Developments in Theory of Bulbous Ships 



bulbs with those from a general ship bow so that all waves are cancelled every- 

 where. Indeed, we have to choose carefully the ship shapes or the source dis- 

 tributions (7) for ships for which we adopt bulbs: namely ship shapes for which 

 the bow waves are either positive sine waves (a2n+i "" ^' (-1)" ^jn '- o) ^^^ ^^® 

 application of a doublet bulb, negative cosine waves (32^ = 0, r-l)" a^n+i " ^^ 

 for a source bulb, or strong positive cosine waves plus weak (positive or nega- 

 tive) sine waves for a doublet bulb combined with either a source (sink) or a 

 quadrupole bulb. 



Since no waves from a finite singularity distribution for the bulb can cancel 

 the bow (or stern) waves completely, the best bulb is such a distribution of sin- 

 gularities which produces waves so as to minimize amplitudes in all directions 

 (statistically). This is equivalent to minimizing the bow (stern) wave resistance. 

 In fact, it is not very difficult to obtain the optimum distribution of concentrated 

 singularities in a power series of z along a finite vertical line at the bow such 

 that minimum bow wave resistance is obtained corresponding to a given power 

 series for the ship source distribution. 



Indeed, the bow wave resistance (23) can be represented in a quadratic form 

 in a^ of (7) and b^ (coefficients in z for the distribution of singularities as in 

 (32), (36), or (38)) with coefficients represented in terms of Bessel functions. 

 Therefore we have only to solve the simultaneous equations, 



(40) 

 n = 1,2,.. . 



for h^ when a^ are given. Since the bow resistance due to sine waves and that 

 due to cosine waves are additive as shown in (23), the concentrated singularities 

 for each case can be dealt with separately. 



The optimum distribution of the concentrated singularities at the finite 

 stern line for several given ship source distributions are calculated (Yim 1963) 

 and shown in Figs, 3-7. These indicate that the strength of the singularities at 

 the deepest point (the same level as the keel) is the largest. Especially for the 

 higher Froude numbers, the optimum distributions appear to be almost concen- 

 trated at the keel. This rather supports Wigley's fourth rule. However the op- 

 timum size of the bulb is extremely sensitive to the Froude number. We notice 

 in Figs. 3-7 almost a linear distribution of the doublet for the low Froude num- 

 bers. If we were given the volume of the bulb, the optimum distribution would 

 be also sensitive to the Froude number and the displacement of the bulb would 

 gradually move from the keel closer to the surface as the Froude number in- 

 creases, since the effect of a bulb is stronger at a smaller depth. This would 

 clarify the difference in the opinions of Wigley and Weinblum mentioned before 

 in our introduction. However, in actual ships, the wave resistance is not the 

 only problem. 



1077 



