphs is obvious: The first one is the arithmetic mean of the fore and afterbody ordinates 7, = ER and the latter one the difference — or Tn la respectively. The computation of the wave resistance due to asymmetry is based on Equation [24]: For the trinomial = 3 5 Teas (agit ONG te Dias) 29 STOPOW GNL Jo SeAang evorty-[euot4oeg - HL sunset YW GOL0'0 = % -j090 peso = °® 08'0 00S2'0 (20), piezo (0),v Z98'0 - z96z 1-14 100! G2Z-SS 10 SO Ob ‘ON |apoW 2p98'0 + 296211 = 4 | | (,9G2bEO + I ILI] -9 9PLL0)+(,9 29EE'0 - ,E5Z01 + ,998891-1) =U = (3),y a 1 — 4 O72'| 99260 = lO- 2 0- ¢£'0- v'O- S'0- 9'0- 20- 8'0- 6'0- Ol- Oo at S Je | 020 Ovo gisoo =°y 7090 rego = @ 1 uoij9as diyspIW |. UJ 0} Pasajay jua!oIyya0q oyDWSIg 080 -,. 00620 (2O)yv SP60- So820 - Toye osbe'o-ge9z1 ='y 00! 2-09 10 SO Ob “ON I@poW wes ; Oral ; OSP8'0 + 8892! = 4 PBll'0 + 9h6S910 - IZI~SO) + (g9EzIZ'O + ,988S0'0 — ,92ESI1-1) = U=(3)V sp | Ee eM = pes n jeetrereenl ge es hop 30 STOPOW GNL JO SeAINO eody-[eucT Neg - G{ sansTy 2 90 oie) v0 xe) zo 0 (o) 'O- zo GO 4170p Gi Oe Ge Oe GP OIF (e) bl190°0 = °3 GEz190 = OP ‘ L —0z'0 960 = ooszo0 _ (0), z * Tiveo ~ (oy 1 6be80- 8021 = ‘4 A ovo | 6veeO+ BOre2! = ' (g20l'0 — 9220 - 92€'0) + (g328E20+ 9696011 - 292I1p90 -1) = U = (a) a a 090 = L e Sos — 08'0 ee PSs SSS see (0011 S.- SSS E Se GZ-S9 10 SO Ob ‘ON |apow L | | | L 02'I Ol 60. 80 20 9'0 sO v0 xe) 20 ie) (o) \'0-__2/0= fe: vO- GO- 910- -20- —-8'0- ~— G6 0- ——OI- 3 ) 9eb0'0 = °3 SIE 06020 = oz0 ‘7 .00szo (eOly 2960 - SOsee coy ¢o1e'0 - siz = '4 ovo eole'o + 2si2i = 5 (,3610€0 — ,928610 + SIEOI0) + (,912v2'1 + ,36160'2 - ZH10SI'0 -1) = U= (3) 09'0 S2-O2 10 SO Ob ON |apOW | eer 080 001 021 31 co 0.707 0.500 0.408 0.354 0.316 0.288 F Figure 16 - Total Wave-Resistance Coefficients and Coefficients Due to Asymmetry of the Four TMB Models Shown in Figure 14 and Figure 15, f/L = 0.125 32 0.25 0.05 0.04 0.20 0.15 0.10 0.01 0.05 Ea rae ce) [Roo er LS Ap ered env ae men co 0.707 0.500 0.408 0.354 F pa oe Pirate aoe Figure 18 - Total Wave-Resistance Coefficients Po of the TMB Figure 17 - Total Wave-Resistance Models Shown in Figures Coefficients r, of the TMB Tanga eet Ol Models Shown in Figures (Mand iSi0 G/l = O25 with the derivative oe AW see é 1 3 5 we obtain = é (vas 2m + 2m™ i inet mm + ! 4 Gos [™, a5 oe ou My a, ae ee Mi eee me, [33] Figure 19 shows the functions a Ra 1” -:- —___...... 02“ Unpec? b*/a 33 0.3 ° 0 0) 2.0 3: es “6.0 7.0 8.0 9.0 0.0 ue a 2Fe co 0.707 0.500 0.408 0.354 i eile 0.288 0.267 0.250 0.236 0.224 = ae Figure 19 - Wave-Resistance Coefficients aaa Due to Antisymmetrical Distributions Following Figure 5 corresponding to the distributions é - g> é- Be and Me shown in Figure 5, where curve i is derived from the TMB body, Figure 14. The "amount" of asymmetry which corresponds to the equation n, =é is very large, but by assuming the strength parameter a, < 1 (Equation [He]) more usual distributions are reached; for these asymmetric terms the wave- esistance curves are obtained simply by multiplying the ordinates of Figure == 34 19 by ae. The resistance curves in Figure 19 corresponding to é - Ee and IE are somewhat similar in the range of the large hump and the ratios of their absolute values are of the order of 0.5. In the range of the second hump the ordinates of both curves are small, but it is characteristic that here a much lower resistance corresponds to the finer line If, rather than to €- €°. We return now to the four TMB models designated by I, II, III, IV shown in Figures 14 and 15. In these figures the line Ax(§) shows the sym- metrical part of a body. The resistance results are plotted on Figures 16, 17 and 18;* in them the lower set represents the contribution due to antisymmetry Ra Toa i UrpeC? bi/a ’ the upper set the total wave-resistance coefficient a R sti R, 0 = Ugpgc? b47a The computations are made under the assumption that the doublet distribution u*(é) = A*(€). With the model number rising from I to IV the prismatic in- creases and the asymmetry decreases. In the important