PULSE WAVES IN VISGO-ELASTIG TUBINGS 



129 



As an application to experimental technique, let us 

 consider the case of a I'emoral artery with a pulse-wave 

 velocity of 800 cm/sec. What will be the reflection co- 

 efficient for a cannula 5 cm long tied into it? For the 

 angular frequency co = 10 or v = 1.59, we obtain 

 from equation 6.45 R = 7-0.031; the reflected wave 

 would therefore amount to about 3 per cent with a 

 phase shift of go°. But with a frequency of o) = 100 

 {v = 15-9) we obtain R = 0.089 + 7 -0.284, 3"^ the 

 absolute value of .^ would be R = (0.089- + 0.284-)"- 

 = 0.296. The reflected wave would therefore be about 

 30 per cent of the incoming wave, and the phase 

 shift will be obtained from tan tp = 0.284/0.089 = 

 3.19 as ^ = 72.6°. 



As a last example, ramifications shall be considered 

 as a source of reflection. For the general case of ramifi- 

 cation into two branches, we have to consider a con- 

 duit of surge impedance Zi splitting off in two 

 branches, with surge impedance Zo and Z3, respec- 

 tively. In order to find the reflection coefficient just 

 before the ramification, we must find the joint imped- 

 ance Z' of the two branches. One might be tempted 

 to put Z' = Zi.Zi,l{Zi, + Z3) as we do in the case of 

 two electrical transmission lines connected in parallel. 

 This would lead to convenient formulas for the reflec- 

 tion coefficients, but our own attempts to check these 

 formulas experimentally have given, so far, only 

 negative results. This is not very surprising, because 

 we cannot construct an actual ramification without 

 creating some angles in the conduit. Such bends give 

 rise to very complicated hydrodynamic conditions, 

 even in the case of a stationary current, which have 

 been the object of many experimental and theoretical 

 studies in applied hydraulics. A brief account of these 

 is given in a paper by Ansgar Miiller (13). Whether 

 or not gentle curves or sharp angles change the im- 

 pedance in a conduit has not yet been systematically 

 studied as far as we know. It seems to us that we may 

 not use knowledge obtained e.xclusively from straight 

 conduits to tackle the problem of ramifications, where 

 changes in momentum occur. 



hydrodynamic theory of wave propagation in tubes 

 with ideally elastic walls and filled with a nonviscous, 

 incompressible liquid. Since then, the theory has been 

 further developed by Frank (3), Morgan & Kiely 

 (11), Lambossy (8), and Womersley (24). Lambossy 

 treats the case of pulsating flows of a viscous liquid in 

 a rigid tube. He obtains a solution for velocity as a 

 function of time and distance r' from the axis of the 

 tube. From this solution he obtains, in addition to the 

 velocity profile, the viscous drag acting on the surface 

 of a liquid cylinder of radius r' . Integration furnishes 

 the total resistance to flow and the deviation from 

 Poiseuille's law, which we have already mentioned. 

 Womersley, independently, obtained the same solu- 

 tion for a rigid tube although in a somewhat different 

 form, and extended the theory to the elastic tube. The 

 rather complicated functions are given in the form of 

 tables in a more extensive paper (25). A common fea- 

 ture of both theories is the important statement that 

 the type of flow depends upon the dimensionless 

 parameter: 



= r(c^p/>;i)"= 



(7.0 



the significance of which, for pulsating current, is 

 similar to that of the Reynold's number for stationary 

 flow. This is of practical importance in the designing 

 of models. Experiments using soft rubber tubes of 

 large diameter and low frequency are more difficult 

 to make, and yield less accurate results than experi- 

 ments on tubes with smaller bore and higher fre- 

 quency. It cannot yet be stated, however, whether 

 this statement holds true for the more complicated 

 conditions present at sharp bends and ramifications. 



We can obtain much information concerning re- 

 sistance to flow {R-^ from the hydrod)namic theory. 

 As we have already pointed out in section 4, 

 Poiseuille's law, and therefore equation 4.4, is no 

 longer useful to us when we change from the study of 

 stationary flow to that of pulsatile flow. 



Womersley finds, for mean flow in c-direction, the 

 relationship 



7. HVDRODYN.^MIG GONSIDER.ATIONS 



In the previous sections we have treated wave prop- 

 agation in an elastic tube as a one-dimensional prob- 

 lem, that is to say we considered total flow only as the 

 product of mean velocity and cross section of the 

 conduit, and did not worry about the velocity distribu- 

 tion over the cross section of the tube (velocity- 

 profile). As early as 1878, Korteweg (6) worked out a 



pi' 



(7-2) 



where the pressure is given by/) = pg'"^<--i'\ $ is a 

 complex function of the parameter a = ricop/rjiY^-, 

 of the ratio k between wall thickness and tube radius 

 (k = a/r), and of Poisson's ratio a which we always 

 take as 0.5. (The function $ appears in Womersley's 

 tables as i -1- -qF 10, where 17 is some other function 

 having nothing to do with a viscosity constant.) 

 In order to find the value /?•>, we start from equa- 



