126 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



and equation 6.7 furnishes 



;. = (P,'/Z)e-y' - (P„"/Z)-«+" (6.28) 



if we use the index to indicate that the amphtude is 

 taken at the origin z = o. From equations 6.27 and 

 6.28 we obtain therefore the impedance at point Z 



^- (6.29) 



= Z{P/ + P/')/(P/ - P/') 



where P,' and P/' are, in general, complex quantities 

 containing the phase shift <p between pressure and 

 flow. We will define the quantity 



R = P/'/P,' 



(6.30) 



as the reflection coefficient and write the impedance 

 at z in the form 



W' 



Z(i + R^)/(i - R,) 



(6.31) 



For R = o, that is to say, for a tube free from reflec- 

 tion, W becomes equal to the surge impedance Z at 

 every point c, as already pointed out. 14'. becomes 

 infinitely large if Rz = +1 or zero it .ft. = — i. 

 The first case corresponds to a tube closed at .:, 

 the second to a tube open at Z- The first case may 

 be set up experimentally by clamping the tulje, the 

 second by connecting it to an open reservoir. 



If we know the impedance H\, at a given point c 

 along the tube, as from recordings of pressure and 

 flow, we can find the reflection coefficient from 



R. = (M'. - Z)/(I-)-', + Z) 



(6.32) 



This possibility is of great interest in hemodynamics. 

 If, at some peripheral point .: along the arterial stem, 

 we can make a simultaneous recording of pres.sure 

 and flow, preferably with an electromagnetic flow- 

 meter which integrates flow over the whole cross 

 section, and if we determine the amplitudes and 

 phases of the different harmonics with an analyzer as 

 described in the preceding section, we shall obtain 

 the reflection coefficient, that is to say, the ratio of 

 the amplitudes of reflected to outgoing waves, and 

 their respective phases as a function of frequency, 

 provided that Z is also known at that point. Z might 

 be roughly estimated at least from the diameter, 

 wall thickness, and Young's modulus E or E' re- 

 spectively, where care must be taken to measure E 

 at the relative extension occurring in vivo.' Experi- 



' A tpchnic similai- to tliat u.scil by Peterson (14) might well 

 be used for this purpose. 



ments of this kind might well be the best way to 

 settle the old question of peripheral reflection. 



When speaking of reflection, we must alwa\s keep 

 in mind that the reflection coefficient is a function of 

 locus, that is to say, of the coordinate c along the 

 conduit. If we talk of the reflection coefficient of a 

 constriction, for example, we mean the ratio of the 

 incoming to the outgoing wave amplitude just before 

 that constriction, but this reflection factor depends 

 upon all the characteristics of the conduit before and 

 beyond the constriction. 



If we wish to know the input impedance of a con- 

 duit which has a reflection coefficient Ri at a certain 

 point z = I 'we obtain from equation 6.29 



W'o = (Po/h) = ZiPo' + Po")/(Po' - Po") 



= Z(Pi'ey' + Pi"e-y')/(Po'ey' - Pi"e-''') 



and when Ri = P" jP{ , we then have 



\\\ = Z-(i + «,«-=•'')/(■ - Rit---'^^ 



(6.33) 



(B.34) 



With this equation we are able to calculate an 

 unknown reflection coefficient at point z = I from a 

 measured input impedance Wo , if 7 = /3 -|- ja, that 

 is, if the damping constant and the speed of propaga- 

 tion are known with sufficient accuracy. On the other 

 hand, (3 and a may be found from input impedance 

 when the reflection factor Rt at z = /is known; for 

 example, when .ft; = -|- i for a tube clamped at c = /• 

 This method proved to be quite useful for the de- 

 termination of /3. Indeed, the /J-values given in figure 

 1 1 were obtained in this way. The determination of 

 a = 01/ V, on the other hand, can be obtained more 

 precisely by direct determination of phase velocity 

 from recordings at diflferent places along a sufficiently 

 long tube which is practically free from reflections. 



The way in which the complex equation 6.34 can 

 be solved for 7, that is to say for a and 0, has been 

 described in Appendix 2. As the result of such a calcu- 

 lation the reflection factor will be obtained in the 

 form R = a -\- jb, which can be stated in the more 

 convenient form R = P" /P' = \ R\-e\j/. This means 

 that the pressure of the reflected wave leads the 

 pressure of the incoming wave by the phase \j/ (which 

 may also be negative). 



We shall now apply these theoretical considerations 

 to a few examples and compare the results with ex- 

 periments carried out by the author. 



If a second conduit of surge impedance Z-2 is 

 joined to the first, with surge impedance Zj , the re- 

 flection just before the joint will be 



/? = (Z, - Z,)/(Z. + Z,) 



(6.35> 



