PULSE WAVES IN VISCO-ELASTIC TUBINGS II3 



to- 10 



FIG. 5. Stretching rubber. Effect of frequency on dynamic 

 modulus (Edyn, top curve, and ordinate at left) and on the 

 rjM product (lower 3 curves and ordinate at right). 



cathode-ray tube when the stresses and strains are 

 transformed into electrical voltages connected to 

 the horizontal and vertical deviating plates of the 

 oscillograph. A method based on this principle has 

 previously been described by the author and his 

 co-workers (i) and has since been considerably 

 improved. Experiments carried out with rubber and 

 pieces of arteries showed that £jyn as well as tj showed 

 a dispersion, that is to say a dependence on frequency. 

 A common feature of all elastomers is the fact that 

 (except for a few extreme cases) £dyn increases from 

 oj = o (corresponding to the static case) at first 

 moderately, and then more slowly with increasing 

 frequency. For higher frequencies it remains prac- 

 tically constant. Figure 5 shows results for Edy„ and 

 ur] obtained simultaneously on a piece of India 

 rubber. The measurements were carried out with 

 three diflferent amplitudes on the same sample. The 

 points for £dyn for the different amplitudes fall, 

 within the limits of experimental error, on the same 

 curve. A dependence of £dyn on the amplitude there- 

 fore need not be considered. On the other hand, the 

 product (j)r] is greatly dependent on amplitude, as 

 can be seen from figure 5. The increase in cot; is 

 more than it would be if it were proportional to 

 A///. We shall see later that this fact explains in 

 quite a natural way some observations on the damp- 

 ing of pulse waves, which otherwise could hardly 

 be understood. 



There are some marked differences between the 

 behavior of arterial wall material and that of India 

 rubber. For arteries, isdyn depends greatly upon the 

 initial state of stretch upon which the periodic stretch 

 is superimposed, whereas for rubber Edyn is practically 

 independent of it. Figure 6 shows a typical example 



150-10^ , ^ cLyn 



100-10' 



50-10 



Al 



10 20 30 tto SO eoVo I 



FIG. 6. Dynamic modulus of elasticity of a dog's aorta as a 

 function of initial extension. 



of a set of measurements, obtained from a ring-shaped 

 piece of a dog's aorta in oxygenated blood at body 

 temperature. (Not much emphasis shall be placed 

 on the absolute values of £dyn for this experiment.) 

 Also for arterial walls, cot/ depends greatly on the 

 amplitude of stretch. Some experiments on a piece of 

 a cow's carotid artery, stretched with a frequency 

 of 1.04 (w = 6.5), yielded the values 4.4- 10^ dyn/cm^ 

 for Al/l = 0.042; — 3.9-10^ for A/// = 0.018; and 

 2.8 10^ for Al/l = 0.007. 



The treatment of dynamic elasticity given above 

 is, of cour.se, highly simplified, and we have retained 

 only the features essential to our problem. The reader 

 interested in more detail and an explanation of the 

 facts on the basis of molecular physics might consult 

 the monograph The Physics of Rubber Elasticity by 

 Treloar (22). 



3. INFLUENCE OF INTERNAL WALL FRICTION ON 

 D.^MPING .\NT> SPEED OF PROPAG.^TION 



O. F. Ranke (16) was the first to introduce wall 

 friction into the theory of pulse waves. In order to 

 describe the visco-elastic behavior of the wall ma- 

 terial, he used a model of two springs, one of which 

 was damped by means of a dashpot (see fig. 7). 

 In the meantime, our knowledge about elastomers 



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