PULSE WAVES IN VISCO-ELASTIC TUBINGS 



n 



V 



1 & S t^ S 6 7 



FIG. 10. Damping constants for pulse-waves versus fre- 

 quency. O— O— O— O = Values obtained from the decrease of 

 pressure-amplitude taken from fig. 1 1 . X— X— X— X = Values 

 calculated from the impedance of the closed tube. 



distances from the source. This can be easily under- 

 stood if we consider that the amplitudes become 

 smaller with increasing distance. As we have shown 

 in section 2, cor; diminishes with decreasing amplitude 

 and consequently /3 must diminish also. Because the 

 wa\es used for this experiment were very nearly 

 sinusoidal, the observed fact cannot be explained by 

 the presence of more highly damped harmonics. 

 If we compute the /3-values for the section of the 

 curves between ~ = 125 and z = 250 cm, we obtain 

 the values indicated on the curves together with the 

 frequency used. 



A method which we shall explain later permits 

 calculation of an "effective" ^ from the input im- 

 pedance of a closed tube (see section 6). We have 

 used this method for the same tube with which the 

 above-mentioned determinations were carried out 

 and which was clamped at ^ = 135 cm. The results 

 are plotted in figure 10 as a function of frequency 

 together with the values obtained with the previously 

 mentioned direct method. The values obtained 

 with the second method, indicated by crosses on the 

 graph, lie approximately on a straight line running 

 through the origin, if we allow for the lack of pre- 

 cision of such measurements. The indicated straight 

 line (dotted line) would correspond to the formula 



= I.I -la-'-v = ul-q- 



2-rpVo' 



or ujt;- 



0.107 



W7) • 0.64 ■ I O"' 



2-I.37' i.i4-6o8* 

 From this we find for ojt) the value i.i-io~Y 



10 



20 



30 



itO 



FIG. II. /3-Values of another tube with smaller diameter 

 and thicker walls (see table 4). 



(0.64-10"^) = 17.2-10^ dyn/'cm-. This fits quite 

 well with the maximal value of 14.10^ in figure 5. 

 Similar measurements have been carried out on a 

 thick-walled tube in which r = 1.02 cm, a = 0.18 

 cm, and vo = 880 cm/sec. The results are plotted 

 in figure 1 1 . The dotted straight line corresponds 

 to the formula 



/3 = I.82IO-' K 

 Thus we obtain in the same way as above 



1.82 10 



TT ■0117-0.18/(2 ■ 1.02 • I.I4-6.75- lo*) 



which leads to a value of 50-10^ dyn/cm- for ojtj, 

 which has at least the order of magnitude expected. 

 Exact quantitative agreement cannot be expected 

 from such experiments, firstly, because the amplitudes 

 of relative stretch are dififerent from point to point 

 on the tube during the experiment and have not been 

 measured, and secondly, because the physical prop- 

 erties of India rubber changes with time. In addi- 

 tion to this, we have not accounted for the viscosity 

 of the liquids, which is another source of damping. 

 We should therefore not be surprised that our theo- 

 retical calculations, especially for the thick-walled 

 tube of small diameter, furnish a higher value for 

 (jiTj than we obtain from direct measurements (fig. 5). 

 In the next paragraph we shall therefore discuss the 

 influence of the viscosity of the filling liquid. 



