The force on the ship may now be computed by integrating the pressure over 

 the wetted hull: 



R = jjpcos (n,x)da = 2 // p(x,y,t(x,y)){ x (x,y)dxdy 



hull S 



(24) 

 = - 2p J"/ [<p t (x,y, £ (x,y)) + K(grad <p) 2 + gy]$ x {x,y)dxdy 



s 



from Bernoulli's equation, where S is the projection of the wetted hull on the center- 

 plane. We now substitute for <p and £ their expansions in power of £ : 



(25) 

 R = - 2p j J [e<p u (x,y,0,t) + e 2 f^ 2 + • • • + eV« + • • • + Ke 2 (grad ^O 2 



s + • • • + gyHidxdy 



= ~ 2pe H gytixdxdy - 2pe 2 Jj <p u £i x dxdy - 2pe 3 J/ [f«5u, + <p 2 < 



5 s s + %(grzd cpj^xdxdy 



+ • • • • 



If one is going to approximate R by computing only as far as f i3 then one may carry 

 out the integrals only over 5 i _ 1 , the portion of the centerplane section below y = -q 

 -f f^ 2 + . . . + s 1 ' 1 7/j-!. Thus, in the first approximation one need integrate only 

 over S , i.e. the part of the centerplane section below y = 0. In this case, the first 

 integral representing the hydrostatic force vanishes. Since this term gives the impres- 

 sion of being the most important, it should be noted that it really starts to contribute to 

 R only with £ 3 , for by the mean-value theorem 



(26) 

 1 1 ytxdxdy = j dx j dyy$ x = / dx { x (x,y ) J dy y = K I dx i? 2 f *(a;,7/o) 

 5 ° ° =%?Sdx v ftUxMx)) + • • • • 



In the important case of linearized boundary conditions the resistance R may be 

 written : 



R = - 2p // <p t (x,0,y,t){ x (x - f l c dt, y)dxdy. (27) 



Before passing on to special cases, we observe that the method for finding the 

 linearized boundary conditions for a thin ship generalizes to the case when several 

 thin ships are moving through the water. In case the ships are moving on parallel 

 paths, the linearized boundary condition to be satisfied by <p on each centerplane sec- 

 tion S {i> will be: 



tp,(x,y,z t ±,t) = =F c£«\x - ^cdt,y), (x,y) mS (i) . (28) 



Since we shall generally be dealing with linearized problems, it will be con- 

 venient hereafter to drop the subscript 1, as we have done just above. 



Potential and Resistance for Steady Motion. If a thin ship moves with 

 constant velocity in the ^-direction, then <p(x, y. z, t) = lp(x — ct, y, z,) = !p(x,~y,~z), 

 where x, y, z are coordinates moving with the ship. Then <p t = — cy x , <p tt = cy xx , 

 and the linearized boundary conditions for ^T become, after dropping the bars over the 

 letters: 



(1) g<P«(x,0,z) + c\ xx {x,Q,z) = 0, (29) 



(2) <p z (x,y,0 ± ) = ± c?*(x,y) for (x,y) on So, (30) 



(3) lim <p x (x,0,z) = 0, (31) 



(4) lim grad <p = 0. (32) 



y-> — oo 



113 