range of Froude numbers 0.50 2 F 2 0.35 the finer models are extremely unfavorable because of the low prismatic as well as because of the very pronounced asymmetry. When comparing the total resistance values a slight departure from symmetry generally is advantageous because of viscous effects. It has also been pointed out that small asymmetric terms do not increase appreciably the wave resistance even in the most sensitive range of Froude numbers, say 0.45 2 F 2 0.35; this is well supported by our present results, for instance by Curve IV. Further, the obvious fact must be once more emphasized that an immediate comparison between symmetrical and asymmetrical bodies—as to their wave-resistance properties—is only feasible when the sectional area of the former A*(E) is the even part of the sectional area of the latter A*(6) = AX(é) + aX(E) It is entirely possible to obtain asymmetrical forms with wave-resistance properties which are superior to the corresponding ones of a poorly chosen symmetrical form, equal prismatics and principal dimensions being assumed. Similar computations have been performed for other depths of im- mersion; some results are listed in Table 2 of Appendix III. Obviously it is not difficult to investigate the wave resistance corresponding to any curve of the family defined by Equation [4e] at the three depths of immersion for which the integrals have been tabulated. *There is a slight error in the resistance curves R of Model III due to inaccuracy in computations, but it does not invalidate the comparison. oo 0.707 0.500 0.408 0.354 0.316 0.288 0.267 ; arr, ape a R Figure 20 - Wave-Resistance Coefficients C= p/2 US Referred to the Wetted Area S for the Four TMB Models I-IV, f/L = 0.125 (For Comparison of Range Single Curves for f/L = 0.25 and f/L = 0.5 are Shown) To check the order of magnitude of the wave resistance and to en- able a comparison with experimental data, resistance coefficients C, of the four TMB models I to IV are shown in Figures 20 to 22, calculated for b/a=1/7 and C = 1.07. In this case the depth of immersion ratios f/L correspond to the technically more familiar f/D ratios as follows: 1B /AL 0.125 0.25 0.5 f/D 0.875 1.75 3.5 Assuming a rather high viscous-drag coefficient (c., = 0.003), the relative importance of the wave resistance at various depths of immersion 36 0.003 0.0005 0.0004 0.002 0.0002 0.00! 0.0001 co 0.707 0.500 408 354 F Figure 22 - Wave-Resistance Coefficients Crs 72 U2S Referred to the Wetted Area SrmTOL/ Or, 0500. 0408 0354 S for the Four TMB Models F I-1V;,,0/l: = 0.50 Figure 21 - Wave-Resistance Coefficients c_ = WO HOES Referred to the Wetted Area S for the Four TMB Models Tomy ty le One and Froude numbers can be estimated for a comparatively wide span of prismatic coefficient 0.71 2¢ 2 0.59. Attention is drawn to the changes in the mutual relations between the curves in Figures 20 to 22. These changes are dependent upon f/L and upon the obvious shift of the peaks towards smaller Froude numbers as compared with Figures 16 to 18, because of the factor U* in the denomina- tor of Cie Considerations of wave resistance may play a significant role when fixing the optimum elongation ratio D/L as long as free-surface conditions are important. Assuming both ¥ and ¢ to be constant, the surface S and there- fore the viscous drag vary only with VL/D while the wave resistance varies 37 with (D/L)? multiplied by a complicated function ro of F. Restricting F to a range ~0.6 2F 2 0.35, oA is monotonically and, on the average, heavily decreasing with decreasing F. Thus any reduction of D/L heavily reduces the wave resistance. 