287 



■'image ' hull point to the field point, i.e, if 

 ^' = (x',y',z_^), then x| = x',y',-z'). The source 

 strengths a„(x) can be determined by applying the 

 hull boundary condition (5) yielding an integral 

 equation 



Since (fn = on z = 0, the region of integration 

 in (34) may be extended to include the hull water- 

 line plane Sq (see Figure 1) , thus forming a closed 

 surface about the volume V inside the submerged 

 portion of the hull, and 



(X) 



4ir 



a„(x')n 



|x-x'i| 



dS 



F '1' = ipnNto 

 n 



S+Sr 



■"ndS 



(36) 



+ n(x) • i\ 



0, X on S 



(33) 



The integral term gives the contribution from all 



source elements other than at the point of interest 



on the hull. The contribution from the source at 



that point is given by the first term, a^ix)/2. 



Equation (33) with n • [Vckn + V<t)p^ ] as a known 



n -'-n 

 input is solved numerically by the generalized 



Douglas-Neumann program [Hess and Smith (1964)]. 



In practice, the hull surface is divided up into 



quadrilateral elements over which a^ is considered 



constant and the integral equation is replaced by 



a set of simultaneous algebraic equations. Care 



must be exercised to insure that the sizes of the 



elements are small compared to the spatial "wave 



length" of the propeller-induced velocity field. 



This is particularly the case for field points just 



downstream of the propeller since the velocity 



components rapidly become proportional to sines 



and cosines of N((u/U x- f ) so that the wave length 



of these signatures is X=2iTU/Na), which, for J-1 



and N = 5, becomes X = 0.4Rq. In order to obtain 



representations of an entire cycle, it is necessary 



to take element lengths of one-quarter of this 



length or about O.IORq. Upstream, the induced flow 



is monotonic in x and the element sizes can be made 



much larger without loss of accuracy. 



It is acknowledged that the above-described pro- 

 cess does not, in principle, completely solve the 

 problem since the feedback of the hull sources on 

 the instantaneous flow experienced by the propeller 

 is not included in the propeller loadings Ap;^. To 

 do this would require joining the integral equation 

 for the propeller loadings (with input from the 

 propeller generated hull sources) to Eq. (33) to 

 form a pair of integral equations for Ap, and Ojj, 

 which, when solved interatively to convergence, 

 would yield the complete solution. For the present, 

 we are content to ignore the hull feedback on the 

 propeller. 



Once the source densities on the hull surface 

 are found, it is convenient to determine the force 

 induced on the hull in terms of simple integral op- 

 erations on these sources . Although the Lagally 

 theorem and its extension by Cummins (1957) is known 

 for submerged bodies, it is necessary to develop a 

 form which is suitable for use for floating bodies 

 beset by high frequency flows. 



The force as given earlier by Eq. (12) may be 

 considered as the sum of two terms Fjj ^""^ ^n 

 given by 



where the symbols ( ) and ( ) ~ are used to denote 

 a quantity evalutated on the outside and inside of 

 the surface of integration respectively. Noting 



that for <}>n'^' given by (32), 



(i.e. the 



fn^"' ^3^-— • "J >--/ , rn "fn 

 potential is continuous across a surface distribution 

 of source singularities) , and using the vector 

 identity n = n • Vx, one obtains 



Fjj(l) = ipnNu 



VxdS(x) 



(37) 



S+S, 



By means of Green's reciprocal theorem applied to 

 the volume V, (37) becomes 



(1) = 



ipnHo) 



dS(x) (38) 



S+S„ 



since V • V (x) = and ^ <i>-n~ = in V. A fundamental 

 property of a surface distribution of source singu- 

 larities relates the jump in the normal derivative 

 of the potential to the local source strength, viz. 



n • V*^ 



(39) 



But since n • V(|)j^+ = on S by virtue of the boundary 

 condition (5) , Eq. (38) may be written as 



Fn'-"-' = -ipnNo) I I x a^ix) dS(x) + 

 S 



ipnNo) J J X T-^ dS(x) 



(40) 



The first term in (40) has the same structure as 

 that derived by Cummins (1957) for submerged bodies 

 generated by internal singularities. The second 

 term arises from the capping of the volume by 

 extending the free surface through the ship (proposed 

 originally by Breslin in 1971) . For the important 



case of the vertical force, F 



zn' 



we obtain 



(1) 



ipnNo) 



ndS 



(34) 



and 



F,(l) 



ipnNo) 



z a^ix) dS(x) 



(41) 



F,(2) = p 



ndS 



(35) 



A similar analysis can be applied to the convec- 

 tive term F,,'^' (see appendix A) to obtain 



