27 



proportions, means a rough if necessary approximation which must be corrected 

 especially v.'hen the ship forms differ widely from "Michell's ship." 



5.2. WAVE RESISTANCE AS A FUNCTION OP PRINCIPAL DIMENSIONS 



If 77 is the equation of a hull and L is kept constant, the basic 

 variations in beam and draft* are given by: 



a. H = const., B variable, i.e., an affine distortion in the direction 

 of the y-axis. 



b. B = const., H variable, i.e., an affine distortion in the direction 

 of the z-axis. 



Two further variations are popular with experimenters: 



c. BH = const., i.e., an affine distortion along y and z. 



d. B/H = const., i.e., similarity distortion. 



3.2.1 . Variation of Beam for Constant Draft H 



From Michell's integral or the symbolic expression [13], it follows 

 immediately that the wave resistance varies with the square of the beam for 

 all Froude numbers, thus 



R = R(B) ~ B^ or R ~ (B/H)^ or R ~ (B/L)^ [1U] 



where H and L are constants. (As Michell's Integral is valid only when L/B>5>1 , 

 2H/B » 1 and —^, -^ are small, this simple result should be checked.) 



The total of relevant experiments is astonishingly small; of these 

 the most important measurements are due to Wigley.^* He found that within 

 the region 1 6 > L/B > 8 agreement between theory and experiment is reasonably 

 good , and that an exponent n somewhat smaller than 2 in the formula 



R(B) > B" [15] 



is more in keeping with experimental results. 



From Taylor's experiment, an exponent n ~ 1 .6 can be derived. An 

 empirical curve n = n(F) was given by Mumford^^ (Figure 9 on page 29). 



Theoretical estimates of the validity of the law R ~ B^ were made 

 using limiting conditions, as follows: 



a. Comparing a spheroid with an ellipsoid of twice the width it was 

 found for P = 0.226 and F = O.5O that the law R ~ B^ holds within the accuracy 

 of computation. A similar result was found using a theorem due to Lamb.^^ 



*Cf., Figure 8, page 28. 



