59 



"Considering all the circumstances" (the approximate character of 

 the hydromechanlc theory and the choice of the hull equation in order to use 

 Ritz's method) "one should not expect to obtain a final solution of such a 

 difficult problem by heaping up approximations..." 



Later it was shown by Pavlenko,^" von Karman^® and Sretensky'^'' that 

 the solution of the problem was hampered by serious mathematical difficulties. 

 All three authors used infinite draft for their final deductions. The con- 

 dition V = const reduces, in the case of Infinite draft, to Ay = const, i.e., 

 the final answer represents the best shape of the water line, which can also 

 be interpreted as the shape of the sectional-area curve. 



Following von Karman, an exact solution of the problem of calculus 

 of variations exists only for a limited range of medium Proude numbers. This 

 statement agrees to some extent with Pavlenko's analysis. L. Sretensky, how- 

 ever, denies the existence of any solution by square integrable functions over 

 the whole speed range. Because of the fundamental theoretical importance of 

 the problem, at present Wehausen of the Taylor Model Basin is reconsidering 

 the matter. 



The naval architect's point of view is somewhat different from that 

 of the mathematician's. Michell's Integral is only an approximate solution 

 even in the case of an ideal fluid, therefore we must check experimentally any 

 optimum form derived from It. The physical meaning of such results is deci- 

 sive; exact solutions of the integral (if they exist) may be less valuable 

 than approximations which yield results within the important range of the 

 theory. 



The practical results reached may be summarized as follows (see Fig- 

 ures 31 and 32): 



The optimum longitudinal and (within restricted limits) vertical dis- 

 tribution of displacement agree well with experimental work, the latter due 

 mainly to Taylor.*^ In particular some optimum values of ^ = C , t, /? = C„ 

 found experimentally were in agreement with theory. "Swanneck" forms were 

 rediscovered and some new features like "small swannecks" (indication of a 

 bulb) found when studying moderately full vessels. Only restricted use can be 

 made of Pavlenko's forms (Figure 32); the extremely blunt sectional- area 

 curves (or water lines) probably indicate that the theory has been over- 

 stressed. On the whole, theory has lagged behind experiment — partly because 

 of the difficulties in principle, partly because of the inadequate effort ap- 

 plied to the subject . The methods of computation have hitherto admitted the 

 use of only two arbitrary parameters, and more widely applicable approximate 

 results can be expected by improving the methods of calculation. 



