PULSE WAVES IN VISCO-ELASTIC TUBINGS 



125 



1.5 



to 



0,5 



FIG. 1 7. Calculated input impedance 

 with damping constants = 0.002 and 

 /3 = 0.0055. 



100 



ZOO 



■500 



500 



800 



small with respect to the current amplitude. At the 

 closed end, of course, the current amplitude vanishes. 

 The conditions are much like those in an organ pipe 

 which is blown at one end and closed at the other. 

 Standing waves can be established only if the length 

 is an integer multiple of X/4. 



A somewhat different picture will be obtained if 

 damping occurs. Two impedance curves calculated 

 from equation 6.25 with damping constants (3 = 

 0.002 and /3 = 0.0055 are shown in figure 17. The 

 oscillations of the input impedance with varying tube 

 length decrease as we go to greater lengths, and the 

 input impedance itself approaches the surge imped- 

 ance as the length becomes very great. Such input 

 impedance curves can easily be obtained experi- 

 mentally if one records pressure and flow simultane- 

 ously at a given place in the conduit, designated as 

 the origin. Figure 18 shows the result of such an ex- 

 periment, where the flow has been recorded with an 

 electromagnetic flowmeter (5). The frequency was 

 kept constant, while the length of the line was varied 

 by clamping the tube at different places. The dotted 

 line corresponds to the function (6.25) with /3 = 

 0.0038. The shape of the experimental curve fits 

 quite well with the calculated one. This is not very 

 surprising, because equation 6.25 holds for any uni- 

 directional wave propagation. The only qualification 

 we have to make is that the dotted curve has been 

 obtained using a constant /3. Because we know that li 

 depends on amplitude, the value = 0.0038, which 

 we had to use to fit the experiment, is therefore a 

 kind of mean effective value. 



With a damped line we can no longer obtain a 

 true standing wave because the sum of the ampli- 

 tudes of all partial waves running in the positive 



FIG. 18. Observed input impedance for the frequency = 

 2.02 (modulus and phase). Dotted line: calculated curve with 

 /3 = 0.0038. 



.^-direction is always greater than the sum of the 

 amplitudes of the reflected waves. At no point on the 

 line, therefore, can the amplitudes cancel each other, 

 and no nodes can be formed in the strict sense of the 

 word. If reflection at the end of the line is incom- 

 plete, the argument holds even more strongly. 



From equation 6. i it follows, if we consider ampli- 

 tudes onlv, that 



P, = P^'e-y + Po 



(627) 



