Newman 



longitudinal and transverse forces acting on an arbitrary floating body in steady 

 translational motion (Wehausen and Laitone, 1960; Eq. 20.37): 



where 



■n / 2 /COS 



2 



H(0)|2 sec''^ d0 , 



Vsin e I 



t'sec^6'(ixQCOs6 +iyQsin0+ Zg) 



dS . 



Here v - g/v^ where v is the forward velocity, p is the fluid density, i> is the 

 velocity potential, and the surface integral may be taken over any closed surface 

 surrounding the body. We note that, as is customary in ship-wave theory, the 

 reference frame is with respect to the forward velocity so that a rotation of the 

 coordinates is required to obtain the following formula for the side force due to 

 a drift angle /3: 



Y = ^L-B. |H(?)I2 sec^^ (tan /S - tan 6) d0 . 



J-v / 2 



For bodies with circulation this equation also can be applied provided the sur- 

 face integral defining the Kochin function v^^d) is taken over the body plus its 

 vortex wake. 



Apparently no equally general formula for the yaw moment has been de- 

 rived; such an extension would appear to require the analysis of nonlinear free- 

 surface effects. 



The above equation is of limited value since generally the Kochin function 

 is unknown, and it is more difficult to obtain rational approximations for ideal- 

 ized body shapes than in the parallel situation of wave-resistance theory, due to 

 the fact that the body is yawed. For example, the Kochin function for a yawed 

 thin ship will involve both source and dipole distributions, and it must be ob- 

 tained from the solution of an integral equation analogous to lifting surface 

 theory. However if the ship hull is assumed to be slender rather than thin, 

 some progress can be made. 



Restricting ourselves to the simplest case of a body of revolution, whose 

 axis lies in the plane of the undisturbed free surface, the following results can 

 be derived: 



H(0) = -V/S'(x) exp(-ivx sec (9) dx 



+ /3V tan ^/[3S'(x) + xS"(x)] exp(-ii^x sec 0) dx 

 + 0(/32, 5^/2) , 



220 



