Because the operating speed of ordinary ships of displacement type 

 is in the range of Froude numbers from 0.15 to 0.30, one may regard that 

 usual ships are operating in comparatively low velocity. If the speed of 

 advance is extremely low and the elevation of the free surface is very 

 small, the flow around the hull is comparable to flow with an undisturbed 

 free surface, and is similar to the flow around a double body fixed in a 

 uniform stream. Then the deviation of actual flow from the double body 

 flow is due to the elevation of the free surface. Since the elevation of 

 the free surface depends on the Froude number, we may employ the pertur- 

 bation expansion of the free surface condition by the Froude number as a 

 small parameter, but it will be found later that this approach is not so 

 simple. 



If our purpose is to formulate the wave resistance at low Froude 

 numbers, we can derive an approximate formula directly from a general 

 expression of the wave resistance. The wave resistance is determined by 

 the momentum or energy analysis of the asymptotic expression of the fluid 

 motion at a great distance from the ship. 



The Green's function G(P,Q) has an asymptotic expression when the 

 point P is brought to infinite downstream x -*• °°. If we assume the case of 

 infinite depth of water, it takes the form 



r^ /2 Y o (z+z')sec 2 



G(P,Q) a4y e sin(Y Q x-x' sec 6) 



-tt/2 



• cos(Y n y-y' sec tan 8) sec 6d8 (25) 



12 

 The fluid motion there is characterized by the Kochin function. In the 



13 

 case of the distributed Havelock sources of density a(x,y,z) on the 



surface S, such as the first term of the right-hand side of Equation (22), 



the Kochin function becomes 



H(k,9) = - I j a(x,y,z) exp[kz+ik(x cos + y sin 8)] dS (26) 

 ^S 



16 