4.4, LIMTTING DEPTH OF IMMERSION It is important to know below what depths of immersion fy the wave resistance can be neglected. This limit can be established from such cross curves as shown on Figure 11; it obviously depends upon: a. The Froude number F or y, = 1/2R= b. The L/D ratio, and @o The dimensionless shape of the body, primarily its prismatic coef- ficient ¢, especially outside of the large hump. However, some additional simple reasoning may be helpful when curves R = R(f/L) are not available. We can consider the wave resistance as negli- gible either when a. It is a small percentage of a given standard resistance, or b. It is less than an absolute small value 6R. Some obvious differences in results due to the different approach have sometimes been overlooked. a. Assume that for f Zit the wave resistance becomes less than a given small fraction e« of the wave resistance Ry at a standard depth, for instance at the immersion of one diameter; to is derived from a ratio of the resistances in question. Comparing bodies of equal length, fo depends upon the Froude number and upon the dimensionless shape of the body, but only very slightly upon the elongation ratio D/L = b/a, since the latter influences only the constant UmC2og b*/a, which drops out in the comparison. b. Assume that the limiting depth fo is derived from the condition that the wave resistance is less than an absolute value 6R independent of the standard resistance Ro: Comparing again bodies of equal length fo now becomes highly sensitive to changes in D/L. A rough idea of the necessary limiting depth Lo of immersion can be obtained from the decline of the water disturbance with increasing depth in a plane sinusoidal wave; this estimate normally gives exaggerated values fo: Denoting the wave amplitude by ha and the amplitude of the distur- bance by h one obtains 2 7. ba eis 38 putting further Ki = = 2nF°L Z WT. SA h=h. elF and prescribing h/h,, » for instance assuming h/h,, < 0.01, one obtains £5 2~0.15a or fo /u > 1.50? This estimate is superficial for many reasons: a. The resistance depends rather on the square of the generated wave amplitudes, b. The actual problem is three dimensional, and Os The body shape is neglected. However, it shows at least that in principle the limiting depth cannot be expressed as a fraction of the dimensions of the body alone, since it depends upon the length of the free wave A or the Froude number F. From practical considerations matters are somewhat different. As mentioned before, at very high Froude numbers the ratio of wave resistance to frictional drag is normally very small. Thus the problem of finding an accurate value of the limiting depth becomes rather unimportant since even grave errors in computing it do not lead to appreciable errors in the total resistance. : 5. BODIES OF REVOLUTION OF LEAST WAVE RESISTANCE 5.1. TWO-PARAMETER FORMS In an earlier paper? endeavors were made to determine distributions of least resistance for given Froude numbers. The results varied with Froude numbers and depths of immersion, which is quite natural in the light of such resistance graphs as represented by Figures 6, 7 and 8. An important feature is the peculiar "swan neck" form obtained for higher Froude numbers —equal to and above F = 0.35. Because of the limited accuracy of these former calculations the problem has been reconsidered here. The present investigation supports the earlier statements. The formalism needed is very simple. Some controversy arose as to how far the application of exact methods of the calculus of variation is consistent when dealing with surface ships;> the results obtained did not 39 lead to reasonable ship's forms. However, when we restrict ourselves to fam- ilies of curves expressed by polynomials with few arbitrary parameters, we really obtain an ordinary minimum problem and do not need to bother about the difficulties connected with the application of the calculus of variations. Take for instance the family (basic form) (ele Ge a tAlGs mG) ets cE) [34] with two arbitrary parameters. The wave resistance R is given as a second degree function in a, and a: R= 4B a®?