^■' 



In using this reasoning, we can expect good agreement between theory and ex- 

 periments for slender bodies only and depend upon experimental checks and cor- 

 rections for ships of nornal proportions. 



Originally Michell developed his resistance formula by computing the 

 pressure exerted on the body.^^ Other methods are based on the computation 

 of the energy of the wave system caused by the irfotion of a body and on the 

 dissipation of energy calculated by means of an artificially introduced van- 

 ishing viscosity term introduced by Havelock.^®' ®®'^° A fourth approach used 

 by Havelock''^ is the so-called method of singularities, which we shall review 

 briefly because of its advantages for calculating forces when the image sys- 

 tems involved are known. 



The method of singularities may be simply explained as the law of 

 attraction applied to sources and sinks: Two point sources (or sinks) with 

 the output Qj^Qg (-Q^, -Qg), attract each other with a force 



Q1Q2 

 K = 4^ [9] 



where r is the distance between the two singularities, while a source and a 

 sink experience a repulsion of the same absolute value. ^^ It is quite aston- 

 ishing that no broader use has been made of this formula in hydrodynamics, 

 which when applied to electricity is familiar to any student.* 



The formula [9] can be rewritten and generalized to give the force 

 experienced by a source Q due to the velocity v of the stream at the location 

 of the source 



K = -pQv [10] " 



where v can vary throughout space, but is steady at any given point. The mi- 

 nus sign indicates that a source is pulled by the stream in the opposite di- 

 reotion of v. (This equation [10] known as Lagally's formula is as Important 

 as Kutta-Joukovsky's formula for a flow with circulation,^'^'') 



When the velocity potential corresponding to a source-sink distri- 

 bution is known, the horizontal velocity is also known, and the resistance X 

 can be written down as the integral of the product of the distribution and 

 the horizontal velocity over the region of the distribution. The method can 

 be generalized for calculating the mutual interaction between bodies (ships) 

 advancing with constant speed in the same direction in tandem or for any other 

 arrangement. The influence of fixed walls can be treated as a special case 

 of this problem. 



*Note the difference in the sign of the force due to charges of the same kind, when dealing with 

 electric and with hydrodynamlc phenomena. 



