PULSE WAVES IN VISCO-ELASTIC TUBINGS 



127 



TABLE 4 



p = Density of filling fluid = 1.14. 



To test this formula, some experiments have been 

 carried out witli two rubber tubes, the data of whicli 

 are given in table 4. The direct determination of Z 

 as the input impedance of a tube, the length of which 

 was 10 meters and practically free from reflection, 

 gave a value of 130 for frequency i and a value of 160 

 for frequency 5. 



If we join the larger tube I to the narrower tube II 

 we obtain from equation 6.35 a value of —0.352 for 

 R. That is to say, the reflected wave is reversed in 

 phase as is the case for an open end, and the amplitude 

 of the reflected wave will be 35 per cent of that of the 

 incident wave. The calculation of the input impedance 

 measured 30.5 cm before the joint gave the values 

 plotted in the complex plane in figure 19 for the fre- 

 quencies I, 2, 3, and 4 cps. The calculations were 

 made according to the method outlined in Appendix 

 2. The "vectors" drawn to the points from the origin 

 correspond to the absolute value or "modulus" of R. 

 Their mean value is 0.46 instead of 0.35 as calculated 

 from the simple and approximate equation 6.35 where 

 only real values for Z have been used. The phase is 

 somewhat smaller than 180° suggested by equation 

 6.35. If we take into account the complicated calcula- 

 tion of the input impedance, which demands an accu- 



rate knowledge of a and /3, and the approximate 

 character of equation 6.35, the agreement with 

 theory is not too bad. 



If, on the other hand, we join the narrow tube to 

 the large one, we obtain a reflection factor of -I-0.352. 

 The experiment gave in this case a mean phase angle 

 of about -|-35° instead of 0°, and the /?- values showed 

 a strong dependence upon frequency, ranging from 

 0.78 at frequency i down to 0.35 at frequency 4 and 

 0.1 at frequency 4.8. Agreement with the theory was 

 therefore quite poor. 



Better and fairly interesting results were obtained 

 with rigid cannulas of different lengths inserted into 

 the elastic conduit. Because these results are of some 

 interest in regard to experiments on animals, which 

 often necessitate the use of a cannula tied into a vessel, 

 and because they furnish some information on the 

 influence of stenoses and other pathological changes, 

 we will discuss them briefly. 



A piece of conduit of length / and surge impedance 

 Z' shall be inserted into a tube line of practically infi- 

 nite length and surge impedance Z. The propagation 

 constants shall be Y = fi' -\- ja' and y = fi -\- ja re- 

 spectively. Z' as well as Z are assumed to be real. The 

 reflection coefficient at the peripheral joint will 

 then be 



R. = (Z - Z')/(Z + Z') 



(6.36) 



and the impedance at the central joint nearest the 

 source of the wave will be 



»"i = Z'-(i -I- R.,e--^'^)/(i - R.-e-^y''} (6.37) 



If the inserted piece is either a rigid cannula or a short 



FIG. 19. Reflection at the junction of a wide tube and a nar- 

 row one, as: 



I 



I 



2000 



3000 



10000 



Complex /{-values for different frequencies. 



FIG 20. Calculated complex reflection coefficients of an 

 inserted rigid cannula of length 1. All the fi's with different 

 parameters co 1 lie on a half circle with radius 0.5. 



