RESISTANCE, PROPULSION AND SPEED OF SHIPS 



239 



M 



agp 



(( e'' cos (U + kx) [ux -l{z- h') \dS, (13) 



Equatidii (1 1) can now be rewritten in terms of F and 

 Mas 



k ^bF k mi 



li = Ha S — -rr 



(ji ot 01 at 



(14) 



In periodic waves the exciting force /•' and moment il/ 

 take the form 



(15) 



Z = Zi, sin (icl -\- a) 

 M = .1/(1 sin (cot + t) 



and from uncoupled chfferential equations of motions, 

 given in Chapter 2, it follows that ship motions can be 

 expressed as 



i" = KFa sin (bit -\- (J + b) 

 e = K'Mo sin (ut + T + () 



(16) 



where K and K' are magnification factors and 6 and e 

 are phase-lag angles for forced oscillations. 



Using equations (15) and (16) in (14) and taking mean 

 values of the quadratic terms, Havelock arri\-ed at the 

 expression for the mean liackward force 



R 



hkKFo- sin 6 + kkK'M,r sin e 



or 



R = i_kF„t» sin S + -i/uil/o(9o sin e 



(17; 



(18) 



The amplitudes of the exciting functions F„ and Mn 

 and the amplitudes of motions j",i and do are essentially 

 proportional to wave heights. Equations (17) and (18) 

 show, therefore, that the resistance added by waves and 

 wave-caused shi]3 oscillations is pruportional to the square 

 of the wave height. These equations show furthermore 

 that the resistance depends on the jjhase lag angles 5 

 and e. At very low frequencies cj the resistance prac- 

 tically vanishes. At the synchronous frequency, sin 5 

 and sin e approach unity and the re.-^istance is at its 

 maximum. 



The nature of the derivation, based on the Froude- 

 Kriloff hypothesis, excluded the hydrodynamic effects 

 of ship's speed. In the later work of Korvin-Kroukovsky 

 and Jacobs (1957) it was shown that ship speed has 

 relatively weak effect on the magnitude of the exciting 

 functions Fo and Mo- Therefore, the I'csistance, based 

 on ecjuations (17) and (18), should be nearly independent 

 of the speed per se, but should be strongly dependent on 

 the frequency of encounter o)„ which governs the phase- 

 lag angles. This is qualitativelj' confirmed by the 

 towing-tank tests, for instance, those shown in fig. 5. 



Equations (17) and (18) were derived in order to 

 demonstrate the dependence of the resistance on the 

 wave-cau.sed exciting forces and on ship motions. Ua\'e- 

 lock (3-1942) warned against using them for actual com- 

 putations because of the uncertainties involved in the 

 use of differential equations of motions. He particularly 

 emphasized the uncertainty of estimating damping 



forces. It also should be remembered that the deriva- 

 tion was based only on the d<p/dt term of the complete 

 Bernoulli equation 



P = po - Opz + p(i<t>/(it 



(19) 



Havelock mentioned that "...the usual approximate 

 eciuations for the m(»tion of the ship are olitained by 

 taking into account also the hydrostatic buoyancy and 

 moment arising from the term gpz in (19)." However, 

 he has not further discussed this. Kor^•in-Kroukov.sky 

 and Jacobs (1957) .showed that heaving and pitching 

 amplitudes and particularly pha.se relationships are 

 strongly affected by cross-coupling of hca\'ing and 

 pitching motions. By virtue of equations (17) antl 

 (18), the resistance in waves also should be affected by 

 cros.s-coupling. 



Hanaoka (3-1957 NSMB Symp.), using advanced 

 mathematical methods. Section 2-6, has derived the 

 expressions for the resistance of a Mitchell-type ship in 

 waves.^ In agreement with Havelock's work, the added 

 resistance caused by waves and ship motions is siiown 

 to be proportional to the square of the wave heigiit and 

 to sines of the phase-lag angles. The calculations were 

 made for an idealized ship identified as Weinblum's 

 (1932) "form 1097." The principal dimensions of the 

 ship are as follows: Beam/length, 0.10; draft length, 

 0.04; midship-section and block coefficients, 0.75. 

 Model resistance in kilograms is shown plotted versus 

 (wave length/ship length) ratio in Fig. 2. The data are 

 given for three ship speeds expressed in terms of Froude 

 nunibers. Excellent agreement between calculated and 

 experimentally measured resistance is demonstrated. 

 However, this agreement is obtained by using an ideal- 

 ized ship form. The experience of Emer.son (1954) and 

 of Kor\'in-Kroukovsky and Jacobs (1954), in calcvUating 

 .ship resistance in .smooth water, indicated that it is more 

 difficult to reach agreement in applying calculations to 

 ships of normal commercial ffjrm. Furthermore, the 

 calculations become tedious in this case. 



Rather small variation of the wave resistance with 

 wave length is shown in Fig. 2. Changes of .ship's 

 course, which modify the apparent wave length ^, can 

 be expected, therefore, to ha\'e rather small effect on the 

 resistance. This confirms the earlier statement of 

 Kreitner (1939) : "Therefore, the course must be altered 

 by about 40 deg or more in order to get a tangible retluc- 

 tion of the additional resistance, whereas the character of 

 the ship's motion will be completely changed with a 

 much smaller deviation. Thus the loss of sea .speed is 

 somewhat independent of the course; hence improvement 

 lies not with navigation but with design." 



In summary, a simple workable expression for the 

 resistance of a ship pitching and heaving in waves is not 

 yet available. The efforts to derive it are recommended. 

 The primary use of such an expression will be in ai^iilica- 

 tion to ships of high block coefficient and low horsepower 



* For additional theoretical work see Haskind (3-1946), Et;gers 

 (3-1960), Iiuii and Maruo (19.57), and Maruo (1960). 



