MECHANICAL I'UUl'ERTIES OF POLYMERS 169 



From the clofiniug relations for the stress strain equation 



Tr8 = I^Sre = 11 



dr 



and the tangential particle velocity due/dl, one can calculate the im- 

 pedance Z per square cm. of cylindrical surface at r = a. This relation is 



Ue CO 



'Jo(ka) _ 2/ 

 J\{ka) ka 



(2) 



Since only the first mode is excited, parameters can be adjusted to 

 keep k quite small, i.e. {ka < .2) and equation (2) can be simplified by 

 using power series expansions for the Bessel functions. Neglecting higher 

 order terms this results in 



Z = ~^'^^^' (3) 



To evaluate the impedance of the liquid surrounding the rod, the tor- 

 sional wave is first propagated along the length of the rod without the 

 liquid, i.e. with Z = 0. Then from equation (3) A; = and from equa- 

 tion (lA) 



■pw 



= Uo + jBo)' (4) 



where Ac and Bo are respectively the attenuation and phase shift in 

 the rod alone. With the small loss in metal and glass rods Ao can be 

 taken equal to zero and 



Bo = co/Vm/p = '^Ao (5) 



where Vo is the velocity of propagation in the rod alone. 

 When the liquid surrounds the rod, however, 



k' = P^-^9' = -Uo-i- jBof + U + jBY (6) 



For the usual case where {B + Bo) » (A + Ao), equation (6) approxi- 

 mates 



k' = {B -V Bo) (- ^B + iAA) = 2Bo (- A5 + j^A) 



where A5 is the increase in phase shift per centimeter and AA the in- 

 crease in attenuation per cm, both directly measurable quantities. The 



