Linearized Potential Flow Theory for ACVs in a Seaway 



The instantaneous surface S Q may be assumed to be composed of the 

 steady surface S together with an additional strip of area S' aris- 

 ing out of the oscillations. 



The strip corresponding to an element of area dtdr; extends from 

 the point L on the boundary of S to L' on the boundary of S . 

 If £[_ anc ^ £r' are tne longitudinal co-ordinates of L and L' 

 respectively, 



in the undisturbed condition when L and L 1 coincide. 



In the disturbed condition points on S are obtained by the vertical 

 projections of points lying in the £ V plane within the displaced, 

 position of the cushion boundary in that plane. Thus, setting f = 

 in equation (3. l) for transformation of co-ordinates, we obtain after 

 linearization with respect to 6 



since the geometry of the cushion boundary in the £ V -plane is 

 unchanged by the displacement. 



We may now write 



// f ( « v)dUv= // f (€ q ) d£dr? + // f ({ , ) d*dq 

 O ^o 



The integral over S' may be written 



/., f 



f ( fir )<M 



and as the length of the strip is small, we may replace the inner 

 integral by 



< « 1/ " « L ' 



f («1) 



* =£ L 



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