Newman 



effects of the free surface are approximated by reflection of the underwater 

 portion of the hull above the surface, either with a positive (in phase) image 

 corresponding to very slow motions or a negative (opposite phase) image corre- 

 sponding to very rapid motions. It follows that there will exist a hydrodynamic 

 force and moment which are strictly linear functions of the acceleration of the 

 body or ship, and which can be lumped with the inertia terms in Euler's rigid- 

 body equations of motion, equivalent to an added mass and added moment of in- 

 ertia of the body itself. Rough estimates of the added mass and added moment 

 of inertia for specific hull forms can be inferred from the extensive data avail- 

 able for ellipsoids. For elongated bodies such as ship hulls quantitative calcu- 

 lations can be based upon slender body theory which for lateral motions is 

 equivalent to the simple strip theory. The sway added mass and yaw added 

 moment of inertia are given respectively by the integrals 



(::> IW- 



) dx 



where m(x) is the added mass of the two-dimensional transverse section at x, 

 and the integral is over the length of the ship hull. The corresponding cross- 

 coupling force and moment are 



Y" = N. = I xm(x) dx . 



This last symmetry relation is confirmed by experiments only in the low- 

 frequency domain (van Leeuwen, 1964). At higher frequencies the inequality of 

 the cross-coupling coefficients is probably due to the shedding of unsteady vor- 

 ticity in the wake. 



Circulation 



The next degree of complexity is introduced because the classical added 

 mass concept cannot account for the side force and yaw moment associated with 

 a steady drift angle. These are analogous to the lift force and pitching moment 

 on a wing or hydrofoil and can be analyzed by regarding the hull and its image 

 above the free surface as a symmetrical (uncambered) lifting surface, with the 

 drift angle taking the role of the conventional angle of attack. In view of the 

 small draft-length ratio of ships, we confine ourselves to the theory of wings of 

 small aspect-ratio, rather than considering lifting-surface theory in its full 

 generality. Here we can choose to treat the simplest case of a flat plate, appro- 

 priate to the case of small beam -draft ratio, or account for the fullness of the 

 ship's sections within the framework of slender-body theory. 



The applicability of these theories, at least to the total force and moment, 

 rests on the validity of a Kutta condition at the ship's stern, as opposed to the 

 more physical boundary layer growth and separation which may be expected to 

 occur. In this context it is important to recall that the low aspect-ratio wing 

 theory is somewhat unreliable in the case of a wing with rectangular planform 

 and especially in the case of a wing whose span is decreasing towards the 



214 



