Yim 



for all e which is in \d\ < 77/2. Therefore the superposition of two waves (26) 

 and (27) becomes 



Tf/2 



^sB "^ [A(^) - B(5)] sin aj(0) 66 



(28) 



and the bow wave resistance is 



r 



[H6) - B(0)] 2 cos- 



Ad 



(29) 



By matching B(0) to A(0) graphically to make [A(i9) - 8(5)] as small as possible, 

 especially for small 6, Takahei (1960) found the most favorable doublet strength 

 /x and the position of the doublet Zj in (27). They built cosine ship models ac- 

 cording to Inui's method, observed the wave patterns by the method of stereo 

 photographs, and tested numerous spherical bulbs faired at the cosine ship. 

 Finally they obtained the models C-201F2 with the so-called waveless bow. 

 Namely, they observed a remarkable reduction in the bow wave heights due to 

 the bulb at the design speed. If we notice in (7) and (11) 



,(") 



(0) 



(30) 



we can readily see in (9) that the bow waves consist purely of sine waves if the 

 source distribution (7) is an even power series and consists purely of cosine 

 waves if (7) is an odd power series. If we consider (7) with only an even power 

 series and in addition. 



(-1)" a^^ > in (7) 



(31) 



(as in the cosine series) the waves will always be positive sine waves. (However, 

 (31) is a necessary but not a sufficient condition.) Yim (1963) showed that these 

 positive sine bow waves due to a source distribution of even power series can 

 be completely eliminated by a doublet distribution along a semi-infinite line 

 x = o, y = o,-co>z>o, with the doublet strength in the negative x direction, 



Kz 



for < z, = -z < 1 



1^ L^ n+l 



n = 



for Zj > 1 



having the relation 



(-1) 



n (2n)! 



(32) 



2n+l 2n 



(33) 



1074 



