Bow and Stern Waves and Small-Wave-Ship Singularities 



n,^"\0). (-l)"m^">(l). (41) 



Hence, 



1 



r ( 2n+ 1 ) ,.n\ 



m(x) m '(x) dx = . (42) 







In the evaluation of the wave resistance by substitution of Eqs. (38) and (42) in 

 Eq. (34) we obtain 



J Bi(x,z, 



6) m(x) dx = 



Therefore in this case the term including Bj(x,z,6') is like a local disturbance 

 and does not contribute to the wave resistance. Now the situation is exactly 

 similar to the case of the uniform source distribution of Eq. (36) mentioned be- 

 fore. Namely, if we use a doublet distribution to cancel the bow waves, the re- 

 sistance due to the source given by Eq. (34) becomes zero, while the resistance 

 due to the doublet given by Eq. (35) remains, and is equal to the bow wave re- 

 sistance, which is equal to the stern wave resistance here. 



If a ship is not symmetric, the contribution of B^ in Eq. (38) to the wave 

 resistance is not zero, and the bow wave resistance is not equal to the stern 

 wave resistance. If we cancel the bow waves by the doublet distribution in this 

 case, the stern wave resistance will be exactly the same as the effect on Bj on 

 the wave resistance plus the bow wave resistance which is exerted on the dou- 

 blet distribution. To sum up, for a symmetric ship, the regular bow waves on a 

 ship hull which are responsible for the total wave resistance can be eliminated 

 by our ideal bow bulb, yet there remains the wave drag of the bulb itself, which 

 is equivalent to the stern wave resistance. 



This means that the regular wave heights along the ship hull is not the only 

 origin of all the wave resistance. The stern wave energy can be propagated with 

 the expense of the force on the bow bulb. 



If we attach the ideal stern bulb to a symmetric ship in addition to the ideal 

 bow bulb, it is known that the entire wave resistance will become zero (2). That 

 is, the ideal stern bulb will have a negative force to cancel the force at the bow 

 bulb. This mechanism can be seen by the same method used for the bow bulb. 

 Namely, we consider a doublet distribution to cancel regular stern waves on the 

 line x= 1, y= 0. Then the regular part of 4>^^ in Eq. (35) becomes 



onx=l + 0, y=0 



XX '^^^ ^° stern waves only without the 

 stern bulb effect onx= 1-0, y= 0. 



The stern waves of a symmetric ship are exactly the same as the bow waves 

 and of the opposite sign of the regular waves due to the doublet distribution. 

 Therefore, from Eq. (35) the force at the doublet distribution at the stern is in 



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