238 



THEORY OF SEAKEEPING 



pitching, and the rusistunce is caused by waves them- 

 selves. 



(b) Longer waves in whieli there is apprecial)ie pitch- 

 ing and heading. The resistance in this case is only 

 partially due to waves themselves and it is caused to a 

 large extent by ship motions. 



2.1 Resistance Caused by Waves in Absence of 

 Heaving and Pitching. The hrst case, (a), was in\esti- 

 gated theoretically by Kreitner (1939) and Havelock 

 (3-1940). Kreitner expressed the force R, exerted bj- 

 wa\'es on a motionless hull as 



H = gpB{h/2)- sin a 



(1) 



where a is the mean angle of entrance between \\ I> and 

 center plane, B is the beam, h is the wa\'e height, and p 

 is the specific gravity of water. By expressing angle a 

 in terms of length/beam ratio of a ship and introducing 

 the concept of "additional resistance per ton of displace- 

 ment," R. he arrived at the expression 



R 



B 

 H 



0.8 



L 



(2) 



where H is a. ship's draft and L is .ship's length. 

 Ha\-elock (3-1940) derived the expression: 



1 



R = - gpa-B sin- 



(3) 



where the last factor is the mean value of .sin a* with 

 respect to the Iteam. Tiiis expression was derived for a 

 body of infinite draft with the extreme assumption of 

 reflection aroinid the front half and a smooth water 

 around the rest of a body. Therefore, Havelock con- 

 sidered that actual ship resistance cau.sed by reflection 

 of waves .should not exceed that given bv the e([uation 

 (3). 



Jacobs and Lewis (1953) applied Kreitner's fornuila 

 (1) to several recently tested ship models. They ob- 

 tained an excellent agreement after making certain 

 empirical adjustments in the original formula. The 

 revised formula is 



R 



0.174 goih'-iyi^ (sin a)"'' 



(4) 



2.2 Added Resistance Caused by a Combined Action 

 of Waves and Ship Motions. Havelock (3-1942) in- 

 vestigated the increase of a ship's resistance cau.sed by 

 the c(jmbined action of waves and of heaving and pitching 

 motions.^ It has been assumed that added resistance 

 due to wave reflection, outlined in the previous .section, 

 is small as compared to the resistance caused by the 

 variation of buoyancy distribution. This as.sumption 

 was verified Iw the results of calculation. Froude- 

 Kriloff hypothesis was adhered to; i.e., the pressures 

 acting on a ship were taken to be tho.se existing in the 

 wave structure when the ship was not there. The 



* Havelock defined a as: '' . a being the angle which the 

 tangent at any point makes with the fore-and-aft-central a.xi.s." 



^ Thi.s derivation was repeated bv Havelock (1945) and bv St. 

 Denis (3-1951). 



following wave properties, taken from Table 1 of Appen- 

 dix A,'* are relevant: 

 ^'elocity potential 



4> = (flg/'o;)e*' sin (a-/ + kx) (5) 



Pressure (exclusive of hydrostatic) 



]) = a.gpe!'' cos {ust -\- kx) (6) 



Pressure gradient in .r-direction 



dp, d.c = agpkc''' sin (uit -\- kx) 

 Pressure gradient in ^-direction 



dp/bz = agpke"^ cos {ut + kx) 



(7) 



(8) 



Assuming the ship to be restrained in its eciuilibritnn 

 position, the instantaneous resistance Ru can he e\-aluated 

 by the integration of pressures p over the ship's wetted 

 surface So or by integration of pressure gradient o\'er the 

 displaced volume r : 



I and n de,signate directional cosines of normals to the 

 elements of wetted area dS in x and z-directions. The 

 force calculated from the foregoing expression is purely 

 periodic with mean value zero (Havelock, 3-1937). 



Suppose now the ship to be in slightly displaced posi- 

 tion due to a hea\('^ t and pitch angle 9. Tlie increment 



of the \'olume added by each svu'face element is 



dV = [nt + nxd - W{z - h')]dSo 



sin (a)« -h kx)dV (9) 



(10) 



whei'e /(' is the height of the instantaneous center of rota- 

 tion above the origin 0. The added resistance can be 

 calculatetl by applying e(|uati<;n (9) to the added volume, 

 .-^o that total wave resistance becomes 



R = R, 



ogpk I 1 c'- sin (wt + kx) 



[iii + nxd - ie(z - h'}]dSo (11) 



Expression (11) can be used directly for the computa- 

 tion of the instantaneous resistance caused by waves 

 when the data on ship motions f and 6 are available. 

 The mean resistance is obtained by integration of the 

 instantaneous resistance over the oscillation cycle. In 

 order to in\-estigate the nature of the added resistance, 

 Havelock endeavored to connect it with the wave-caused 

 heaving force and pitching moment, and with the re- 

 sultant ship motions. 



On basis of eciuations ({>) and (9) the wavc-cau.sed 

 buoyant force and pitching moment are 



F = -agp ff r'--cos {oit + kx)ndSo (12) 



' Except that symbol ^ is used for vertical displacement instead 

 of )/. f//w used instead of c, and oj instead of At. Opposite direction 

 of the wave travel is taken. 



^ The use of symbol f for the heaving motion of a ship is limited 

 to the present section. 



