196 



For large Vcalues of ./; (real, positive) H^(%vy — i) approaclies 

 asymptotically to 



X 



»/^ JtJr 



Ji 



K2 



therefore for {k R) sufficiently large: 



kr 



aR «~i^ 



«r 



cos I pt \- -' (f) 



(«) 



where (f = argument H^^^^c R). 



From (8) it appears that, damped waves ai-e propagated from the 

 cylinder to infinity, the velocity of propagation being 



P P V^ 



/9. 



^Iv/'i 



2pii 



and the wave-length 



2 rr V 2jr 1/2 



k z= vT = z= -'^ = 2 .T 



2ji 



op 



p k 



The frictionai moment on the wall of the vibrating cylinder is 



(8') 



2jrfi72" 



Ö(0 



ö7 



where a> = — . First we determine 

 R ^t 



or 



R 



from (7) 





= R 



a . H^'ixR) . 



R ^ H,^'^){cR) 



(9) 



For the reduction of the 2'"' part on the right hand side of (9) 

 we make use of the well-known recursion-formula of the cylindrical 

 functions : 



By its application (9) obtains the form 



'd« 

 dr JR 

 giving for the frictionai couple 



= R 



__ e'pt 4 ac — ? — ^ — - e'P^ 

 R ^ H^i-^){cR) 



K= 2jt^R* 



dm' 

 ö7 



d 

 = — ijriiR' to+R- 



R dt 



2jTuR'ac^^-~—^^'P^ 



(10) 



(ii; 



For an infinite time of swing, i.e. p:=0, but with a rotational 



velocity differing from 0, |c| = 



QP 



becomes 0. In that case the 



