PULSE WAVES IN VISCO-ELASTIC TUBINGS 1 23 



and for current 



, = I'ei^'^y- + /"«/"'+7.- (6.2) 



With 4. 1 we obtain from these expressions 



dp I 



dz (I 



{josp + R^JU'ei"'--" + I"ei'"+'") (6.3) 



But dpi dz can also be obtained by differentiation of 

 6.1 with respect to c, and we get 



dp 

 'dz 



= P'-ye'"'-^' - p"ye'"" 



(6.4) 



Comparing the coefficients on the right of 6.3 and 

 6.4 we obtain 



P'-y = (I'/(l)-(jc^p + R,) 



-P".y = (r/(l).iju:p + /?,) 



(6.5) 



Writing Z for tlie complex quantity (jup + R2)/{Qy), 

 we obtain for 6.2 the equation 



P' P" 



(6.6) 



Then, if we divide by e'" , that is to say, if we consider 

 amplitudes only, we obtain the important equation: 



P' P" 



I = - ■ e—" «+T' 



Z Z 



(6.7) 



which relates the amplitudes of pressure and current 

 along the conduit and is much used in electrical en- 

 gineering, for transmission lines. If we replace the 

 exponentials by the hyperbolic functions 



sinh^ = Yiie' — e~') cosh 4: = l-oie' + e'') (6.8) 



we obtain for 6.7 



/.- = /o cosh (7c) — — • sinh (7c) 



(6.9) 



where /o denotes the current amplitude and Pa the 

 pressure amplitude, at an arbitrarily chosen origin 

 within the tube, and /. the current amplitude at a 

 point c. A similar equation holds for the pressure 

 amplitudes: 



P, = Pocosh(7..) - /„Z sinh (7c) 



(6.10) 



Equations 6.9 and 6.10 permit calculation of the 

 amplitudes at a given point c from the input ampli- 

 tudes at ^ = o. An analogous pair of equations is 

 used for the inverse problem 



P. 



/ii = /.. cosh (7^) -\ sinh (7.J) 



Pii = fj cos (7^) 4- /^Z sinh (7^) 



(6..1) 



(6.12) 



We can terminate a conduit of finite length / with 

 any real or complex resistance W i . A purely real 

 resistance might be difficult to realize in practice. As 

 an example of a complex resistance we might use a 

 rubber balloon at the end of the tube line. (See 

 Appendix i.) By definition we have 



/>■ = /ilfi (6.13) 



If we make especially M'l = Z, we obtain 



/o = /,[cosh (7-/) -t- sinh (7/)] = /i •(■!■' (6.14) 



and likewise 



Po = Pic-'' (6.15) 



In this case the complex input impedance P0//0 is 

 equal to the output impedance — using the terminol- 

 ogy of electrical engineers — and is independent of 

 the length of the conduit, which might even be made 

 infinitely long without altering the reasoning. In 

 other words, this means that a line terminated by an 

 impedance Z behaves like an infinitely long conduit 

 without reflection. The impedance Z is called the 

 surge impedance of the conduit. It depends only 

 upon the characteristic data of the conduit, such as 

 the radius, wall thickness, elasticity, etc., but not 

 upon its length. 



Because many sources of reflection at constrictions, 

 junctions, marked curvatures, etc., are present in the 

 arterial system, evaluation of Z is of great interest. 

 Let us therefore look at the expression Z = 

 {Ri -\- jwp)/{Qy). For the thick-walled tube which 

 we used for a previous example, R2 was found to be 

 0.6 + o) 0.06 for w = 15 (see 4.5), that is to say, for a 

 frequency of 2.4 it would be 1.5, whereas cop would 

 be about 15, so that R^ is about ten times smaller 

 than cop. If we neglect the friction of the liquid in a 

 first approximation, we obtain 



Z =ya>p/(<2-T) 



(6.16) 



from 3.13 we have 



7' 



a- + 2ja0 = i-cj-vo- + w'C)/(i)„< -I- u=C2) 



and therefore 



7 = jw[(vc' +iwC)/{v„<' + w^O)]* (6.17) 



