Linearized Potential Flo" 1 Theory for ACVs in a Seaway 



= S. 



+ S! 



+ S! 



where S 



is the equilibrium position below the load water plane 



z' = which is known from design considerations, S. ' is the strip 



° » 



between z 1 = and the undisturbed water surface z = and S* 



'o 

 is the additional strip between z = and the actual free surface 



z =f . 



Let us first consider the contribution of the last term in 

 (III. 3) which represents the hydrostatic pressure of the water to the 

 horizontal and vertical forces given by (III. l). This pressure is the 

 same on both sides of the longitudinal plane so that we may combine 

 h. and h , and considering first the horizontal force 



- 2 8 p g / / z + z' cos 



x' sine - h (1 



- COS0) 



in0 



dx'dz' 



which may be transformed by Stokes' theorem into 

 - 2 8pg cosd(h h(x', z') 



2 5 p g si 



- 2 8 p g sin 



z + z' cos 6 - x'sin# - h (1 -cos 6) 



G 



dz 1 



h(x',z') dx'dz' 



z + z' cose - x'sine - h ( 1 -c 



OS0) 



dx' 



+ 2 8 P g sine cose / / h(x', z') dx'dz' 



where the line integrals are taken along the boundary of the region 

 Si + S. , i. e. the line of intersection of the plane y = b with the 

 plane z = and along a line from the stem to the stern running 

 along the keel. 



Along the upper boundary, i. e. along the waterline, 



z + z' cose - x' sine - h (1 - cose) = z = 



G 



191 



