130 



HANDBOOK OF PHYSIOLOGY -^ CIRCULATION I 



(T^ 



^ — r-^AAAA/^-^fln — 



T 



n/. 



FIG. 23 



tion 4. 1 , vvhicli we write, for the present purpose, in 

 the form 



R2U = 



dp 



du 

 dz 



(7-3) 



If we use for p the complex term from above, R^ will 

 also come out as a complex quantity. Dividing by the 

 exponential, we obtain 



i-Ri-P/v = joiP/v - j-^u>P/v 



and, finally, we get for the real part, whicii is the only 

 part of interest to us. 



or 



R2 = /?c|;wp(*-i - i)l 

 = -ni(aV'-'')-Im<!'-' 

 R". = — i;i(a-A-)-Im|*~» - 



(7.4) 



With r = 1.02 cm, r/i = 0.078, p = 1.14, A: = 0.2 

 (this is the set of values in the tables next to the sample 

 calculation a/r = o. 193/1.OQ = 0.189, which corre- 

 sponds to our tube model), we obtain for a = i, 

 which corresponds to the very low frequency v = 

 a)/(27r) = 0.0105 * ~ 0-OI55 + ; 0.1235 ^'""^ 'h^ ''^" 

 ciprocal $~^ = 1.025 ~ / 8.13. Finally Ri becomes 

 0.078-8. 1 3/ 1. 04 = 0.608. That is to say, we obtain 

 for this very low frequency practically the same value 

 as we would obtain from equation 4.4, namely, 

 R2 = 0.078-8/1.04 = 0.600.* In a similar way, we find 

 for a = 10 Rt = i.oi. 7?2 as a function of angular 

 frequency or the parameter a is plotted in figure 22. 

 Because Womersley calculated his tables only up to 

 a = 10, the dotted part of the curve is merely a 

 guess.' Using a very crude approximation, R« can be 



' In a more general way this result can be obtained mathe- 

 matically by expanding the function Fit, = 2 Ji (a 7^"')/ 

 [a y^ J a (a 7^'^)], where Jo and Ji are the Bessel functions of 

 order zero and one. In this way we find, for small values of a, 

 that Im j(i + F,o)-M = -S/r^and therefore R-. = i)-8/r2. 



' In this case, also, the function F\ can be appro.ximated 

 for large values of a by using the semi-convergent series for the 

 Bessel functions Jo and J\. This leads to the asymptotic formula 

 R-. = ('), oA^) (2- \/2/*), where x = 2.74 for a = 0.5 and 

 k = a/r = 0.2, using Womersley's equation 40 with the + 

 sign suggested by the preceding case a > o. A graph of Rt 

 values over a more extended frequency scale shows that the 



represented by a linear relation. (Compare the dis- 

 cussion in section 8.) For the mentioned example, we 

 obtained R2 = 0.6 -|- 0.06 to. We used this result pre- 

 viously in section 4, when discussing the influence of 

 liquid friction on velocity of propagation and damp- 

 ing (see equation 4.5). 



8. ELECTRICAL ANALOG OF THE ELASTIC TUBE 

 AND ITS LIMITS OF APPLICATION 



A reader familiar with the theory of electric trans- 

 mission lines will see the striking resemblance between 

 the propagation of pulse waves and electric waves 

 along a line. This relationship was first mentioned by 

 Landes (9). Before making extensive comparisons, we 

 should first determine which type of electric conduit 

 corresponds to the mechanical tube model. Let us 

 consider a homogeneous electric line, for example a 

 telegraph wire stretched out at a certain distance 

 above the reconducting ground, or a one-core cable 

 in a grounded reconducting sleeve. The cable or the 

 telegraph wire has a definite resistance R, self-induc- 

 tion L, and capacity C per unit length {dz). The loss 

 per unit length due to incomplete insulation may be 

 represented by the conductivity G. We can therefore 

 replace a part of the line dz by the scheme shown in 

 figure 23.4. For the tension u and the current i we can 

 easily derive the differential equations 



(8.0 



(8.2) 



di du 



— = G-» -I- C — 



dz dt 



We look now for siinilar equations for our tube model. 

 Equation 8.1 obviously corresponds to equation 4.1 

 in the form 



+ /?■:■'. 



(8.3) 



dp p di 

 ~ dz " dl 



Now can we also find the mechanical equation corre- 

 sponding to equation 8.2? Solving equation 3.5 for 

 di/dz, w'e obtain (dropping the inde.x z) 



di 2Trr^ dp R^-r^ d'i 

 dz Ea dt E'a dzdt 



(8.4) 



Equation 8.4 has no term with p, but only its deriva- 

 tive against time. Therefore the term including u in 



rise of the fij-curve diminishes with increasing m so that the 

 ^2 value, say for a = 23, would be about 1.9 instead of 23 

 as indicated by the dotted line in figure 22. 



