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THE BELL SYSTEM TECHNICAL JOTJRNAL, JANUARY 1952 



— t 



LOG ,0 OF SOUND FREQUENCY, f 



Fig. lA — Velocity and attenuation for a medium with one relaxation frequencj-. 



wavelength curve. If the variation in this relaxation mechanism is 

 studied as a function of temperature and chain length, the type of seg- 

 ment may be determined. If, however, the attenuation of the medium 

 is so high that its wave properties cannot be determined, some informa- 

 tion can still be obtained by determining the loading, or mechanical 

 impedance, that such a wave exerts on the driving crystal or trans- 

 ducer. If all the relaxations occur in the stress-strain relation, it can be 

 shown that there is a reciprocal relation between the propagation con- 

 stant r = A + JjB, and the characteristic impedance per square centi- 

 meter Zo given by the equation 



Zor = (i^ + jX){A + m = icop 



(1) 



where A is the attenuation and B the phase shift per centimeter, R the 

 mechanical resistance and X the mechanical reactance per square centi- 

 meter, CO is 27r times the frequency and p the density of the medium. A 

 typical two relaxation mechanism is shown by the curves of Fig. IB. 

 By assuming values for the stiffness and dissipation factors and fitting 

 a theoretical curve to the measured values, the relaxation frequency or 

 frequencies can be determined. 



1 All the relaxation mechanisms discussed in this paper are represented in 

 terms of equivalent parallel electric circuits in which the resistance terms repre- 

 sent viscosities and the inverse of capacities represent shear elastic stiffnesses. 

 In mechanical terms these correspond to a series of Maxwell models as discussed 

 in a paper by Baker and Heiss to be published in the next issue. 