+ 4B a? + 8B aa + 24Ba + 2UHBa +B [35] 222 44 4 24.2 4 2.2 4 4 O where wo tt m,, - 6m. + 9m ae 1a 55 Baa 5 Mh © Vai. % Hie Bon is ane % ze i: IM, fi 6M, | Be Sih Hi) 2 15 55 B =2m_ - 3m 4 35 55 5 z 50M | differentiating R partially with respect to el. and als one obtains the min- imum conditions OR aA al = a Bot. + B ts 3B 0 2 [36] ORL = Oa, Bots i BBs + 22, : whence 3(B B =) By Bie A ee 2.44 424 = Bion Biceien Be 2e 44 24 [37] 3(B. Bo - BB) :] orca e 4 22 2 24 < BiB ieeBe 22 44 24 These equations lead to results which are not applicable to practice when Vise 1 and of restricted interest when oS 2 (f/L is assumed equal to O51 25})) HO Tiss T Ts f/L=0.125,7 =1- 7.1462 + 18.9364 — 12.7966, 6 = 0.58, t= 15.30 oo f/L = 0.25, = 1-11.95¢" + 31.126* — 20,176°, 6 = 0.36, t= 20.44 Ss GS IN 2 ne nc aaa eet eo 0 0.1 0.2 0.3 5 0.6 0.7 0.8 0.9 1.0 Figure 23 - Doublet Distribution for Least Resistance, Two-Parameter Forms, F = 0.403, Uae 3 When Mens 3 the distributions shown on Figure 23 are obtained. We note again the difference in the shapes when f/L = 0.125 and f/L = 0.25. Ex- tending the calculations to Piss 4 and ¥5 = 5» curves of more and more "rea - sonable" character are obtained as shown in Figures 24 and 25. The apparent failure of the theory to yield useful results in some cases, is often due to lack of suitable conditions imposed. There is no reason, for instance, to expect a solution which leads to a "normal" prismatic coefficient if no restrictions as to this coefficient are made. On the con- trary, it is rather fortunate that one obtains results which meet other re- quirements of practice (i.e., are "reasonable"), without this restriction in certain ranges of Froude numbers. | 5.2. ISOPERIMETRIC PROBLEMS, ONE-PARAMETER FORMS Introducing a condition ¢ = const we obtain an isoperimetric pro- blem. Then Equation [34] retains only one arbitrary parameter. This can be f/L=0.125, n= 1- 3.1976 + 6.6676* - 4.4704°, ¢ = 0.628, t= 6.55 =0.25, 7=1- 4.61767 + 10.915 ¢* — 7.2984°, d= 0.601, t = 8.36 Figure 24 - Doublet Distribution for Least Resistance, Two-Parameter Forms, F = 0.3544, Ue 1- 2.4476? + 3.46124 - 201466 Figure 25 - Doublet Distribution for Least Resistance, Two-Parameter Form, F = 0.316, i D 42 interpreted, for example, as Taylor's tangent value t. The resulting equation is of the type n(g) = nl) - go' [e2 - Set + Leel [38] Here n,(é) is a given polynomial complying with the condition ¢ = const; its tangent value to may be chosen in such a way that the equation No is as simple as possible. The function 5) MUNG Satie, alent | ms - gfe? - ete te’ = 4.06) [38a] has the properties: dA, (1) ee 0€ 2 A, 7 (0) A,n(1) =0 1 Ba A. n(é)dé =0 Je t! is the variable tangent parameter, the resulting t of the Equation [38] being obviously t = t' + to: Assuming ¢ = 2/3, Pa Seale ee Sh- -2¢— pti [e - Bee + 70°] one obtains pat tre R= 2t A+ 3tia + 4m [39] al = 2 tA + 3A =0 [39a ] is 400 BAG, _ £0) “ LG 3 9 Mee 49M 3 Als 6 se Uivtes 3 Ines snipe 20 ia ey somes eas allel ye hence Ses tl = +3 [39d] Another isoperimetric problem is given by t = const and ¢ variable. Although this problem looks somewhat artificial since there are no technical reasons to keep the tangent of the sectional-area curve rigidly fixed the a) f/L = 0.125, n= 1—- 2 + 2.02442 — 264 + &°), gd = 0.821, t=2 f/L = 0.25,n= 1-62 + 5.954 (e — 267 + 6°), g= 1.122, t=2 Figure 26 - Doublet Distribution for Least Resistance, One-Parameter Forms, F = 0.408, 1 = eee t a ee SAS 2 6 4 6 = = Helis h, f/L = 0.125, n= 1- €° - o212(e? — e&) — 0576 (4% — 6), g= 0.783, t= 4 Wa f/L = 0.25, 7 = 1 6+ 3.134(6? — 6°) — 7.268 (6% — €°), 6 = 1.038, t= 4 Ne Figure 27 - Doublet Distribution for Least Resistance, One-Parameter Forms, F = 0.400, Ui Yl results are interesting. Figures 26 and 27 show that assuming rather differ- ent t values optimum ship lines with a similar trend may be obtained. We notice that the optimum area coefficient @ for a medium depth of immersion f/L = 0.25 is much higher than for a slight immersion f/L = 0.125. This might have been inferred from the shift of the resistance curves follow- ing Figures 6, 7 and 8. The necessary formalism is again very simple: assuming as before a curve 7, with the fixed t = ty value, for example as before, ie = 1 - foe ty = 2, and