Yim 



be used to improve the bulb form used to decrease the cosine wave heights as 

 well as to cancel the sine waves. 



Another idea to cancel negative cosine waves is to use a source line. In the 

 same way as we found the infinite doublet or quadrupole line to cancel sine or 

 cosine ship waves, we can find the line source distribution 



00 n+ 1 "" 



b„z , 



"^^1^ -- L^-^^Ti- f°^ < z^ < 1 



n = 



00 



-Ll^l [.r'-(^, -!)"*■] for z, >1 



(38) 



with 



(2n+ 1)! k"""^ , , 



b - r-n" - - ^— a (39) 



which completely eliminates cosine bow waves due to the source distribution (7), 

 of odd power series. Of course, we have to take care to employ a sink distribu- 

 tion at the ship afterbody in order to have a closed body. 



Bulbs at ship sterns can be dealt with exactly in the same manner as for 

 ship bows in an ideal fluid , neglecting the effect of propellers and other attach- 

 ments. However, the influence of the viscosity and the wake near the stern is 

 so important that the stern problem should really be considered separately. 

 Therefore we deal here only with bulbous bows and bow waves. Henceforth we 

 may omit the word "bow" except to avoid ambiguities. 



In all three kinds of bulbs mentioned above, the strength of concentrated 

 singularities along the vertical line increases with the depth, starting with zero 

 strength at the free surface. This suggests the shape of a bulb to be used for a 

 practical ship. 



PRACTICAL APPLICATION OF THE THEORY OF 

 WAVE CANCELLATION 



In understanding the mechanics of cancelling regular ship waves through 

 the concept of elementary waves and for the practical application we can note 

 here three important characteristics of an elementary wave in each direction of 

 propagation between the angles -77/2 and 77/2: (1) the point where the wave 

 starts, (2) the phase of the wave, (3) the amplitude. In general, regular bow 

 waves consist of elementary waves which have different characteristics in each 

 direction of propagation, despite the fact that point or line singularities by them- 

 selves produce negative sine elementary waves (pt. doublet) and cosine elemen- 

 tary waves (pt. source or quadrupole) in all directions of propagation from the 

 point of the singularity's location. Therefore it is impossible to match in all 

 directions the aforementioned three characteristics of elementary waves from 



1076 



