Pa'idoussis 

 - 1 pU^ [c,S(L) . c,f |W dx]|^ . n. IJ = 0, (9) 



X 



which is the equation of small lateral motions. For a uniform cylin- 

 der this equation becomes 



where the diameter, D, and M = pS are now constant. 



We note that in the absence of frictional forces, (10) becomes 

 the governing equation for small motions of a cylindrical beam con- 

 taining flowing fluid [ 3i] , where we interpret M as the mass of the 

 contained fluid per unit length. The physical similarity between the 

 Internal and external flow cases is striking, albeit that In the former 

 case fluid friction does not enter the problem. We shall refer to this 

 later. 



We finally note that Eqs. (9) and (10) also hold to describe the 

 motions of a towed flexible body. If we Identify U as the towing 

 speed, provided the tow-rope forces are taken into account as part 

 of the boundary conditions. 



III. BOUNDARY CONDITIONS 



Clearly the boundary conditions will depend on the mode of end 

 constraint. Let us consider the case of a towed flexible cylindrical 

 body shown In Fig, 2. The body consists of a uniform cylinder ter- 

 minated by a rounded 'nose' and a streamlined, tapering 'tall', Incor- 

 porated to provide reasonable cixlal flow conditions over the body. 

 We assume that the towing craft moves horizontally In a straight 

 course with uniform velocity U, so that the tow-rope In Its undis- 

 turbed state lies along the x-axls; we also consider the assumptions 

 made at the beginning of §2 to hold. 



We may use Eq. (9) to analyze the system, together with 

 boundary conditions stating that (a) at the downstream end, x = L, 



986 



