93 



Havelock has converted Michell's integral into the form 



7r/2 



R = iSn K^^pi (p2 + Q^)sec^ede [20] 



K = As- where 

 v^ 



P =jj <Texp[iKQ X sece - K^zsec^Gjdxdz [21] 



V 9v 

 substituting ''' ^ 27r ax 



Havelock's form of Michell's integral^^' ''^ is obtained 



R = !^E^ J (p2 + Q2)sec3 ede [22] 



TTV^ 



P =Jj|| e''^/^'^"'%os(|J sece)dxdz [23] 



Compared with Michell's formula the direction of the z-axis Is reversed (up- 

 wards). Havelock's form has advantages from the point of view of computation 

 because of the finite limits of the integral. 



79 



Putting secG = coshu a third form has been introduced by Havelock. 

 The same result can be deduced as follows: 



2. Using Lamb's method based on Rayleigh's frictional coefficient fi 

 (frictional force proportional to velocity). With the inclusion of the fric- 

 tional term In the equations of fluid motion, energy is dissipated at a rate 

 equal to 2 times the total kinetic energy of the liquid and this must be equal 

 to the product Rv. This approach is efficient, but highly artificial.^®'®® 



3. Establishing the connection between the wave profile at a grekt dis- 

 tance aft of a moving body and the wave resistance of the body. This method 

 represents an extension of the usual theory of group velocity. The rate of 

 work done on the fluid by the moving body (otherwise expressed, the power ex- 

 pended) is equal to the rate of work done across a fixed vertical control 

 plane minus the rate of flow of total energy across this plane. ^** 



4. By the method of singularities (Lagally's theorem) sketched In the 

 main text. The resistance is computed from 



R = k7Tp{( a{x,z) u(x,z)dxdz [25] 



