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10. WHOLLY SUBMERGED BODIES 



The wave resistance of wholly submerged bodies deserves a short 

 treatment even within the scope of the present report . We mentioned the prob- 

 lem when treating ships of least resistance and the influence of beam on the 

 resistance of ships with the purpose of elucidating the conditions valid for 

 surface vessels. 



More generally, the theory of wave resistance of wholly submerged 

 bodies moving horizontally near to the surface represents an interesting 

 study in hydrodynamics capable of wide applications; it has a fundamental 

 bearing on problems of submarines and torpedoes . 



An important problem in tank work is to reproduce conditions for a 

 body moving in an unbounded fluid. In this case we try to establish the min- 

 imum depth necessary to avoid wave phenomena or to reduce them so drastically 

 that their influence can be eliminated by rather small correction factors, 

 similar to the well known procedure in aerodynamics. Such considerations were 

 sometimes neglected in earlier experiments and led to doubtful results. 



From Michell's (Havelock's) integral a resistance formula can be im- 

 mediately written down which is valid for a totally submerged system of singu- 

 larities distributed over the vertical centerplane and which is suitable to 

 generate a submerged body like a submarine, etc., provided the total output 

 is zero. Particularly, distributions symmetrical with respect to a horizontal 

 plane can picture double models which are valuable for resistance research. 

 For a first orientation we may confine ourselves to bodies of revolution, 

 whose image system is given by a line distribution. 



The theory is based on the assumption that the depth of immersion f 

 is great compared with the radius b of the resulting body; f/b>?>V. Under 

 this condition the wave resistance of a very elongated body of revolution 

 gives a fair approximation to the resistance of more general bodies of the 

 same length and sectional-area curve, provided the vertical and horizontal 

 maximum dimensions (height H and beam B) do not differ too much. This state- 

 ment is supported by some calculations made earlier and by a remark due to 

 Havelock and to some extent by Lamb's formula.^® Prom a formal point of 

 view the calculation of wave resistance for a body of revolution is simpler 

 than for a surface ship, the same auxiliary functions being used. A matter 

 of primary importance for practical work is to determine the limiting value of 

 the ratio f/b above which the theory may be expected to yield a reasonable 

 result . 



