112 HANDBOOK OF PHYSIOLOGY ^ CIRCULATION I 



/^ 



Fig. 3 



g 



AAAAAAAA 



^ 



lie on an ellipse (see fig. 4) and the values of So and 

 ip can also be found from this ellipse. For the points 

 of maximal and minimal strain Si and ^2 the ex- 

 pression lat -\- <p will take on the values (3'^)-7r, 

 (32)^- • • and we obtain from equations 2.5 and 2.6 



S, 



Si = 2*0/(1 + tan- 1^)' 



/ = 



2Aro 



' cos ip 



2Aro//cos ,p (2.7) 



(2.8) 



The quantity 2.V(j of our model corresponds to Al 

 of the actually measured characteristics (fig. 2C) 

 and the difference .S'2 — 6"! , to AS. For the restoring 

 force of the measured sample we obtain therefore 



/ = {Al/AS)-cos ip 



(2.9) 



In analogy to Hooke's law we define the dynamic 

 modulus of elasticity by equation 



AS L 

 Eiya = — ■ — COS ip dvn/cm^ 

 Al ?„, 



(2.10) 



where ?„ stands for the cross-sectional area which 

 corresponds to the prestretched state of length /„ . 

 The phase angle tp can be read from the ellipse by 

 drawing a vertical line through its center. This line 

 cuts the ellipse at points 3 and 4 (see fig. 4). At these 

 points co< takes the values o and tt and we obtain 

 from equation 2.5 



15*4 — Sz = 2 • 60 ■ sin (p 



(2.1!) 



Since the amplitude So is identical with ()^)(6'2 — ■S'l), 

 we obtain 



Because Si — .S3 and S2 — Si are well-defined lengths, 

 sin <p ov ip can easily be found from the e.xperiments. 

 In order to determine the second essential quantity 

 R or, better, the product u>R, we substitute for / in 

 equation 2.6 the value found from 2.9 and obtain 



oR = — • sm <p 



(2.13) 



Just as we did with the elastic modulus, we define in 

 this case a specific constant ri making R = riqm Im and 

 obtain instead of 2.13 



ASL 



Al q,„ 



sin 1^ 



(2.14) 



The quantity 7; has the dimensions of a viscosity 

 constant {g cm~'sec~^) and can therefore be looked 

 upon as the viscosity of the examined material. 

 The symmetrical equations 2.10 and 2.14 determine 

 its visco-elastic behavior. 



It is now necessary to comment on experiments 

 and their results: the above-mentioned elliptical 

 characteristic can be produced on the screen of a 



sin ,p= I^S,- S,)/{S, - S,) 



(2.12) 



FIG. 4 



