Bow and Stern Waves and Snnall-Wave-Ship Singularities 



From the composition of the no-wave point singularity and its image, it can 

 be easily seen that any such three-dimensional point singularity with finite 

 strength located on the free surface is nullified by its image. Thus, in the case 

 of the zeroth-order no-wave point singularity, the flow field produced by a dou- 

 blet in the x direction located on the free surface is exactly the same as that 

 due to a vortex element at the same point. This flow field is also equivalent to 

 that due to a pressure point on the free surface. 



If we consider the strength of a no-wave singularity increasing inversely 

 proportionally to the distance between the singularity and the image, we will ob- 

 tain a higher order no-wave singularity (derivative with respect to z) located on 

 the free surface. In this case, with a single no-wave singularity on the free 

 surface, the linear free surface condition is exactly satisfied without needing 

 any image. This is the similar situation as of a vortex on the free surface in 

 the case of infinite Froude number. 



ANALYSIS OF FORCES DUE TO A ZERO-WAVE BOW 



It is well known that the ship wave resistance can be obtained by pressure 

 integration around the ship hull or by applying Lagally's theorem to the source 

 distribution. We consider as an example a wedge ship for simplicity which has 

 bow waves and shoulder waves. If we cancel the bow waves completely with a 

 bulb, there will remain shoulder waves, yet the pressure integration on the 

 wedge hull seems to give no resistance from the front of the wedge ship. In 

 connection with this problem, it will be interesting to see the actual resistance 

 force distribution of bulbous ships. For the simple case of a submerged point 

 source and a point doublet, the analysis has been performed (6), and it was 

 found that the negative interference due to the doublet on the source contributes 

 to reduce the total resistance. 



Lagally's theorem shows that the force at the source distribution is 



AttpJ lcp^(x,z) m(x,z) dxdz (34) 



and the force at the doublet distribution is 



477P I ^xxC''' ^) M(x, z) dxdz , (35) 



integrating over the whole source and doublet distribution respectively. 

 If we consider a source distribution 



m(Xj,Zj) = Bg (36) 



ino<Xj<a, 0<Zj<H,y=o and the doublet distribution 



Ml = bzj (37a) 



693 



