Murthy 



Peters and Stoker \?) have indeed considered a flat planing type of 

 hull and a yacht type of hull which is a combination of a thin vertical 

 hull and a thin horizontal hull. However, the utmost that has been 

 achieved in these cases is the derivation of integral equations for the 

 potential with singularities at the edges of the hull. No method of solu- 

 tion of the singular integral equations, or even the possibility of a 

 solution has been indicated, as the equations contain singular kernels 

 and are therefore not of the classic Fredholm type. 



In the case of an amphibious hovercraft, however, we had 

 managed to derive an explicit integral representation for the poten- 

 tial in the form of a source singularity distribution over the free sur- 

 face directly below the cushion opening together with a distribution 

 of line sources and line doublets along the boundary of this region. 

 This happy position had come about because the two boundary condi- 

 tions for this boundary value problem for amphibious hovercraft free 

 from water contact were of identical nature, both relating to the pres- 

 sure on the free surface, and therefore constituting a Dirichlet pro- 

 blem. In the case of bodies floating on the water surface there is a 

 pressure condition on the free surface not occupied by the floating 

 body, namely that the pressure is constant (taken as zero for conve- 

 nience) and a velocity condition on the immersed part of the hull, na- 

 mely that the normal velocity of the hull and of the contiguous water 

 particles are equal. In other words, the flow is tangential to the hull 

 when boundary layer effects are ignored. There are, of course, the 

 usual conditions at infinity and at the ocean bottom. This is therefore 

 a Neumann problem. 



In the case of the ACV we are now considering, having an air 

 cushion of the type previously studied but with the addition of a pair 

 of parallel side hulls of arbitrary immersion, the boundary conditions 

 are of a mixed nature. The two pressure conditions for a freely hover- 

 ing air cushion are still present together with the normal velocity 

 condition for floating bodies just discussed. It will be seen presently 

 that an explicit integral representation for the potential due to the 

 hulls is possible on the assumption that they are "thin" (a common 

 and necessary assumption in the theory of ship motions) and with a 

 sufficiently large separation so that the effects of mutual interference 

 may be ignored. However, it has been found possible only to derive an 

 integral equation for the potential due to the air cushion with the ker- 

 nel containing the "jumps" in the potential across the boundary. Al- 

 though the presence of the air cushion does not appear to affect the 

 potential for the motion and oscillations of the side hulls in calm wa- 

 ter, the influence of the side hulls on the potential of the air cushion 



104 



