Submerged Two-Dimensional Bodies 



Certain features of Tuck's results, however, indicate that the validity of 

 this second-order theory may be questionable. The three following points will 

 be mentioned here: 



1. At a submergence equal to twice the diameter and at Froude number 

 U/\/ib = 0.55, the second-order theory gives twice the wave resistance obtained 

 by linear theory, and at an even smaller Froude number, U/Vgb = 0.47, his work 

 predicts the resistance three times the linear theory. Results of this nature 

 are not in agreement with the assumed converging perturbation series 



w = ew^ ' '> + e^w( 2^ + ... 



and in this speed range at least, it is therefore doubtful if his theory is appli- 

 cable. 



2. The fact that the difference between the linear and second-order theory 

 increases as the Froude number decreases may indicate that the second-order 

 wave resistance approaches infinity as the Froude number approaches zero. 

 Unfortunately, no data are shown by Tuck for Froude numbers smaller than 0.47. 



3. The largest submergence investigated by Tuck was twice the diameter of 

 the cylinder. When the body is as close as that to the free surface, the waves 

 created by the cylinder will break, resulting in a highly nonlinear phenomenon 

 which cannot be treated by second-order perturbation theory. 



In addition to these points, it should be mentioned that the circular cylinder 

 lends itself nicely to mathematical treatment and is therefore very useful for a 

 preliminary mathematical investigation of nonlinear free-surface effects. But 

 we must keep in mind that when assuming inviscid fluid flow past a circular 

 cylinder we should not expect the mathematical solution to be "a good approxi- 

 mation to the true wave resistance" (as hoped by Tuck). 



The object of this work has been to investigate the accuracy of second- 

 order wave theory by a comparison of analytical and experimental results, and 

 also to clear up some of the uncertainties in previous work. To perform such a 

 comparison the author has applied second-order wave theory to a streamlined 

 two-dimensional body and conducted experiments on an 11 -foot-long strut of 

 13 -inch chord length. Wave resistance data have been obtained by three tech- 

 niques: (a) drag measurements, (b) wave survey, accounting for second-order 

 effects, and (c) second-order theory. In addition to these data, the wave profiles 

 have also been measured and computed for the selected body shape. This is the 

 first time wave profiles correct to the second order in wave amplitude have 

 been computed for flow past a body. 



MATHEMATICAL FORMULATION 



An infinitely long cylinder is supposed to move with a constant velocity U in 

 a direction perpendicular to its axis and at a fixed distance below the free sur- 

 face. The problem is to determine the surface waves and the wave resistance. 



597 



