Linearized Potential Flow Theory for ACVs in a Seaway 



factor of e ,<r to the terms containing the potential and again take 

 the real part. The two real parts are then multiplied together. This 

 procedure is implicitly understood when a factor of e ,a is indi- 

 cated by an asterisk. 



It will be seen that the cushion stiffness terms denoted by 

 the parameter SjQ m enters in the higher order vertical force of 

 0( <5 2 /3,5/3 2 , 5/3a , (3 2 a) although the longitudinal force of these orders 

 is free from- this effect. In actual fact, the effect of stiffness is 

 contained in the longitudinal force of O(80at, /3 a8) although these 

 forces have not been written down. 



Also, it may appear a little surprising that the dynamic 

 pressure of the water obtained from the potential <f> does not enter 

 in the expression for the vertical force at all, whereas the horizon- 

 tal force contains the potential in all orders. This is due to the 

 approximation contained in our expression 



£ as = - ( r i + r f - ft) dxdy 



x y 



which is valid only for an infinitesimal slope of the water surface in 

 the x- and y-directions . It appears, therefore, that the actual 

 shape of the IFS determined by <j> is irrelevant as far as the vertical 

 force is concerned so long as the slope is very small. The situation 

 is therefore much the same as over wavy ground as the hydrodynamic 

 properties of the water surface do not seem to matter for small slo- 

 pes. On the other hand, in the case of the horizontal force, which is 

 proportional to the actual slope, however small, the dynamic pressure 

 determined by the motion of the water is very relevant. This also 

 applies to the moment as will be seen presently. 



Turning now to the moment due to the action of the surface 

 pressure on the cushion hull 



•// 



M p = - // p s : xn ) 



S 



and since the position vector with respect to o' of an element dS 

 of the IFS at P (see sketch) may be written 



179 



