[28 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



constriction, 7' or / are small. In this case the expo- 



nential can be expanded {e 



-tin 



ly'l) and we 



obtain 



Wi = z'-[i + RAi - ■2y'l)]/[i - R.(i - 27'/)] (6.38) 



Taking the value Ro from equation 6.36 this leads to 



W, = Z'[2Z' - -iVKZ - Z')y[2Z' + ay'KZ - Z')] (6.39) 



The reflection coefficient at the central joint meas- 

 ured from the orisfin is 



R, = (W, - Z)/{W, + Z) 



(6.40) 



Taking for Wy the value of equation 6.39 we obtain 

 finally 



Ri = WHZ' - Z)(Z' -f- Z)']l\:iZ'Z -f- yl-{Z' - Z)2] (6.41) 



As we have already mentioned, Z' = py'/Q, holds also 

 for the rigid cannula; v' is then the speed of propaga- 

 tion of a sound wave in the liquid of density p. Its 

 order of magnitude is i o^ cm/sec, whereas the propa- 

 gation velocity for the elastic tube is of the order of 

 magnitude of 10 cm/sec. If we use a cannula of the 

 same cross section as that of the elastic tube Z' ^ Z, 

 and we obtain for ^1 , after some transformations, the 

 approximate formula 



R, = y'l-l\:iZlZ' -h 7'-/(i - 2^/^')] (6.42) 



because 7'/ is very small, ly'l-Z/Z' will be small of a 

 higher order, and we may write 



R^ = y'l/Wl + 2Z/Z'] 



(6.43) 



For equal cross sections, the surge impedances are 

 proportional to the propagation velocities, 2(Z/Z') = 



2-v/v'. For a glycerine-water mixture of density 1.14, 

 the sound velocity will be 163,000 cm/sec, whereas 

 the pulse wave velocity for the tube used was about 

 590 cm/sec; 2(Z/Z') was, therefore, 0.065. If ^^'c 

 ignore damping in the rigid cannula, we can write 

 7' = jw/v' and obtain from equation 6.43 



.fti = jwl/ijoil + iv) (6.44) 



Separating the real and imaginary parts, we get 



R, = wT-/(w'-r- + 41:') + j-2wlv/(wr- + 4;'=) (6.45) 



It can easily be seen that all points R lie on a circle of 

 radius 0.5 in the complex plane. The center of the 

 circle lies on the real axis at x = 0.5. The positions 

 of the points R on the circle are functions of the 

 parameter ccl only. Figure 20 shows such a half-circle 

 with some points corresponding to the indicated 

 parameters. The v value used was 590 cm/sec. This 

 kind of representation (space-curve construction) is 

 much used by electrical engineers. 



Let us take, for example, the relatively small value 

 <<)/ = 100. Figure 20 gives for this a reflection coeffi- 

 cient of 0.09. A cannula of 10 cm length would then 

 produce at frequency 10/(2^) = 1.6 a reflected wave 

 with an amplitude equal to 9 per cent of the incident 

 wave with a phase shift of nearly 90°. A cannula of 

 5 cm length would reflect only about 5 per cent under 

 the same conditions. 



Measurements with three different cannulas of 10, 

 30, and 125 cm lengths, inserted into a rubber tube 

 of the same cross section, showed in general the be- 

 havior predicted by the theory outlined above (fig. 

 21), taking into account the lack of precision of such 

 measurements. 



o.r 0,2 0,5 o> o,s 0,6 0.7 o,s 0.9 



FIG. 21. Experimental values of R obtained with three dif- 

 ferent cannulas of lengths 1 = 10, 30, and 125 cm. 



3,0cm' g sec' 

 ./ 



t 



".i^z 



FIG 22 



-2.0 





/^ 



^ 



/ 



1.0 







u^ 



10 



20 



3,0-10 



z,o-io: 



i.o-io"'. 



30 



-1 — ' — 1 — 



is 



^^ 



X 



FIG. 22. Rz\ R; and /?„ as functions of frequency. R2 has been 

 computed from VVomersley's theory, R.- and Ry from the ap- 

 Droximate formulas cauation 8. 1 o. 



