Yim 



the integrated singularity system will be a representation of the flow around the 

 Inuid in the horizontal meridian plane in the limiting case of zero Froude num- 

 ber. 



Influence of Froude number on wave pattern is only by affinity through a 

 factor kg ' . In particular this wave pattern comes out to be independent of the 

 choice of singularity system to represent the Inuid in unbounded flow, whereas 

 the uncorrected wave field and wave resistance are not.-'' 



For a symmetrical Inuid, the corrected flow will be symmetrical as well 

 and therefore even the third order wave resistance component R< ^^ as given in 

 my Eq. (51) in the preceding paper will be zero. This is due to a peculiar selec- 

 tion among all correction potentials making the second order wave resistance 

 vanish and differing by wave free potentials. 



If now, by tracing streamlines, we could for any k^ determine an associated 

 hull form, starting with the Inuid for zero Froude number, we would have a class 

 of zero wave ships with form dependent continuously on k" ^ But theory tells 

 us that no such forms of finite extent exist. However, I hope that the approach 

 sketched above will after some intuitive modification lend itself at least for de- 

 termination of hull changes to achieve low wave making. The expressions for 

 the velocity field are much simpler than for those of the uncorrected wave sys- 

 tem due to a source. 



REPLY TO DISCUSSION 



B. Yim 



I appreciate Dr. Eggers' interest in my paper and his cordial comment on 

 it. The theory shows that if we are required to cancel the entire regular ship 

 waves for a certain Froude number, we can do it, but with some features we do 

 not want in practice such as infinite volume or infinite local disturbance. How- 

 ever, if we consider the dependence of the wave height on such parameters as 

 the depth, the Froude number, and derivatives of the waterline y coordinate 

 with respect to x, we can easily see that the deeper part of the concentrated 

 singularity corresponding to the bulb attachment contributes mainly to the local 

 disturbance rather than to the regular waves with which we are concerned; and 

 the higher order derivatives on the waterline are significant for regular waves 

 only when the Froude number is sufficiently large (>0.3). Thus it is clear that 

 in practice even the finite bulb works, under such a theoretical basis, with the 

 image systems as are explained in the present paper. 



'■'K. Eggers, discussion following Dr. Pien's paper in "Fifth Symposium on Naval 

 Hydrodynamics: Ship Motions and Drag Reduction," Office of Naval Research, 

 Department of the Navy, ACR-112, 1964. 



700 



