Yamazaki 

 where 



-l<v<l, rB<r<rp, ^(r)>0, ^=1,2, N, 



and ^(r) and 0u(r) are respectively the quantities related to the half chord length 

 and the skew of the blade. The mean surface Sp can be approximated by a helical 

 surface with pitch 27Ta(r) and rake angle e , that is, we have 



Xb(r,v)^a(r) [^^C r ) + ^ ( r ) v] + re, (6) 



where the more precise designation of 277a(r) will be noted later on. Further 

 the blade thickness at the point (r,v) on Sp is denoted by t (r.v). Then the mean 

 surface Sp of the rudder is expressed by using the parameters y^ and u as 

 follows: 



''i = '^mCvi) + '^(yi) " • yi = yi • ^i = zrCyi-u) , (7) 



where . ' - '^v.- v 



-1 < u < 1 , y^ < y^ < y^ , x (y ^) > , x^Cy^) - x (y^) > , 



and X (yj) and ^uivi) ^^^ respectively the half chord length of the rudder and the 

 distance between the propeller and the rudder. We can get approximately 



^R(yi'") ^ • i-^-' ^ - or 7T . (8) 



Further the rudder thickness at the point (yj.u) on Sp is denoted by t^(yj,u). 

 Then the lines v or u = i and - 1 on Sp or Sg indicate the trailing and leading 

 edges of the propeller blade or rudder respectively. Equation (4) may be con- 

 sidered to represent either the original hull form or the hull form modified so 

 as to include the effect of the boimdary layer thickness, and in this paper, for 

 simplicity, we carry forward the former. 



Since no lift acts on the ship hull under the assumptions of the Introduction, 

 the hull form can be represented hydrodynamically by source distribution on the 

 surface Sjj. We denote the strength of sources at time t in the elemental area 

 on S^, whose projection on the x^y^ plane is dxjdyj by m^(x^,yj,t) dxjdyj, where 

 the subscript k refers to Eq. (4) and the value of mj(Xj,yj,t) is not always equal 

 to that of nijCxj.yj.t) because of the presence of the propeller. Thus the velocity 

 potential 4)^ due to the hull can be obtained from the source distributions. The 

 propeller can be represented hydrodynamically by appropriate vortex systems 

 and chordwisely distributed doublets. The former consist of the bound and free 

 vortices, and the later represent the effect of the thickness of blade sections. 

 The bound vortex is arranged approximately along the line v - constant over all 

 the surface Sp, and the free vortex shed from the bound vortex flows with the 

 velocity of the water at its position. However, as the result of observations on 

 the experiments in cavitation tunnels, the geometrical form of the free vortex is 

 not greatly disturbed by the presence of the hull and rudder, so that the free vor- 

 tex can be assumed to extend rearward in a helicoid without contraction retaining 



20 