denoting ¢= Po coils Pens ei allay (= 2 Zi. ae 4) [40] Ais) = We eNGS |= 25> a 5") [40a ] complies with lho PMO) = A\c(0))) = 0 1 : 2 [4 .n(s)as = 1 04 n(1) : og from se = -26 + 26.25g'[é - 4e9 + 36°] we obtain Bi o' = 36.25 At a] 2 with iz nl 3 + VO ele Caan Cele aus Al =m, - 4m, + 3m, 6. RESISTANCE CURVES OF THE FAMILY (2, 4, 6; ¢; t) A systematic survey of resistance properties of ship forms can be obtained by a different approach, i.e., varying the parameters of a given family of ship lines and plotting the corresponding resistance curves. Re- stricting ourselves to a two-parameter equation (28s CaCH ear ic ee ee [34] 2,4,6 45 r) Equation [35] can be used for calculating the resistance, or still simpler, eS 2 a 2 2 5 ‘las as Hay mes i Be nes ty #28, en t 6a 2, ee ii Wee 2. Weed [42] The parameters a. a, and a, are connected with the basic form co- efficients gand t by the equations = 10 3 a=9-oer+e t aS 15+ 1Pg-2t [43] ais 1 - a, - a, Table 3 contains wave-resistance coefficients Te bog eS OG a 25 5) Ehovel 0.68 2¢ 2 0.56 with an interval of Ag = 0.02 within a range of Froude num- bers 12 F 20.25 (for t = 0 additionally = 0.50, 0.52, 0.54) at a depth of immersion ratio f/L = 0.125. The corresponding curves spaced Ad = 0.04 are shown on Figures 28 to 35 grouped following t and @. The main purpose of these plots is to dem- onstrate the dependence of the wave resistance upon t for ¢@ = const; it is interesting to note that the peak values (cf page 24) differ as much as ~15 percent for t = 0 and t = 3, in close agreement with results known from stu- dies of surface ships and the tendency exposed by the minimum calculations. One should, however, remember that theory tends to overestimate the favorable interference effects and that viscosity precludes the realization of excessive angles of run. On the other hand, for very high Froude numbers the relative importance of asymmetry decreases, so that forms with steep slopes at the bow and moderate slopes at the stern may be advantageous. SUMMARY Using Havelock's basic work and some former investigations by the present author, a systematic synopsis is made on the wave resistance of bodies of revolution. Tables evaluated by the Bureau of Standards and graphs are given which allow the investigator to estimate immediately the wave resistance of a wide class of bodies of revolution defined by doublet distributions along the axis expressed by polynomials. Some discussions refer to the relations between this distribution u*(é) and the sectional area of the body A*(é). For "normal" shapes of dis- tribution the usual assumption is made that there is affinity between u*(é) 46 % and A*(é). In extreme cases the shape of the body can be calculated by methods due to Landweber and Amtsberg; no corrections, however, are given for the influence of the free surface on the shape. Within the first-order theory the resistance can be split up into a main part due to a symmetric distribution with respect to the midship sec- tion and a part due to asymmetry, which can be investigated independently. . Large amounts of asymmetry can influence the resistance detrimentally in some ranges of the Froude number. The investigation of the resistance as a function of the body form leads to conclusions which sometimes are contrary to those derived for sur- face ships. The choice of appropriate prismatic coefficients varies deci- sively with the range of the Froude number, as is clearly illustrated by the numerous graphs. The same applies to the influence of. the tangent value t. Ceteris paribus the resistance is approximately proportional to the square of the midship section. The dependence of the resistance upon the depth of immersion is in- vestigated; this dependence is best explained by the ratio f/X, where pete is the length of the free wave. Thus for common prismatic coefficients the wave resistance decreases rapidly with increasing f except in the range of high Froude numbers (largeA values). In the range of high F the calculation of forms (distributions) of least resistance leads sometimes to results bare of practical applicability; by introducing suitable restrictions such diffi- culties are avoided. These investigations show important peculiarities of the distributions. A set of resistance diagrams calculated for the family (2,4,6;¢;t) gives a survey of the resistance properties of a class of normal bodies. Acknowledgment is made of the valuable help provided by Messrs. Samuel Lum and David Rego and Miss Janet Kendrick of the Taylor Model Basin. The author further wishes to express his gratitude to Dr. Alt, Dr. Levin and Mr. Blum of the National Bureau of Standards. Finally, it is a pleasure for the author to thank his colleagues, Mr. Cummins and Mr. Landweber, for re- viewing the present report. 6 aie ! _| j 0.316 0.288 0.267 0.250 mh i Sy a se ERE Figure 28 - Wave-Resistance Coefficients Po UC? pg b Ja’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = goes cata care: Over 7, = 1/2F* and F = U/VgL t = 0 0.7 0.05 0.6 0.5) Y, L ll ls qe 4 ——— 1 4] o 0707 0500 0408 0.354 0.316 0.288 0.267 0.250 F = a [ae Sate eas Riek 2 Figure 29 - Wave-Resistance Coefficients irs Un? pg b! /a’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = at amle Goal aca cs Over 7%, = 1/2F* and F = U/VgL iG Sy 48 (ee 1 ese as ha | o 0.707 0500 0408 0354 0316 0.288 0.267 0.250 F si NE GRR Figure 30 - Wave-Resistance Coefficients ae U2) g b*/a’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = 1a Gy Goa Over y, = 1/2F° and F = U/VeL t=2 [Maar te | | EASE ce Eee) < 0.707 0500 0408 0.354 0.316 0.288 0.267 0.250 F es atc ae a Figure 31 - Wave-Resistance Coefficients oe UnC2pg b4/a’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = 1-a é7-a é*-a é® Over y», = 1/2? and F = U/VeL 2 4 St = 3 [@) 49 Lia ! ! a | xc =r 0.707 0500 0.408 0.354 0316 0.288 0.267 0.250 F f a R Figure 32 - Wave-Resistance Coefficients Bees lnC2pg b*/a’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢@;t) = Ieaieona eacane | Over y, = 1/2F2 and F = U/VeL % = 0.56 | 2 ae | 1 i L IL } °o 0.707 O500 0408 0.354 0316 0.288 0.267 0.250 F : : en R Figure 335 - Wave-Resistance Coefficients 1S Ng In0"p g bp /a? of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = aa GFea Gna. Over y, = 1/2F° and F = U/VeL % = 0.60 50 SSS f°) | 2 3 5 6 7 8 een ee | ec «0.707 0.500 0408 0.354 0.316 0.288 0.267 0.250 ; ws R Figure 34 - Wave-Resistance Coefficients ee ImC%pg b*/a’ of Symmetrical Bodies Belonging to the Class (2,4,6;6;t) = 1-a €?-a é*-a_€° Over » = 1/2F? and F = U/VeL FIN LES Cts 206 Gum Re IAN ER YOO RUS ee co 0.707 0500 0408 0.354 0.316 0.288 0.267 0.250 F Y Bs R Figure 35 - Wave-Resistance Coefficients i lnC pg b*/a’ of Symmetrical Bodies Belonging to the Class (2,4,6;¢;t) = 1-a €2-a é*-a &° Over ». = 1/2F2 and F = U/VeL BA Nic Me OMEN 51 APPENDIX I APPROXIMATE CALCULATION OF THE SURFACE S OF A CLASS OF ELONGATED BODIES OF REVOLUTION From Guldin's rule S = 2m | yds = 2m {“y Vi + (y*")2 ax =a oe S = 2m | “yax + mf y yl? axe mA + af “y yl? ax [29] a —a Mm —a the main part of the surface is given by m times area of the meridian section An plus a correction term neglecting higher order terms. With y = bH b OH i Seria 10k the correction term becomes +a 2 +1 2 2 12 = oes Ou = bf ei ae Ces mab L, Ble) & rab 7 [30] i.e., the correction term is equal to the area of an ellipse with the axes a, b multiplied by the square of the elongation ratio and a numerical value I dependent upon the equation of the curve. To get an idea, with obvious denotations, t= Wes H=1 - é i 280i Hi=sieres we 2n® Ly n Tt) > (@-1) Gn-1) Veet The next term in the expansion of S 7 +a 4 im mab b* +1 4 ote Vay Oe sie wl, Wad [31] with H' = @H/Ogé is obviously of the order b*/a*. However, taking H=1 - é” the factor gt eis aR SeeNeR Onn! ( HAN Os = 5 ines(ones) grows with n° when n is large. Provided b/a is not too small, say ~1/7, the error in neglecting all terms except the first (Equation [29]) is only per- missible as long asn < ~5. 