Ogilvie 



A: + d): = , in the fluid region; (3- 28a) 



|^ = nj, on z = d(x,y); (3-28b) 



(j), - vf^j = , on z = 0, (3-28c) 



where v = co^/g. In the present case, the functions satisfy the 2-D 

 Laplace equation and a 2-D body boundary condition, and they must 

 satisfy the linearized free- surface condition. Instead of the previous 

 simple condition at infinity, we must impose the 2-D outgoing- wave 

 radiation condition and a condition of vanishing disturbance at great 

 depths. Thus, the boundary -value problem is much more compli- 

 cated than in the infinite -fluid case, but, thanks to the slenderness 

 assumption, we have only 2-D problems to solve, and, thanks to the 

 small -amplitude assumption, the problems are linear. 



The actual velocity potential function can now be expressed: 



(3-29) 



(j)(x,y,z,t) ~ Re ) ito^j (t)^j(x,y ,z) 



j=3,5 



It must be observed that each «^j is complex, because of the radi- 

 ation condition. It is necessary to devise an appropriate numerical 

 scheme for solving these problems. Both mapping techniques and 

 integral- equation methods have been successfully applied. Note, 

 incidentally, that the heave /pitch problem requires solution of just 

 the ^2 problem, since the slenderness assumption allows the 

 approximation to be made that <^g— - x^_. 



The result of this analysis is a pure strip theory, that is, 

 the flow appears to take place in cross sections as if each cross 

 section were independent of the others. It is consistent to follow 

 the solution of this problem with a computation of the pressure field 

 at each cross section, from which force-per-unit-length, then force 

 and moment on the ship can be found after appropriate integrations. 

 We obtain the following formulas for the force and m.oment on the 

 ship resulting from the motion of the ship: 



Fj'^Ct) = - p r ds ni[g(e3 - xeg) + {i^hi^^^+i^%)] , (3-30) 

 So 



where j = 3 for heave force and j = 5 for pitch moment, and the 

 symbol Sq denotes that the integration is to be taken over the hull 

 surface in its mean or undisturbed position, which is specified by 

 Eq. (3-18). The first term, involving g, is just a buoyancy effect. 



754 



