:i6 



HANDBOOK OF PHYSIOLOGY ^^ CIRCULATION I 



^' = K^^')/ 



2 -^ + 2(1 +<r)- 

 r 



+ (l +a) 



m 



(3.'9) 



where a denotes Poisson's ratio. For rubber, as well 

 as for arterial walls, this ratio is almost exactly 1/2. 

 Equation 3.19 can therefore be written as 



(-0/ 



i>5 + 3 - + ',5 



(")■ 



(3.20) 



From the theoretical point of view, v can only become 

 equal to ;'o according to equation 3.15 if cjC and 

 therefore ojtj vanishes for v = o. Whether or not this 

 is the case cannot be deduced from our experiments. 

 Nevertheless, it can be shown that in any case the 

 influence of co?) on the speed of propagation is very 

 small. For a ma.ximal value of 14-10' dyn/cm- for 

 w-q, according to the results of figure 5, we obtain 

 for coC = uir\a/{2rp), with the data given above, the 

 value 0.48- 10' and for co-C' the value 0.23- 10^°. On 

 the other hand z^o^ = 14.7-10'". The quantity co'C^ is 

 therefore quite small compared with wq'', and we 

 obtain for the roots (j'o^ + oj-C-)''- in the equations 

 3.15 and 3.16 the value 1.008 v^ ■ The difference 

 between v and va is therefore too small to be measured. 

 Thus, a wall viscosits' of the assumed value has no 

 measurable influence on the speed of propagation 

 which conclusion is in fair agreement with the 

 measurements of A. Miiller (12). 



The damping constant 6, on the other hand, be- 

 haves quite differently. In this case, also, the factor 

 before the bracket in equation 3.16 depends very 

 little upon cotj. But the bracket contains a difference 

 between unity and a quantity which is but little 

 less than one. Because co-C- « s'o'' the bracket can 

 be expanded to read 



I - i-oVC^o* + w^t'^) 



/2 ^ , _ I 





/ arC-\ orC- 



~' \ 2fio-/ 2ro' 



and we obtain (or /? the simple approximate formula 



/3 = coC= oj = K'o 



(3.21)' 



' Morgan & Kiely (11, equation 56) find for low viscosity of 

 the liquid in our present notation. 



Vo \_r \2a)p/ \ 4/ 2E_] 



(i -0,0027 



fi- 0,0029 

 (iz 0,0032 



(i^Q.OOU-l 

 litO,OOU8 



6.88 ^(i^O.OOSO 

 f^: 0,0060 



100 



zoo 



300 



t400 cm 



FIG. 9. Decrease of pressure amplitude in pulse-waves of 

 different frequencies. 



With a)C = 0.48-10^ and j'o = 608 cm sec, K be- 



comes 1.08- 10 ^ or 



6.75-10 *. The damping 



constant is therefore roughly proportional to the 

 frequency, if cojj is constant. 



The damping constant /3 can be found when we 

 measure the pressure amplitudes in a tube at different 

 distances from the source of the wave. Care must be 

 taken that at the recording site the waves are truly 

 sinusoidal in shape. Such determinations were 

 carried out first by A. Miiller (12) and have been 

 repeated several times by the author. Figure 9 gives 

 an example of such a determination, where the 

 natural logarithm of the pressure amplitude is 

 plotted against the distance from the origin of the 

 line for a number of frequencies. Whereas the data 

 from Muller's paper give a nearly exponential de- 

 crease with c, this is not the case for the present 

 example where the distances z are considerably 

 greater. Damping decreases markedly for larger 



Neglecting viscosity of the liquid altogether, we obtain /3 = 

 0) (oiTj/a 2'o E). It can be seen at once that this is identical 

 with our equation 3.21 if we replace 0/(2 r p) in the latter by 

 i\lE. 



