94 



where <r(x,z) Is the source-sink distribution over the vertical center plane 

 and u(x,z) the x component of fluid velocity at the point x,z, which must be 

 calculated from S^Sx.''^ Evaluating [25], Michell's integral is again ob- 

 tained. However, the method of singularities appears to be far superior to 

 any other known, when determining the forces experienced by singularities in 

 a given flow; it enables us to calculate the lift and the transverse force as 

 well as the resistance, and is especially powerful when systems of bodies are 

 investigated. Havelock's and Dickmann's work furnished beautiful examples of 

 its application. 



We mention finally formulas developed by Kotchine^^ which have been 

 successfully applied by him and by Haskind''-^'^ to the calculation of forces on 

 floating or submerged bodies. 



C. Sretensky's Formula for the Wave Resistance in Shallow Water 



Sretensky has developed a formula for the wave resistance of "Mi- 

 chell's ship" in shallow water. The theory is valid under the same assump- 

 tions as Michell's integral provided the draft/depth ratio is small. The de- 

 duction is based on methods of potential theory. 



R = S^rpg r !! "^ ^^ "» dm ^26] 



v^ J l/m^ - sm t;anh mh cosh mh 

 ™o ' V 



where 



P = j j cosh m(2 + h) cosl/-^ tanh mh x ff(x,z)dxdz [27] 



Q = ff cosh ra(z + h) sin^-^ tanh mh x a(x,z)dxdz [28] 



<r(x,z) is the soxiroe-sink distribution which can generate a ship. Using the 

 relation between the normal velocity component and source strength: 



the integral can be easily rewritten in terras of the surface equation y(x,z). 



The lower 11 

 given by the equation 



The lower limit of the integral ra is of special interest; it is 



tanh m^h = -p^ m^h [29] 



tanh m^h = F^^ m^h [29a; 



