132 HANDBOOK OF PHYSIOLOGY ^^ CIRCULATION I 



\L 



theor 

 0,5 cm'' g 



- 0,'/ 



-0,3 



0.2 



0.1 

 Z 



L_ 



8 



10 12 1U 



_i I 1 — 



FIG. 24. Calculated values of Z, from equations 8.1 1 following 

 Womersley's theory. 



Of course, the proportionality of R; with frequency 

 is, again, a rough approximation, first because /3 is 

 proportional to the frequency only in a general way 

 and second, because Z, too, still depends a little upon 

 frequency, showing a slight increase with it. L and C 

 are practically constant. We might be tempted to 

 believe that the resistance R; is caused only by fluid 

 friction and R„ onh' In' wall friction. But this is not 

 true because the damping constant &, and therefore 

 q, is a function of both viscosities t/i, and r\ of the 

 liquid, as well as of the tube wall. L is not absolutely 

 constant. This might ije explained from equation 8.10 

 by the variability of the surge impedance. In reality, 

 there is a small interdependence between frequency 

 and the oscillating mass of a definite part in the tube. 

 It is also possible to obtain an expression for L from 

 Womersley's theory. Taylor (21) obtains the following 

 formula : 



— Re^- 



(8.1 1) 



where $ is the function already mentioned in the 

 previous section. In figure 22 we have drawn not only 

 the curve for Ri calculated from equation 7.4, but 

 also the function R. = Zqtxi and 



R„ = VZqKq- +/<=)] I /co 



with/) = 1.20- io~' and q = 2.9- lo"''. These values 

 have been obtained from measurements with the 

 thicker-walled tube used as an example several times 

 before. For Z the real value 307 has been taken. In 

 figure 24 the value L calculated from equation 8. 1 1 

 has been represented as a function of the dimension- 

 less parameter a. It increases a little from the value 



p/(r'ir), passes through a flat maximum, and again 

 approaches the initial value. That means that in a 

 certain range of low frequencies, a little more of the 

 liquid mass is in oscillation than for the higher fre- 

 quencies. 



A more detailed analysis has thus shown that the 

 analogy between the elastic tube and the substitute 

 electric circuit shown in figure 23/? is only formal. To 

 use such a system as the basis for analog computer 

 studies on pulse waves, we would have to build re- 

 sistances with the frequency characteristics postulated 

 by the previous analysis. But these resistances should 

 exhibit an ohmic character, which means that no 

 phase shift should occur between current and tension. 

 This might be achieved with certain expensive elec- 

 tronic equipment, but the whole circuit would be 

 rather complicated. If someone succeeds in realizing 

 such an analog, it might be used, for example, to 

 study the propagation of a single pulse as a transient 

 phenomenon, which is practically impossible by 

 means of mathematics alone, even with the help of 

 Fourier integrals or Laplace transformations. The 

 study of single impulses in the mechanical model, 

 using a rubber tube and with the help of recording 

 manometers, is not of much use, because it is almost 

 impossible to give the input pulses any specific desired 

 shape. On the other hand, this is easy to do with elec- 

 trical input pulses. Also, recording tension with a 

 cathode-ray oscillograph is much simpler than with a 

 recording manometer. The same is true for the re- 

 cording of electric current, whereas the recording of 

 flow in the tube model with the electromagnetic flow- 

 meter involves rather expensive apparatus. The study 

 of reflections due to ramification with the help of an 

 electrical analog would not be of any use, because it 

 would be based on the same false suppositions men- 

 tioned at the end of section 6. As a whole, not much 

 insight can be gained from the electrical analog, 

 because it provides only a different picture of things 

 which are already understood. Ncv'ertheless, its use 

 has had merit, since it has acquainted physiologists 

 with mathematical methods and such concepts as 

 impedance, resistance, and .so on, which have proved 

 to be very useful in electrical engineering. The reason 

 for our devoting so much space to all this lies in the 

 fact that the use of analogs has become fashionable, 

 and there is great danger that they may be used im- 

 properly. On the other hand, the recognition of 

 common features in the diff'erent fields of science is 

 always pleasant to the human mind. 