52 Using Equation [30], various changes in S can be easily estimated within the range of validity of the formula. For instance, the influence of an asymmetric term can be discussed as follows: Vales H, qr Hy t +1 2 +1 B +1 y= I, (H, + H, ) (Hj + H) dé = i + [ HH} dé + ie ie lest aé [32] Where I, refers to the even part following [30]. When the meridian curve has vertical tangents at the bow and stern (or bow or stern) the pre- ceding reasoning can be applied in principle to a range 1 - en Z2éED-(1 - € and the remainder is calculated as the surface area of a segment of the sphere generated by the radius of curvature at the nose or stern. Such an approach is, however, only useful when the integrals involved are of simple ae type. 53 APPENDIX II EVALUATION OF THE AUXILIARIES INTEGRALS* The integral to be computed is given by: ™ cell as palibisollinas (y)M,(y)d ae i= = e 0 M.(y¥)M.(yv)dy ") ae VO /paieat | mk) ky? The functions e 7 and M; (y)M, (7) are well-behaved in the entire interval of integration. However, the algebraic function (v/y,) / Viv/yy—1 causes some difficulty at the lower limit of integration, i.e., at y= Moe In the neigh- borhood of y= Vo? the contribution of I is far from negligible and therefore an investigation was carried out to determine the asymptotic behavior of the integral as a function of the upper limit. Specifically, the following func- tion was examined: Y%) (1+ €) i) = IP M(y)f(p)dy € > 0 % where _ ky? M(y) =e My (y)M,(y) and (ip) (i Op ea It was found that: “(Grae 6 M,(y)M,() ¥, vee {1 + fet ote?) This asymptotic expression was used to determine the interval of integration, Ay far a numerical integration. This interval was too small to be practicable, evén allowing for subsequent changes in Ay. A new approach to the problem was sought in a suitable transforma- tion. The following transformation very quickly presented itself: 2 We Cad ised 2Z AZ dy The original integral was transformed as given by: *By J. Blum, National Bureau of Standards 54 2 2 2 2 (z+ ») (Z Ley 2 1-5 | e 7 oO (Maa oy OZ % Jo Vaeere : J e In this form, the integrand behaves properly, (there is no longer a singularity at » = oe) and the integral converges rapidly. The integral was actually computed by using the form in [5]. The numerical integration was performed once using Simpson's rule and a second time using the trapezoidal rule—for checking purposes. The interval AZ was taken as 0.1 and the range extended from 0.0 to approximately 3.5. The M functions were computed from previous tables by using 4-point Lagrangian in- terpolation. The exponential function was computed from tables and the use of the approximation e * =1 - x + x?/2, x <0.01. The algebraic function in the integrand was computed in straight form and fashion. All of this work was done on the IBM electronic calculator (type 604) and the auxiliary IBM punch card equipment. All the IBM operations were checked— independently wherever possible. €]000L8°91 2/009 1L0"h i ¢ UUTIMZ G PO[YUCOY z poyvucnt y popuyoin = peeve ep epee gie weype OOhHL*9 L| £} 00000" Lz} ¢}oongl Le] £}002S1°9¢] 2] 009S1'°S | 4 (83 dln F ai ti yo oe nies y hee ire ii 2a, s 12,2...) 0.482176 | 0.478585 | 0.399765 | 0.287017 | 0.177414 | 0.0931007 | 0.0399619 | 0.0130724 0.00291654 |} 0.000501923 | 0.0006541 86 | 0.000906827 | 0.000754301 | 0.000409008 «241061 -501587 | .487737| .397167] .275175] .161702 0787014 0299938 00802265 00142954 -000915184 | .00153311 00154011 00101414 000444701 256458 -5274u8 | 497324) .394953} .264013] .147151 0659474 0219018 00472942 00135409 00224750 00289584 00237037 00132932 - 000493900 272409 541757 | .507347} .393122} .253531| .133762 0548385 0156858 . 00319270 .00269019 00449889 00474239 00339760 -001 69992 «000556605 288913 -562515 | .517805} .391676) .243731 | .121534 0453746 «0113457 = 00341 247 -00543785 00766933 . 00707276 - 00462181 -00212591 - 000632815 305971 -528699| .390614] .234610] .110468 +0375558 00888162 | .00538875 - 00959706 0117588 00988694 00604300 00260729 -000722531 +323582 540029] .389936} .226170| .100563 | .0313821 00829351 | .00912153 | .0151678 0167674 0131849 00766116 | .00314406 | .000825753 341747 551794} .389642] .218411 | .0918200] .0268536 | .00958139| .0146108 . 