LMO 



IIM)K()1)\ NAMIC.S l.\ Mill' 1)1.,SU,.\ 



Sec. WJ 



second ami third ti-iins in tlic luarkrls, hut in such 

 a manner that it ftiriiuHl a t|unntity cakuhitod in 

 the same way as tlie friction resistance and 

 addetl to it. More is said of tliis uuusiud feature 

 hiter. Rankine callwl the conil)iiiation of tlie 

 product of the length L, the mean girtli, and the 

 terms within the lirackets "tlie augnientod 

 surface." 



The year following the publication of this first 

 formula, and after developing his ncoids and 

 lissoneoids, Rankine derived from them a similar 

 formula but by a (|uite dilTerent approach, taking 

 "into account not only the direct resistance caused 

 by the longiludinal component of the friction, but 

 the resistance caused indirectly through the 

 decrease of pressure at the bow, and diminution 

 of pressure at the stern, assuming the vertical 

 disturbance (of the water surface) to be unim- 

 portant" (italics and comments in brackets are 

 thase of the present author) [Phil. Trans., 18()4, 

 p. 384). Rankine's reasoning here is not too clear, 

 especially as to the source of the reduced pressure 

 at the stern in a region of potential flow, but 

 there is no douljt that he was thinking intensely 

 on the subject. 



The second formula, given "as a probable 

 approximation for lissoneoids" [Phil. Trans., 

 1864, p. 390], takes the following form when put 

 in the notation of this book: 



« = C,|,S'r{l+4(^y] (.50.ii) 



where .S is the wetted surface with no obliquity 

 correction and AV, measured abreast the miiklle 

 of the bfxly, is the increa.se in the "velocity of 

 gliding" over the .speed of the vessel, due to the 

 potential flow around the .ship. 



If the sin* i^ term of Eq. (oO.i) Ls neglected, the 

 expressions within the brackets in these two 

 equations become e.s.sentially similar in that both 

 are indirectly functions of the beam. Increasing 

 the beam incrca.scs both sin" !« and (AV/V)' 

 and at the .same time increases the resistance. In 

 fact, the inclusion of the.sc terms to the second 

 power may not be too distantly related to the 

 conclusions derived many decades later, to the 

 cfTect that under similar circumstances the cal- 

 culatetl resistance due to wavemaking increiuses 

 as a function of tho.sf|uare of the beam [Il.uvclnck, 

 T. H., IN.\, 1018, p. Jdll. 



Despitr- the inr-on)[)lctc knowli-dge <if phy.sical 

 phenomena upon which they were i)a.se<l, I'](|s. 

 (.jO.i) and (.")().ii) bear a striking likeni-.ss to those 



in u.se in many quarters today, as may be noted 

 from Sec. .'JO.7 of this part of the book aiul from 

 SX.VMI'] Technical and IJe.search Bulletin 1-2, 

 Marcli l'.).J2, page 3. If Rankine did not choo.se to 

 emphasize those factoids which have since become 

 important, he must at least be given credit for 

 outspoken discussion of the subject and profes- 

 sional honesty, still highly prized. In the referenced 

 18(14 paper, on page 390, he headwl the section 

 containing his expression for Eq. (50. ii) with the 

 frank statement "Provisional Formula for Re- 

 sistance." On page 296 of his later 1871 paper a 

 similar heading reflected his increased confidence 

 in the formula by reading "Probable Law of 

 Resistance." 



Despite his lack of knowledge of the laws of 

 friction resistance Rankine brought out the 

 following important points: 



(a) The use of the longitudinal instead of the 

 tangential components of the friction drag in the 

 direction of motion 



(b) The use of the velocity with whiili the water 

 actually moved over the surface of the ship, as 

 contrasted to the overall speed of the .ship. 



During the period from 1858 to 1863, and prcjb- 

 ably at other times, Rankine (with J. R. Napier) 

 used the formulas of Eqs. (50.i) and (oO.ii) "with 

 complete success in practice, to calculate before- 

 hand the engine-power required to propel propo.sed 

 vessels at given speeds" [Phil. Trans., 1863, p. 

 136]. 



Other engineers and scientists, working inde- 

 pendently on the problem in Great Britain, 

 endeavored to find formulas for calculating both 

 the friction and the pressure resistance [Phipjis, 

 G. II., Inst. Civ. Engrs., London, 8 and 15 .Mar 

 1864; Jour. Franklin Inst., Jan-.Jun ISiit, \'ol. 

 XLVII, Third Series, pp. .308-312]. 



As with many other things of that period the 

 .ship-resistance formulas have long since been 

 forgotten, at least in their original form, but their 

 purpose remains as valid as when it was expres.sed 

 nearly a (•(•ntur\- ago. 



50.3 Modem Developments in the Calculation 

 of Pressure Resistance Due to Wavemaking. 

 The early history of the analytit^ allaik on the 

 problem of ship resistance, sketchetl in Sec. 50.2, 

 shows (irst some groping and spasmodic efforts, 

 then the beginnings of an active campaign that 

 in the period 1920-1955 has become .systematic 

 and has shown increasing promi.se of practical 

 re.sull.s. 



