PULSE WAVES IN VISCO-ELASTIC TUBINGS 



131 



the electric analog is dropped, which means that the 

 conductixity G must disappear. On the other hand 

 we find, in the mechanical equation, a term with the 

 composite derivative [{Rir^/Ea)-(d''i/dzdt}] which 

 has no analog in equation 8.2. We will therefore try 

 to work with a modified circuit according to figure 

 235 with G = o. In series with the capacity we con- 

 nect a resistance R„. The decrease of current in the 

 element c/z of the transmission line corresponds then 

 to the diagonal current driven by the tension u 

 through the combination CR,,. This tension is the sum 

 of the tension Ug^^ across the resistance R„ and the 

 tension Uc across the capacity C, and we have 



w = «R„ + "c (8.5) 



The term —di/dz in equation 8.4 corresponds to the 

 diagonal current iy per unit length dz of the line, that 

 is, to the current going through the side branch 



R. 



TABLE 5 



C of the substitute circuit; or, as a formula: 





= iy = C- — = C- 



" 9/ 



- c 



dt 



(8.6) 



The voltage drop Ur^^ is, however, iyRy = —Rydi/dz 

 and thus we get the electrical equations correspond- 

 ing to the mechanical equations 8.3 and 8.4: 



9.V 

 ai 



ai 



+ R." 



9« d^i 

 C 1- C R„ 



dz dl dzdl 



(8.7) 



(8.8) 



The difi^erence between these equations and the me- 

 chanical one lies only in the different meanings of the 

 constants. Therefore, the substitute circuit (fig. 23) 

 that we chose was practical. To simplify for the 

 reader the change from one system to the other, we 

 have put corresponding values together in table 5. In 

 the first column, we find the terms for electrical values 

 and some of their combinations. In the second col- 

 umn, we see the corresponding mechanical values as 

 obtained by means of comparison with equations 8.7, 

 8.8, and 8.3, 8.4, respectively. The difference between 

 the third and the second column is only the introduc- 

 tion of the velocity vo for the undamped case. The last 

 column contains the dimensions of the mechanical 

 constants for the tube line. For the electrical analog, 

 as well as for the elastic tube, the product RC of re- 

 sistance and capacity, respectively, and the product of 

 viscosity and reciprocal elasticity i/E have the di- 

 mension of time. 



Our electric model has, however, a great practical 

 disadvantage, as previously acknowledged by Taylor 

 (21). From the second column of table 5, we see that 



* For low frequencies only (w — > o), 



there is an obvious dependence of C upon frequency, 

 if the modulus of elasticity depends upon frequency. 

 Nevertheless, this alone would not be too bad, because 

 in most cases E depends very little on frequency, ex- 

 cept in cases of very low frequencies not important in 

 the physiological range. But matters are much worse 

 for the diagonal resistance, Rj, because ri is approxi- 

 mately inversely proportional to the frequency oi, the 

 product ojTj staying more or less constant. Finally 

 Ri ^ R-t is also inconstant, as we see theoretically 

 from the Womersley papers mentioned in the pre- 

 ceding section. Taylor (21) shows that one may ex- 

 press the values Rj, R„, L, and C in a more general 

 way by Z and 7, that is to say, by the measurable 

 surge impedance Z and the propagation constant 

 7 = /3 -|- ja. He obtains the expressions 



R= 



Re(Zy) 



ReiZ/y) 



L = {\/o>)-Im(Zy) l/C = -<^Im(Z/y) (8.9) 



Re and Im, before the brackets, mean that one must 

 take either the real or the imaginary part of the value 

 in the bracket. Again we will first discuss the general 

 course of the values considered. Z, as we know, is 

 practically constant and real, a = oj/y is almost ex- 

 actly proportional to the frequency and the same too 

 is approximately true for ji. Therefore, we take a = 

 p-(j) and 13 = q-ic and we have 



R, = Zgo, Ry= (iM-Zq/{f+p^-) (8.10) 



L = Zp = constant C = (p- + q-)/(Zp} = constant 



