^Bs 



Developments in Theory of Bulbous Ships 



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I |S,(0) [S,(l) COS (k, sec 9) + S^(\) sin (k, sec 0)\ 



- 82(0) [SjCl) sin (k J sec (9) - S^(\) cos (kj sec d)]\ K^ cos^d^ 68 (25) 



K = 8 [1- exp (-k^ sec 2^)] . 



From (23) we can see that the bow wave resistance consists of the sum of 

 the wave resistance due to sine elementary waves and that due to cosine elemen- 

 tary waves. The same is true of the stern wave resistance in (24). The expres- 

 sion for the interference resistance (25) shows that there is no interference be- 

 tween the elementary sine waves and the elementary cosine waves starting from 

 the same point either at the bow or the stern. The humps and hollows of the 

 wave resistance are due to the interference resistance, and this is usually very 

 difficult to evaluate. However, if the bow or stern wave resistance is small, the 

 interference resistance is also small. The idea of bulbous bows or bulbous 

 sterns is therefore to reduce the bow or stern wave resistance. 



MECHANISM OF BULBOUS BOWS 



We consider the bow wave (9), and the bow wave resistance (23) due to a 

 sine ship with its source distribution 



mj(x) = cos (7Tx) in 0<x<l, 0>z>-l (26a) 



which has no cosine elementary waves but only positive sine waves from the 

 bow in all direction of propagation. Namely 3^(0) - in (9) and (23) and 

 Si(0) > 0, or we may write 



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^sB = ^(6*) sina;(0) dd (26b) 



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with A(0) > 0, for \e\ < 7t/2. Now we observe the regular wave height due to a 

 point doublet of strength -fx at (0,0, Zj), which was calculated by Havelock (1928), 



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^3 -^ - 4k^ fj. exp(-k^Zj sec'^6) sec'^6 sin [k^ sec^i9 (x cos d + y sin 0)\ d6 



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3(6) sin co(0) dd . (27) 



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Inui, Takahei, and Kumano (1960) noticed these doublet waves also consist 

 of sine elementary waves and that the amplitude function B(t') is purely negative 



221-249 O - 66 - 69 



1073 



