192 



livi)K()i)\.N \Mu:s i\ Mill' nisic;\ 



Sec. 19.6 



l>y Ml). ('l'.).iiiV 'I'lic forimila for this operation is 



dx 



Bdr, 

 Ldt 



(49. xi) 



It corrr.spoiiils to Weinblum's E(|. (,2t)) on |)iigt' 

 84 of TMB Roport 710. 



Somewhat unfortunately, in Appenilix 1 of hi.s 

 1915 paper, D. W. Taylor gave the name of 

 "acceleration" to the second derivative d'lj/dx'. 

 It might well have resulted in some further 

 development in the field of .ship form, in the four 

 decades following that paper, had he comhincd 

 the first and second derivatives to obtain the 

 well-known radius-of-curvature equation, given in 

 Sec. 49.9 as Eq. (49.xxi). Much more might now 

 be known of the hydrodynamic cfTects of surface 

 curvature and the best way of working curvature 

 into a hull. 



49.6 Summary of Dimensionless General 

 Equations for Ship Forms. In TMB Report 88(5, 

 i.s.sued in June IO.j.j, Weiublum derived and set 

 down for convenience a number of general 0-diml 

 equations not mentioned in the foregoing. These 

 are listed hereunder for convenience, as adapted 

 from pages 8 and 9 of the referenced report, 

 accompanied by some explanatory notes. Wein- 

 blum's notation is modified slightly in some places, 

 to bring it more nearly into agreement with the 

 ATTC and ITTC standards, but this sliould not 

 inconvenience the reader. 



All etiuations listed are developed from the 

 basic hull ecjuation (49.i), namely y = ±y(x, z), 

 and all are dimensionless. It is a.ssumcd that the 

 maximum section area occurs at midlcngth. 



Coordinates and offsets 



{(ksi) = £ , 7,(eta) = I , f(zeta) = J (19. ill) 



2 2 



Hull equation 



r) = ±j){k, f) (49. iv) 



Waterplane c<|uation 



V = ±({. 0) (49.V) 



Midlength section equation 



i?=±7;(0, r) (I9.vi) 



Centcrplane or profile equation 



= ±7)({, f); f = f ({,()) (I9.xii) 

 Arca-of-ucction equation 



.l(i) = 2 f ' „(f, f) f/f (49.xiii) 



Section-area curve ecjuation with unit ordinate at 

 midlength 



Midlength section-area coefficient 



^- = 7^ = 1'"^"'^)'^^ (■^^•-) 



l.iiail wali'rplanc cnclliiMcnt 



Prismatic coefficient 



Block coefficient 



49.7 Limitations of Mathematical Lines. It 

 is pos.sible to make excellent and profitalilc use of 

 geometric shapes and mathematical equations for 

 small "pieces" of ship .surfaces. Examples are 

 portions of cones for the local enlargements 

 around single shafts emerging from the trailing 

 ends of skegs, the elongated barrels of bo.ssings, 

 where the conical axis need not coincide with the 

 shaft axis, and the basic portions of bulb bows, 

 such as the one illustrated in Fig. 07. H. There is 

 a limit, however, where the expenditure of time 

 and labor in this process is not justified by the 

 hydrodynamic improvement in the ship or by 

 other advantages enumerated previously. In 

 general, the field of usefulness of the mathematical 

 line, as it is \nth the accurately faired line to be 

 described presently, centers principally around 

 those traces of the hull which are parallel to the 

 direction of water fiow. Tli(> statements in Chaps. 

 24, 25, 27, and 28 indicate that neither unfair- 

 nes.ses in the transverse sections, nor fore-and-aft 

 corners and discontinuities, have too great a 

 detrimental effect ui)oii resistance. It is unnec-es- 

 .•iary and futile as well as impo.ssible, to try to 

 mak(! a hyperliola of the 2()()th or the .WOth 

 order lit a perfectly acceptable midsection com- 

 posed of a vertical side line, a horizontal bottom 

 line, and a circular-arc bilge corner. Where the 

 nuithematical lines offer an advantage, u.se them. 

 Where they d(j not, furgcl them. 



