299 



and the vertical force acting in the hull, from 

 (10), is 



p(x,t) 



k dS 



(B-2) 



a* 



HP 



^1 



— (X^,t) - VgCx) • V(l)pjj(X^,t) 



Now if Vg = (U + Ug, Vg, We) > the symmetry of Vg 

 is such that Ug and Vg are even in z, while Wg is 

 odd in z. It follows that 



In the high frequency approximation, (|) = on the 

 free surface, z = 0, and this condition can be 

 satisfied by constructing an image of the hull 

 surface and a negative image of the propeller in 

 the upper half space and allowing the fluid domain 

 to extend to infinity in all directions. The 

 negative image propeller is identical to the propel- 

 ler proper, but rotates in the opposite direction 

 and the signs of the dipole singularities represen- 

 tating the effects of loading and thickness are 

 reversed from those of their images in the lower 

 half space. 



The image hull surface, Sj 



is identical geomet- 



rically to S, but the signs of the singularities 

 on Si required to diffract the unsteady flow from 

 the "two propellers" will be reversed from these 

 on S due to the symmetry. The magnitudes of the 

 singularities at image points will be equal. 



VgCx) • V(l>pjj(x^,t) = Vg(x^) • V(t)pjj(Xj^,t) 

 and hence 



p{x,t) = PpH(x,t) - PpH(Xi,t) 

 in which 



^PH 



8t ^^S 



•"PH 



(B-6) 



Thus, the unsteady pressure at points on the hull 

 can be obtained from calculations, or measurements, 

 of pressures at image points on the double-hull, 

 with the double-hull and propeller deeply submerged . 

 Turning now to the formula (B-2) for the vertical 

 force , we obtain 



low Froude number approximation will sajtisfy the 

 rigid wall free surface condition Vg • k = 0. In 

 this case, the steadily moving hull can be reflected 

 into the upper half plane with a positive image 

 singularity system, i.e., the singularities on the 

 image surface, S-^ will be of the same sign as the 

 singularities on S to diffract the velocity iU. 



Because of the assumed linearity, the unsteady 

 potential may therefore be considered as the sum of 

 contributions from the propeller and hull and their 

 respective images . 



(B-3) 



where 



"Pi 



■^Hi 



F, (t) 



Ppjj(x,t)n • k dS 



p (x. ,t)n • k dS 



(B-7) 



But since n (x) • k = 

 written as 



F^(t) = 



-n(x.) • k, (B-7) may be 



p (x,t)n(x) • k dS 



*g(S,t) = - *Hi(%'t) 



(|>p(x,t) = - ())p^ (Xi,t) 



x^ = (x,y,-z) 

 for all (x,y,z) outside the surface S + S,- 



define 



"PH 



I'H 



then it follows that 



'(x,t) 



''PH 



(x,t) 



•^PH 



(Xi,t) 



all X 



(B-4) 



If we 



(B-5) 



Therefore, the complete unsteady potential in the 

 fluid beneath the zero potential free surface can 

 be obtained entirely from consideration of the 

 propeller and the double-hull in an infinite fluid. 



The unsteady pressure at a point on the hull 

 surface S is now given by 



P(x,t) = 



PH 



(x,t) 



Vg(x) 



V(j)pjj(x,t) 



p„^(x. ,t)n(x, ) • k dS 



HP 1 1 



(B-8) 



or, since the image hull S^ is geometrically iden- 

 tical to the hull proper. 



F^(t) = - 



Pjjp(x,t) n • k dS 



S+S- 



and consequently the unsteady vertical force on the 

 hull can be obtained from force calculations, or 

 force measurements, using the double model and 

 propeller deeply submerged. 



