(In comparing this conclusion with experimental results given as resistance coefficients, 

 one must take care that the coefficient is based on a length squared and not on wetted 

 area or displacement %.) 2. The resistance is independent of the direction of motion. 

 3. For any given hull shape the symmetrized hull, 



U*,y) = Y^{x,y) + r( - x,y)], (52) 



has less resistance than £ if £, =^z £. (In fact, the symmetrization can be carried out 

 with respect to any plane perpendicular to the x-axis, so that even a pair of ships in 

 tandem can be so produced.) The displacement remains, of course, the same. Wigley 

 [4, 5] and Weinblum [3] have made experiments to test the validity of these conclusions. 

 1. is satisfied approximately; however, an empirical determination of n in a law 

 R r-; B n would show n somewhat smaller than 2 (see Weinblum [10], p. 27ff.). As 

 might be expected, conclusions 2. and 3. are not well satisfied. The reason lies in the 

 neglect of viscosity (see also Weinblum, loc. cit., pp. 61-65). 



The behavior of the wave resistance for small and large values of the Froude 

 number F — c/{gL)^ can be found as an asymptotic expansion. Wigley [9] has 

 derived results of this sort; Inui [1, pp. 67-68] finds several terms in the expansion for 

 small F and a special class of hulls and uses it in some computations; Kotik, in unpub- 

 lished work, derives the expansion for a very general class of hulls and includes the 

 effect of sharp angles in the hull. A method for obtaining such an expansion can be 

 outlined fairly easily for an "elementary" ship. Let us introduce dimensionless variables 

 by measuring lengths in terms of L and let / — gL/c 2 = F~ 2 . Then we may write 



8 f x \2 



R = %pc*L* -PI d\ [P2 + Q*], (53) 



T J l V A* - 1 



P = f dx X'(x) cos f\x • } dy Y(y)e~^ = C(f\) • W(f\*) (54) 



-h -H/L 



and similarly for Q = S(fX) • ^(A 2 ) with sine replacing cosine. If X(x) has no 

 corners, i.e. if X' has no discontinuities, integration of C(/A) by parts yields: 



i i n 



C(/\) = — [X'QQ + X'(~ JO] sin %f\ / dx X"{x) sin /Ax, (55) 



A A J-i 



where the last term is 0(l/f 2 \' 2 ) (i.e. f\ times the integral remains bounded as 

 fX -^ co ) if X" is of bounded variation (a standard result from the theory of Fourier 

 coefficients) . Similarly, one may obtain 



i i n 



S(f\) =—[- X'OQ + X'(- %)] cos %fX + — / dx X"{x) cos f\x (56) 

 A A J-\ 



and 



i i r° 



W(f\*) = — [F(0) - Y(- H/L)e~^ H/L ] / dy Y'(y)e»\ (57) 



A 2 A 2 J-f/l 



where the term in W(fX-) is 0(1// 2 A 4 ) if Y' is bounded. In order to carry the 

 asymptotic expansion further, one performs further integrations by parts, assuming 

 that the higher derivatives behave properly. Finally, we note that if X(x) had corners 

 at, say, x ls x 2 , . . ., x„, 



- % < .ti < r 2 < • • • < x„_i < %, (58) 



with jumps in X' given by 



AX'(xi) = X'(xi+) - X'(xr), (59) 



121 



