134 HANDBOOK OF PHYSIOLOGY ^^ CIRCULATION I 



coy CJy 



CJ= CJ, 



cj <ajj. 



FIG. 3A 



complex equation like 5 can be split into two real equations. 

 If we use Re as an abbreviation for [QJpo} — {kQ;/o}V)], 

 and if we replace c"' by cos (p + j sin ip, according to Euler's 

 theorem we obtain: 



'0 f* = 'o(cos ip+ i- sin tp) = po/(R + jR, ) 



(8) 



In order to separate the real and imaginary parts of the 

 right side of the equation, we multiply numerator and 

 denominator by the conjugate complex R — jR, of the 

 latter. This leads to 



!o(cos ,p + /-sin ^) = A,(7? - jR,)/(R' + R,n (8a) 



From this we obtain the first real equation for the phase 



tan ^ = -R./R (9) 



The real part of equation g furnishes 



;„ -cos ^ = p„R/(R,? + R'-) ( i o) 



With cos i^ = 1/(1 + tan- <p)''- we get the second real 

 equation 



i„ = p,/iR'- + RJ') 



(11) 



The highest possible amplitude of current or flow will be 

 obtained if /?„ = 0. This is the case when the angular 

 frequency becomes 



m" 



(12) 



The frequency given by 1 2 is the resonant angular frequency 

 of the rubber balloon with neck of length / and cross sec- 

 tion Q, 



Equation 6 permits a geometrical interpretation. For dif- 

 ferent frequencies, all the points W = x + jy in the com- 

 plex plane lie on a straight line parallel to the/v-axis which 



TT 



CO 





FIG. 4A 



cuts the .v-axis at the distance R from the origin, the cor- 

 responding frequency being the resonant frequency oj^ 

 (fig. 3A). The distance from the origin to the point W is 

 equal to the absolute value W of the impedance. 



Figure 4A shows how the phase angle ip depends on the 

 frequency. For the undamped case R = o, the phase angle 

 jumps from ( — ^2)-ir to +('2)"" at the resonant frequency 

 (jir . For R > oh passes continuously from ( — 'a)'"' to 

 i + l-i) •"■ with increasing frequency. 



The first term within the bracket in equation 6, Q_lpui, 

 is called inertance, while the second term, K()^/(Fa)), is 

 called compliance. If these terms are properly chosen, a 

 device similar to that of figure i might be used to match 

 the surge impedance of a rubber tube, which would then 

 behave like an infinitely long tube free from reflection. 



APPENDIX 2 



DERIVATION OF THE COMPLEX REFLECTION COEFFICIENT R 



In order to find the complex reflection coefficient R for 

 a given y = ^ + ja from equation 6.34, we put R in the 

 form R = (■-<<■+"•' and try to find a and h. If we insert 

 this form for R into equation 6.34, we obtain with 7 = 

 + ja 



Z 



I _|_ (,2(a+j6)j-2(«+,a).i , _j_ gHx+iy) 



I _ f2(<i+l6).f-2(«+|<.).l I _ j2(I+;V) 



Using X for a — ffl and y = b — al. With the definitions 



cotanh .v 



we find 



— = —cotanh (.v + ly) — 



f ' 4- e~ 



= tanh .V 



tanh ( — .V — jy) = u -\- jv 



When Z/li'o is given as a measured quantity u and v are 

 also known. Solving for ,v and v we obtain 



