Forces on an A.C.V. Executing an Unsteady Motion 



I OO CO 



R =— — = /c( r ) dr / dw / du w 2 . (P 2 + Q 2 ) . 



** PgJ J J 



-oo 



. cos)\gk'tanh-kd) • (t - r ) ( • cos { w(s(t) - s( t ))| , (25] 



■where 



::=i ■//, 



qS;5 =//p(-v) £<wx + „y)dxdy (26) 



S 



The range of the u integration in Eq. (25) may be halved for a pres- 

 sure symmetric about the x axis. 



Eq. (25) is similar to that for a thin ship obtained by Lunde 

 (1951b). His formula included an additional integral which was sim- 

 ply proportional to the instantaneous acceleration. This extra term 

 is zero if the singularity distribution (Eq. (l)) lies on the free surfa- 

 ce - as for a pressure distribution. 



The steady-state wave resistance can be derived from Eq. 

 (25) by allowing the velocity of the craft to be constant for a long time. 

 If the velocity is suddenly established at a value c , then one obtains 



oO 



-§ /w 2 dw / < 



7T pa J J 



R = ™ /w" dw / du (P + Q ) 



4 7T pg 



sin] (7+wc)t[ sin{(7 -wc)tf 



7 + wc 7 - wc 



As t — » oo , the oscillations in the integrand increase so that there 

 is only a contribution from the second term, and this occurs when 



7- wc = (27) 



This is the relationship between the transverse and longitudinal wave 

 numbers for free waves travelling at the speed of the craft. The 

 analysis is simplified if we use polar coordinates : 



w = k cos 6 



and 



u = k sin 6 (28) 



where k is the circular wave number and 6 is the wave direction. 

 The limit process is carried out for a similar case by Havelock 



45 



