124 HANDBOOK OF PHYSIOLOGY "-^ CIRCULATION I 



Inserting this expression in 6.16 we obtain finally 



Z = -- W +>C)"2 



(6.18) 



If we further neglect wall friction, C becomes zero 

 and we get for Z the very simple formula 



Z = p»o/(i 



(6.19) 



This formula was first proposed by Landes (9). It is 

 also valid for the propagation of sound waves in 

 rigid tubes. In this case v, as we have mentioned in 

 the first section, is conditioned by the elasticity of the 

 liquid and Z becomes very high in comparison with 

 the surge impedance of a soft rubber tube. Because 

 coC is, in most cases, much smaller than vo-, equation 

 6.19 is a fair approximation, as demonstrated by our 

 experiments. 



If Z is real, that is to say if equation 6.ig can be 

 used, pressure and current are in phase when no 

 reflection is present. Experiments show, in most 

 cases, that current leads pressure in phase by about 

 5 to 10 degrees. If we introduce liquid friction again, 

 we obtain instead of 6.19 the equation 



Let us again use wC = 0.48-10 , co = 10, R2 = 1.2 

 (according to equation 4.5, vo = 880 cm/sec and 

 p = 1. 1 4. Then tan <p = —0.0714 corresponding to 

 an angle of about —4°. That is to say, current leads 

 pressure by about 4°, in fair agreement with the 

 afore-mentioned experiments, because / = PI Z = 

 P/Ze-^"''". 



With the help of equations 6. 1 1 and 6. 1 2 we are 

 now able to find the input impedance of a line which 

 is closed at the end c = /• In this case, the output 

 impedance will be real and infinite or in mathematical 

 expression 



Pi/Ii = 



(6.23) 



From this we obtain the input impedance Po/Io = Zq 

 at the origin of the line with the surge impedance Z 



Po/h = Zcosh (-,/)/sinh (7/) 



(6.24) 



If we suppose Z to be real we obtain, by separation 

 of the real and imaginary parts. 



P0//0 = Z-[e'-«' 



- 27 sin (2a/)]/ 



2 cos (2a/)] 



(6.25) 



Z = ii/jaQ)-(R2 + jwp){iv + juCy- (6.20) For tlie undamped line with /3 = o we would have 



Because ;a>C generally amounts to not more than 10 

 per cent of vcr, we can expand the root and get 



Z = (i/y«Q)-j)o'-[i + jo,C/(2v,'')]-{Ri + jojp) (6.21) 



Separating the real and imaginary parts, we can put 

 Z in the form 



Z = A + jB or Z = Ze'^ 



where tan tp is given by 



tan .p = B/A = (oi-pC — 2V(rR2)/(R<o>C + iviroip 



(6.22) 



Po/h 



-}Z sin (2al)/[i — cos (2a/)j 



(6.26) 



that is to say, the input impedance is purely imagi- 

 nary. The current leads the pressure by 90° if sin (2co/) 

 is positive and lags behind the pressure if sin (2co/) is 

 negative. Figure 16 shows the input impedance 

 plotted against the lengdi of the tube. The ordinates 

 are the absolute \alue | Po/Ia | (solid line) and the 

 phase angle (dotted line). Z has been taken as unity 

 for I Po /o I- If, for example, / = X/4, X being the 

 wavelength, the current is maxiinal at the origin of 

 the conduit and the pressure amplitude is infinitely 



FIG. 16. Calculated input-impedance of 

 a closed rubber tube. (Modulus P^/P and 

 phase ip) as a function of the tube length. 



f 



■so' 



90° 



