Unfortunately, the Froude-number region in which this expansion has its 

 greatest validity is one in which the wave resistance is not of great practical importance 

 and also one where the influence of viscosity on wave making may be presumed to be 

 great. However, it does give information of the sort one hopes to get from a theoretical 

 investigation, a statement concerning the dependence of wave resistance on some simple 

 parameter of the form, in this case the tangents of the angles at the bow and stern at the 

 waterline. If a sharp turn at the shoulder may be taken to approximate a corner, the 

 second formula indicates that this also has an important wave-making effect at small 

 Froude numbers. 



At the other extreme of infinite Froude numbers, or small /, an asymptotic 

 expansion in the general case does not seem to work out as neatly. However, by 

 working with upper and lower approximating functions, one can see that the expansion 

 starts out like 



C w = AP + Bf\ogf+ ■ ■ ■ , (67) 



which is also confirmed by Havelock in his discussion of Wigley [9]. The coefficient A 

 appears to be proportional to the square of the volume divided by the distance between 

 the surface and the centroid of the midship section. In view of the fact that the method 

 used is awkward and can almost certainly be improved, it would be unpleasant to 

 commit it to paper. In the case of the long vertical strut one may write out the 

 asymptotic expansion immediately from the known expansion of Y (x) for small x 

 (see below). It is of the same form as for the ship of finite draft and begins with the 

 term 



8 



- f 2 { y + log %f} { jdx f 0) } 2 , 7 = Eider's Constant (68) 



Observation indicates that the higher the Froude number, the thinner a "thin" ship 

 should be in order not to violate the condition of small disturbance of the water. Thus 

 the applicability of an expansion at infinite Froude number to normal ship forms seems 

 dubious. 



In the Froude number range 0.2 to 0.6, important in practice, studies of 

 influence of hull form have to be chiefly computational or experimental. Havelock 

 [1-5] has made a systematic study of the effect of varying various aspects of the hull 

 form; many of his results are summarized in [10] and in Wigley [1]. His computations 

 have included varying the form of the cross-section of an infinitely long vertical strut 

 while keeping its area constant, adding parallel middle-body to such a strut, and cutting 

 the strut off at various depths. Wigley [4-10] and Weinblum [2-9] have made extensive 

 investigations, both theoretical and experimental, of the interrelation of ship form and 

 wave resistance, with the aim of both testing the predictions of Michell's integral and 

 of explaining theoretically known facts about ship forms. The results don't lend them- 

 selves to a brief summary; one should consult the original papers and also summaries 

 by the authors themselves, e.g., Wigley [1, 2, 3], Weinblum [3, 10]. The benefits to 

 be gained from further studies of this sort have certainly not been exhausted. Newly 

 developed computational methods and tables to facilitate them will make such studies 

 much easier (see Guilloton [1] and Weinblum [11]). 



It would be a gross exaggeration to state that these studies have had any sub- 

 stantial effect on ship design. However, most naval architects are aware of their 

 existence and that they do give insight into the wave-making properties of ship forms. 

 As shown by both Wigley [7] and Weinblum [7], the advantages to be obtained by use 

 of a bulbous bow under certain conditions could have been predicted theoretically from 

 Michell's integral. Unfortunately, the theory followed the discovery in this case, but one 

 can hope that it will also happen the other way occasionally. 



Ships of Minimum Wave Resistance. Once a theoretical expression for the 

 wave resistance in terms of ship form is found, it is natural to try to use this expression 



123 



