Understanding and Prediction of Ship Motions 

 Xj - ^ (n .ipp dS - (n •ij)d^ p dx3 I 



p 4(X,,X,,t) 



(n-ipd^ pdx3 



where L is the line of intersection of the ship hull at any instant with the vindis- 

 turbed free surface, and l,{^^,-s.^,t) is the free surface elevation. I have as- 

 sumed that the ship is wall-sided near the free surface. It can now be recog- 

 nized that the quantity in braces is just an integral over that part of the hull 

 surface which is wetted when there are no waves and no ship motions. It has 

 the same shape and size as S^, but it is displaced from the equilibrium position. 

 The direction cosines, (n • i-), in fact have the values appropriate to S^ itself, 

 but the pressure must be evaluated on the actual position of s. However, we can 

 use the Taylor expansion of the pressure to convert it into a function to be eval- 

 uated on S . The quantity in braces is then just: 



J Cq ij) {p + (^ + ^'^x) -vp + ...} 



dS 



The correction term, the integral over L, is expressed partly in terms of 

 each of the two coordinate systems. It is perhaps easier to retain the quantity 

 (n • ij) as it stands, and so we must express the inner integral in terms of the 

 primed coordinates. From the geometry, it is found that this correction term 

 can be written: 



(n-ipd-e dx3 (p + (f + ^x X) -Vp + .. .} , 



where L^ is the intersection of the undistrubed free surface with the hull sur- 

 face, the latter being in its undisturbed position. The upper limit of X3 is in 

 error by a small quantity which does not affect the result. If we now work out 

 this integral, systematically keeping only lowest order small quantities, we 

 obtain 



^vj d^(=-ii)fe{-i 



1 ocp -u o^ 



Bt g 3xj 



J x, = 



The complete result for x^ is: 



^\-- P \ dS(n.i.) j-gx3-gC^3+x/j-x,^2) - ir+ v|^- Vcp^-Vcp, 



.V[l.(^.^xx).v]^j-pvjd^(n.ij)^{ 



g 3t g Bx/^^3+''2^1 '^1^2)r- 



71 



