Muvthy 



dG(x,y, z; £,V,o ) 



]df' 



oV 



<=-b 



It will be observed that we have derived an intepral repre- 

 sentation for the hull potentials in the form of a source singularity- 

 distribution over the two longitudinal planes of strength equal to the 

 normal velocity in the case of steady motion (the classical Michell 

 potential) and a source distribution of density equal to the normal 

 oscillating velocity together with a distribution of doublets oriented 

 longitudinally and of strength equal to the product of the normal 

 oscillating displacement and the forward velocity. 



In the case of the air cushion, we have derived integral 

 equations for the potentials and an explicit integral representation 

 is only possible under some additional assumptions. 



During steady motion there is a primary distribution of sources over 

 the steady position of the lower boundary of the cushion with a strength 

 equal to the longitudinal gradient of the basic pressure distribution 

 with an additional line distribution of sources, doublets and quadru- 

 poles along the bow and the stern, the strength being equal respecti- 

 vely to the "jump" in the longitudinal velocity of the water particles 

 and the "jump" in the potential itself. As these jumps are caused by 

 the discontinuity in the pressure it may be assumed that the line 

 distribution will vanish if the cushion pressure is such that it is 

 diffused to a zero value at the boundary and with a zero value of the 

 longitudinal gradient there. In addition, there is a distribution of 

 doublets oriented laterally over the longitudinal planes of the side 

 hulls of strength equal to the jump in the potential across the planes. 

 The oscillatory potential is given by a similar distribution of sources 

 and doublets over the IFS, along the boundaries of the cushion, over 

 the longitudinal planes, and along the waterline. 



These potentials are discussed in further detail in Sections 

 6 and 7. 



The Interference Potention <t> 



The derivation of the interference potential <|> 11Q is slight- 

 ly more involved as all terms of 0(5/3 ) have now to be taken into 

 account . The method depends upon finding first a particular solution 

 of Laplace's equation satisfying the inhomogeneous equation repre- 

 sented by the boundary condition (V. 3). The homogeneous function 

 denoting the difference between the actual potential <f> 11Q0 and this 



242 



