340 



THEORY OF SEAKEEPING 



On the surface of the body wliere R = r (and since 

 c- = g\'2w) 



Pbw — 



{ — 2nr cos a)/X 



(17) 



(In the 1955 paper an error was made in sulsstituting 

 R = r before differentiating with respect to time, since r 

 is a function of time.) 



The corresponding component of the force due to 

 body-wave interference is obtained by substitution of 

 equation (17) in eciuation (10) 



(W 

 (l.r 



( TT , , . 27r , ^ 



= ■ - -i -r P&nr- sm — (.r - d) 

 \ X X 



, qhrf 2tr , A 



_ 4p y__ cos — (x - d) \ 



c X I 



X cos-Q:e<--'"'™*"'''^rfa 



Jo 



The series expansion of the exponential is 



(-"^rcosaVX 1 27rr 2TrV= COS" a 



X 



x- 



and the integral is evaluated as 



»rr/2 



Jo 



C'OS^ae (2-»^cosa)/X^;^ 



TT Airr Sir'r- 



■4 "~ sx" "8X2" 



Neglecting cubes and higher powers of the small quan- 

 tity r/X in the first term and also r-/X^ in the coefficient 

 of the r-term 



(f ).. = --''''' IV2X 



\h'>' 



c\2 





«^- eos^^.r-.n (18) 



The pressure due to the potential of wave motion 4>w is 

 from equations (11) and (15) 



At the surface of the body 



"/'^ sin — {x - d) 

 X 



Pu, = pghe^ --'•■'""' ">'"^ sin — (.r - d) (19) 



X 



The corresponding component of the force is obtained by 

 substituting equation (19) in (10) 



— = 2pghr sin — [x - d) 

 ax /„ X 



X 



/•»/2 

 COS 

 



,g^g(-2,rcosa)A^^ 



where the integral is ecjual to 



TT-r , 4ir-r- 



2X 3X2 •• 



Again neglecting cubes and higher powers of r/X 



dx ;„ "^^ V 2X 3 X- 



X sin ^ (.1- - cO (20) 

 X 



The first term of equation (20) is seen to be the dis- 

 placement force resulting from the wave rise or fall and 

 the acc(>mpanying increase or decrease of volume. The 

 second and third terms represent a modification of this 

 force, due to the (approximate) exponential variation of 

 pressure with depth, and this modification is known as the 

 "Smith effect." The entire equation represents the force 

 acting under what is usually referred to as the "Froude- 

 Kriloff hypothesis." This force which is exerted by the 

 waves on the body is reduced by the body-wave inter- 

 ference effect indicated by equation (18); for a semi- 

 cylindrical section at zero speed this amounts to ap- 

 proximately doubling the Smith effect. 



Designating by /3 the angle between the longitudinal 

 tangent to the body surface and the positive .r-axis, the 

 derivative r is evaluated as 



dr d^ 

 d^ dt 



- V tan /3 



(21) 



(In the 1955 paper the sign was erroneously taken as 

 positive.) With the substitution of equation (21), the 

 sum of equations (18) and (20) which is the total exciting 

 force, becomes 



'.r \dx J,r \dx J 



dF 

 d. 



2pglir 



X 



sm — (.1- — d) 



X X2 X 



F I IT 



-\- - tan a ' 

 c 



Swr 

 T3X 



cos — (.r — en > 



0} (22) 



This expression replaces equation (26) of the 1955 paper. 

 It differs from it in the sign of the velocity-dependent 

 terms, which are small, and in the value of the coefficient 

 of the (r tan /3)/X-term, which is also small. 



Two steps remain to be taken : (o) Equation (22) must 

 be generalized for ship sections other than semi-circular, 

 and (b) corrections must be introduced for free-surface 

 effects. 



