450 '"- 



APPENDIX . 



The A ttra c ticn of a !<it:id Disc of uhich Part is Movinp . 



The attraction of a rigid dis: of which the centre portion moves with a given velocity while the 

 outer annulus renuini fixed is of somt practical interest as tioing rather like the case of a fixed rigid 

 Box model with a rijid "skirt" in which tht torgt plates moves as a result of the pressure in the water. 

 This case will no* De solved forncilly, thought it has not Deen found possisie to reduce the solution to 

 a simple finite form suitable for comoutation. The case wh. re the rigid "skirt" extends to infinity has 

 been treated tjy Tempcrlcy. 



Let the velocity of any point of the disc he v. Only the case of radial synmetry is considered 

 and over the surface of the disc r = o, the other spheroidal co-ordinate s is a function only of the 

 radial distance p of the point from the axis. The r.lation is 



p = c [l - s^ ' 



Hence if v is a given function of p it may be written as a function of s and, with the usual restrictions 

 on the form of the function, may be expanded in a seri._s as follows. 



sf(s) 



P (5) 



(lU) 



J" t f(t) P^ (I) dt 



(15) 



The potential (ti„ fits tne boundary condition that the normal velocity over the surface 

 is everywhere equal to v is 



'h 



ic 2: 



?„(ir) P„(s) 



(16) 



This potential is to be adcied to <p = (p, + «, to give the complete potential for the disc and the 

 unit point source. The "attraction coefficient" in u-quation (13), i.e. the term in the first bracket, 

 is replaced in this case by 



3 X j at X c 3 r^at r = d/c = r^, 

 = y a oj (ir) 



(17) 



t (io) 



The special case of a piston of radius f? moving with velocity v in a fixed finite circular baffle 

 of radius c is given by 



f(s) = 



k<\ 5 I < 1 



o< I s I <X 



where /\ 

 In this case the coefficients a to bo inserted in equation (17) are 



(1 -\^)v 



-^oS^-^P * 2 (^) 



2B-T-r'^n 

 for n = 3, 5, 7, 



2n 



IJn - T1J2rr?~3T 



K (^) 



T^rr-17 ^n 



(\) 



(18) 