0221501 0226950 - 0169668 -00947630 | .00373623 | .000942480 360465 .563994| .389732] .211332] .0842379] .0239701 .0127452 - 0218566 «0305440 -0295417 0212324 - 0174884 - 00438379 - 00107271 0.379736 | 0.608993} 0.673040 | 0.576630 | 0.390207 | 0.204933 | 0.0778173 | 0.0227317 | 0.0177851 | 0.0308589 0. 0403404 0.0373074 0.0259818 0.0136975 0.00508674 | 0.00121645 0.267895 | 0.446802] 0.515401 | 0.474195 | 0.358427 | 0.224501 | 0.114702 | 0.0456125 | 0.0126175 | 0.00194853 | 0.000726062 | 0.00116444 | 0.00106146 | 0.000538598 | 0.000167800 | 0.000108211 - 284159 -470487} .536077] .484K20} .357237] .215331} .103345 . 0367938 -00833174 | .00157001 -00227915 - 00302235 00231546 . 00108809 -000313147 | .000156728 .300976 | 494580) .557203} .495080} .356432]) .206841} .0931487| .0296200 00592792 | .00294800 00524380 | .00579932 00405329 | .00183479 | .000513887} .000218750 31834 -519081 | .578777 | .506176] 2356011 | .199032| .0841138} .o240974 00540008 | _.00608246 -00962000 | .00949529 00627493 00277833 000770019 | _.000294278 336271 -543990| .600799| .517707] .355974| .191904] .0762402] .0202079 00674822 | .0109735 «0154077 0741104 . 00898039 00391889 00108154 00038331 354748 | .569307} .623271 | .529674) .356321 | .185456} 0695280] .0179695 | .00997234} .0176210 .0226071 0196446 .0121697 0052561 00144846 | .000485850 0.373780 | 0.595031 | 0.646192 | 0.542075 | 0.357052 | 0.179688 | 0.0639773| 0.0173762 | 0.0150724 | 0.0260250 | 0.0312179 | 0.0260978 |0.0158427 | 0.00679090 | 0.00187077 | 0.000601895 0.263667 279690 296267 313397 33108 0.429466 «453492 477926 502768 528019 0.489245 | 0.442482 | 0.325335 509839} .452473| .324402 530882} .462899] .323854 -552374 | .473761 | .323690 574315 | .485059| .323909 0.000882234 | 0.00102788 - 000746938 | .00104870 .000667035 | .00108303 000642525 | .00113087 -000673406 | .00119221 0.197385 188846 .180987 0.0972121 | 0.0375030 | 0.0103302 .0867242} .0293293 00611066 -0773976} .0228007 -00376711 | .00130715, -173809 | .0692325| 0179172 | 00329953 | .00384335 -167312 | .0622288} .0146788 - 00470793 | .0081 3606 349318 -553677} .596705| .496792] .324513] .161495] .0563865] .0130855 -00799232 | .0147 853 0163319 0121886 .00602758 00189870 000759681 | .00126705 0.368109 | 0.579743] 0.619544 | 0.508961 | 0.325501 | 0.156358 | 0.0517057| 0.0131373 | 0.0131527 }.0.0219910 | 0.0237482 | 0.0173292 | 0.00862431 | 0.00275892 | .0.000901347 | 0.00135540 0.259724 | 0.410802] 0.463289 | 0.412207} 0.293847 | 0.172184. | 0.0812897] 0.0305101 | 0.00884174 | 0.00185989 | 0.00178332 | 0.00181881 |0.00246152 | 0.00344350 | 0.00384322 | 0.00331561 0.00150426 - 000527452 0.000423770 | 0.000271083 | 0.000301165 | 0.000553898 .000782298 | .000816457 | .000478817 | .000428905 00255238 00228089 00114028 000500888 00573401 00466439 00228557 000769848 . 0103272 . 00796694 . 00391467 00123578 -275507 ~435169| .483801 -421964} .293171 .164276| .0776716| .0229815 - 00468247 | .000284792 | .000947282 | .00105165 .00156282 00264391 00342728 - 00330874 291843 -459945| .504762) .432158] .292880] .157049| .0632150] .0170980 -00239918 | .000466200] .00152280 001 20355 .00114793 . 002041 29 00306673 . 00331538 308733 -485128]| .526172] .442786| .292972| .150501| .0559197| .0128596 -00199186 | .00240411 00350987 - 00227450 00121685 00163565 .00276158 00333552 326176 | .510720| .548031 | .453850] .293449] .144635) .0497859} .0102663 00346053 | .00609853 00690849 00426453 00176959 00142698 | .00251182 | .00336916 344173 -536719] .570338] .465349} .294310] .139449] .0448135} .00931806] .00680518| .0115494 0117187 00717361 00280615 00741529 00231745 00341 631 0.362723 | 0.563126] 0.593095 | 0.477284 | 0.295555 | 0.134943 | 0.0410024} 0.0100149 | 0.0120258 |0.0187569 |0.0179404 ]0.0110018 | 0.00432652 | 0.00160058 | 0.00217848 | 0.00347697 58 REFERENCES 1. 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