MECHANICAL PROPERTIKS OF POLYMERS 



171 



and the tangential velocitj'^ may be obtained as before and the complex 

 impedance over the cylindrical snrface determined to be 



Z = 





■M 



dHr\kay _ HV\kr)' 

 dr r 



/j^H'i'^kr)' (11) 



Noting that for plane waves Zk = \/ p \i , one may eliminate \i fi-om 

 (11) and evalnate Z at r = a with the result 



^ = i 





{kaY {ka)'_ 



Z, 



(12) 



The above equation can be used to obtain a solution for Zk in terms of 

 the measured value of the cylindrical impedance Z for any set of para- 

 meters that may apply. Except for very heavy loading of the rod, how- 

 ever, the results of eciuation (8) may be used directly with little error. 

 For example calculations indicate that for | ka | ' = 10, the multiplier 

 of Zk of equation (12) is approximately 1.07 Z — 7.4° for phase angle of 



{ka)' of 



Zo 



For {ka)' < 10, the correction multiplier rapidly becomes 



important. This same correction is applicable for the torsional crystal, 

 but since this is only used for dynamic viscosities less than 10 poises, a 

 correction is seldom necessary. 



