PULSE WAVES IN VISCO-ELASTIC TUBINGS 



119 



TABLE 



TABLE 2 



Frequency 

 0l/0o 



2 cps 



1.58 



10 cps 



1.41 



31. g cps (oj = 200) 



■■31 



less viscous mixture for which p = 1.087 ^nd rj] 

 = 0.033 poises. 



On the other hand, damping is appreciably in- 

 fluenced by friction within the liquid. If we consider 

 the rate lit l^o of the damping constants with and 

 without friction in the liquid, we obtain, for the 

 frequencies used above, the values in table 2. It is 

 seen from table 2 that the rate fii/lio decreases with 

 increasing frequencies. Miiller's attenuation curves 

 for fluids with diflferent viscosities show this effect 

 clearly. Also, Morgan & Kiely (11) obtain a reduction 

 of the propagation velocity as a consequence of liquid 

 friction. With the symbols used by us, their equation 

 38 takes the form 



=''[-°'''i(tX] " 



ith a- = 0.5 



where the small influence of wall friction has not 

 been considered. For the damping constant they 

 obtain 



|3 = -l-l — )'" 0.562 + 



■2E 



where both wall and liquid friction have been ac- 

 counted for. With Vf, = 880, ojr) = 14-10=', E = 

 100-10^, and 171 = 0.078, we obtain for the fre- 

 quencies 2, 10, and 31.9 cps the /3-values 0.0013, 

 0.0057, 2nd 0.0188, which are smaller by a factor 

 0.78 than the corresponding values 0.0017, 0.0076, 

 and 0.0299 of table 2. 



To terminate these first sections, we are justified 

 in saying that the simplified treatment which we 

 have given furnishes a fairly complete picture of all 

 the phenomena occurring with purely harmonic 

 wave propagation in conduits free from reflection, 

 even though it is not quantitatively exact in every 

 detail. 



A supplementary remark must be made. We have 

 so far considered the tube wall to be infinitely thin. 

 In realits-, we always have to deal with walls of 

 mea.surable thickness. When a rubber tube is ex- 

 panded, the volume of the tube walls remains con- 

 stant, as we have already mentioned. As a result, 

 the tube shortens at the places where it is radially 

 expanded. If the tube is freely suspended, longitudinal 

 oscillations can be observed when a wave runs 

 through it. If, on the other hand, the tube is submitted 

 to a longitudinal constraint, which is the case with 

 blood vessels, conditions are somewhat different 

 and the velocity of propagation is no longer the 

 same. As we have already pointed out in section 3, 

 Edyn must be multiplied by a factor given by equation 

 3.19 in order to obtain a good agreement between 

 the measured velocity for low frequencies and the 

 Moens-Korteweg equation. Equation 3.19 holds 

 for the freely moving tube. For a tube with complete 

 longitudinal constraint, equation 3.19 has to be 

 replaced by the similar equation 



-(-:), 



r 



+(i + 



''Cr 



(4-6) 



or with u = o. 



E' 



""^ + ;), 



2 + 3- -+ ■ 



■=■ (:)' 



For the example used in section 3 with a = 0.107, 

 r = 1.365; Eiyn = 90- 10^, and p = 1.14, we obtain 

 E' = E^yn 0.93 instead of E' = Edyn 1195 for the 

 freely suspended tube. In the case of longitudinal 

 constraint, ;' turns out to be somewhat smaller than 

 indicated by the Moens-Korteweg formula. Moens 

 accounted for this by the semi-empirical formula 

 ^ = 0.9[is-fl/'(2rp)]"2. 



With this we close the discussion of pure harmonic 

 waves in conduits free from reflection. In the following 

 sections we shall deal with more complex phenomena 

 of nonharmonic waves in tubes which are not con- 

 sidered to be free from reflections. 



