Sec. 49.4 



MATHEMATICAL LINES EOR SHIPS 



189 



Unfortunately, Taylor's method was described 

 in a publication not conveniently available to the 

 average naval architect. There are reprints of this 

 paper but they have been given only limited 

 circulation. Since the equations of waterlines and 

 sections are not linked together into equations of 

 surfaces in which the hull is treated as a whole, 

 it is considered preferable by those who have 

 studied this problem to develop a broader system 

 of ship-hull equations than to make Taylor's 

 work of 1915 available to a more extended group 

 of readers. 



The broader purpose envisaged by Weinblum 

 some fifteen years later, explained in references 

 (8) and (9) of Sec. 49.17, of representing the whole 

 ship surface by single equations, was followed 

 by his more recent work at the David Taylor 

 Model Basin, published in the following TMB 

 reports: 



710 "Analysis of Wave Resistance," written 

 jointly with J. Blum, Sep 1950. This report is in 

 the category of must reading for anyone studying 

 the subject of mathematical ship lines. 

 758 "The Wave Resistance of Bodies of Revolu- 

 tion," May 1951 



840 "Investigations of Wave Effects Produced by 

 a Thin Body— TMB model 4125," Nov 1952. 

 Written jointly mth J. J. Kendrick and M. A. 

 Todd. 



886 "A Systematic Evaluation of Michell's 

 Integral," Jun 1955, especially those portions 

 having to do with ship lines. 



Weinblum has developed mathematical expres- 

 sions, to be described presently, which: 



(a) Are suitable for delineating an entire ship 

 form of given characteristics, based upon an 

 origin amidships. This improves upon the Taylor 

 procedure of treating the forebody and afterbody 

 separately, mth origins at the two ends. 



(b) Will produce a series of forms -with scientific 

 systematic variations in these characteristics 



(c) Will provide a basis for the systematic investi- 

 gation of wave-resistance characteristics of ships. 

 This will, it is hoped, lead eventually to forms of 

 low if not least resistance. 



(d) Will provide a basis for the calculation and 

 prediction of the flow pattern and the pressure 

 distribution around a ship 



(e) Will form a more general foundation for 

 research on other problems of naval architecture 

 involving maneuvering, wavegoing, and behavior 

 in shallow water and restricted channels. 



49.4 Mathematical and Dimensionless Repre- 

 sentation of a Ship Surface. In the Weinblum 

 references of Sec. 49.3 the problem of the dimen- 

 sionless dehneation of a ship hull is generahzed: 



First, by considering the entire underwater 

 boundary as a surface, rather than as a series of 

 intersections of that surface by three sets of 

 parallel planes at right angles to each other, long 

 customary in naval architecture. In other words, 

 instead of defining the shape by offsets of water- 

 lines, of bowlines and buttocks, and of section 

 lines, usually at equidifferent intervals from three 

 given planes of reference, it is defined for the 

 normal case by the y-offsets from the centerplane 

 or the plane of symmetry for amj point on the hull 

 surface having the coordinates x and z. 

 Second, by expressing the ^/-offsets and the x- and 

 0-coordinates not as dimensions in well-known 

 length units but as non-dimensional ratios of the 

 respective offsets and coordinates to the length, 

 breadth, and draft dimensions. These 0-diml 

 ratios are given presently. 



Third, by placing the origin of the coordinate 

 system in the surface waterplane, in the plane of 

 symmetry, and at midlength of the immersed 

 form or underwater hull. The length of this hull 

 is L, the waterline length. While the vertical 

 measurements on an actual ship are usually made 

 upward from the baseplane, in the mathematical 

 system they are made downward because this is 

 the positive direction of the 2-axis of the ship, 

 described in Sec. 1.6 and indicated in Fig. l.K. 



In the discussion which follows it is assumed 



that: 



(a) The ship is a simple one which may be con- 

 sidered symmetrical forward of and abaft the 

 midlength station 



(b) The ship is symmetrical with respect to the 

 centerplane, as is customary for real ships. The 

 definition sketch of Fig. 49.A is an isometric 

 diagram of the outline of such a ship, correspond- 



Non-Dirnensional Distances 



Fig. 49. a Definition Sketch of a Simple Ship 

 Surface with a Single Origin 



