Newman and Tuck 



respectively, and for the rigid free surface condition, and where S(x) is the 

 sectional area, Zo(x) is the vertical coordinate of the center of buoyancy at 

 each section, and B(x) is the beam of the section at the waterline. We note that 

 B22 and Bgg are 0(e'*) while B^^ = O(e^), and as indicated in Table 2, all three 

 are of higher order compared with other terms in the equations of motion. 



Pitch and Heave in Head Waves 



We shall illustrate the above theory by considering in more detail the im- 

 portant case of pitch and heave motions in head waves. If ^3 and ^5 denote the 

 (complex) amplitudes of heave and pitch, the boundary condition on the ship hull is 



^ = ^ (01 + <p^) - - iw^3 COS (n, z) + ioii^ [x cos (n, z) - z cos (n,x)] , (2.15) 



or, for the disturbance potential, 



30g r T r T 



— — - - cok exp I K(z + ix) cos (n, z) + i cos (n, x) 

 dn L J L 



- i'^^3 cos (n, z) + i<^^c [x cos (n, z) - z cos (n,x)] 

 = - a;[Aeil^'' + i^3 - ix^g] cos (n,z) + OCe) . (2.16) 



The flux function is thus 



QC'^) = "^ d'^ = ajB(x) 



3n 



iKx 



Ae + iC, - ixCj 



(2.17) 



The wall force f^^*^^^> can be analyzed from classical slender body theory. 

 Thus the potential 0<*all) jg given by 



0(*^LL)(x,y,z) = 4>^^''\y,z-K) ^ f(*ALL)(x), (2.18) 



where 0^^°^ is the two-dimensional "strip theory" potential satisfying the 

 boundary condition (2.16) on the contour of the hull section and the rigid wall 

 condition on the free surface, and the interaction term f(*ALL) ^g 



( WALL) 1 r 2|x - ^1 



f (X) = - ^J log sgn(x-a Q'(^)d^- (2.19) 



L 



From (2.10) the wall force is 



(WALL) r (2D) -iait CC (WALL) 



F. = dxF. (x) - iojpe J) ^ (^"l cos(n,x.)dS 



= -we 



idJt C - icot CC (WALL) ,_ _„. 



J m..(x)dx- icope JJ f (x) cos(n,x.)dS, (2.20) 



142 



