170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



final working equation is then given by 



Z ='^X 2B,[^A + j^B] = ^ (AA + JAB) = ^ (AA + jAB) (7) 



Since A.4 and Afi are the attenuation and phase shift changes per unit 

 length, then if f is the length of the rod, covered the total attenuation 

 and phase shift changes will be AA and A5 multiplied by 2f. Hence if 

 A.4o and ABo are the measured attenuation and phase changes, the 

 impedance Z becomes 



Z = '^^ (AAo + jA£o) (8) 



This derivation neglects the change of phase occurring at the inter- 

 section between the rod having no liquid and the rod surrounded by 

 the liquid, but it can be shown that this is small and moreover, the 

 change in the wave on leaving is equal and opposite to that occurring 

 on entering and hence this correction cancels out. However, if the 

 liquid is viscous enough there is a correction due to the fact that the 

 measured impedance of equation (8) is for a cylindrical surface, whereas 

 the desired impedance is the characteristic plane wave impedance. 

 Obviously if the radius of cur\'ature is sufficiently large no cori-ection to 

 equation (8) need be made. To obtain a suitable criterion one may con- 

 sider waves propagated into the litiuid from the surface of the rod and 

 solve for the impedance per scjuare cm. of the cylindrical surface. This 

 neglects the variation with z, but since the wavelength along the rod is 

 quite large, little error results from neglecting variations with z. 



An outgoing cylindrical w^ave in the medium may be represented by 



n, = [MkrY - jY,(kr)y^' = H['\kry e'"' (9) 



where the primes refer to the wave in the liquid and 



/ 2 /2 2 / 



(k) = ^ = '^ or ka = -^r- (10) 



M Zk Zk 



where Zu = \/mV is the plane wave impedance of the liquid. 

 The shearing stress 



TtB = p-'Sre = P-' 



dlle Ue 

 dr r 



