tPPENDt X 



Motion of Circular Plates uith Air-Vater 

 backtn^s. 



Let the plate be of radius a ana let it be situated in a rigid wall.. Then if v Is the 

 inward velocity of any element ds of the plate the velocity potential at any point at distance r 

 from ds due to the motion of the plate is 



1 



V ds/r 



(1) 



where the integration is carried out over the surface of the plate. If the water is compressible 

 and c is the velocity of propagation in the water the value of v which determines (^ at a time t 



should be that which holds at a time t - - . 

 incompressible. 



The pressure p ut r \s - p ^ so that 



It will be assured, however, that the water is 



P = 



^ 



ds 

 r 



(2) 



suppose the motion symmetrical about the centre and therefore at radius x write 



^2 "7 ■*« -I 



(3) 



The pressure i: now 



P ^ 



5 I 



x2"ds 



{■*) 



The determination of the pressuo: distribution over the surface of the plate due to its own motion 



is now reduced to the determination of the integral 



"" where r is the distance of a point 



on the plate at radius x to anothar point (at radius^ say) where the pressure is required. This 

 integral is equivalent to determining the potential due to a distribution of matter whose density 

 varies as x ". 



For a ring of unit linear density and of radius x the potential at radius^ in the plane of 

 the ring is 4 - K f i ) i f 5^ > x and i*K ( i ) if ^ < x whore K (u) is the elliptic 



UK I i 1 if ^ < X whor 



.2n 



integral of the first kind to modulus /j.. Hence if the density is proportional to x the potential 

 at ^ is 



(5) 



K (fj.) a^/fi- 



and p = 



i'" * ' <t^ (S'a) say 



J, 1 



2n 



2n 



(6) 



The 



