PULSE WAVES IN VISCO-ELASTIC TUBINGS 



121 



P 



3 



T 



FIG. 14 



represents a highly schematic pulse curve. For the 

 systolic hump we have u.sed a sin-shaped half wave 

 whose "base" corresponds to a third of the whole 

 period. For such a function, the Fourier coefficients 

 can easily be calculated and we obtain, if we take 

 the height of the "systolic hump"" as unity, the values 

 given in table 3. 



Figure 15 shows the three first harmonics drawn as 

 functions of time (thin lines) and their sum (heavy 

 line). The period is assumed to be i sec. We shall now 

 see how this sum-function representing a given pres- 

 sure function, recorded at, say, -t = o of a rubber 

 tube, will be distorted when we make the recording 

 at any other point .: on the tube-line. A distortion 

 will occur for the following reasons: a) The three 

 component waves will run along the line with differ- 

 ent speeds, changing the phasic relations between the 

 components. This is much like the distortion observed 

 many years ago on telephone lines, when speech was 

 blurred at long distances, b) The amplitudes of the 

 components are damped in a difTerent way, because 

 the higher frequencies undergo a higher damping 

 than the low ones. The lower curve of figure 1 5 show 

 the result of distortion for the two distances .; = 1 00 

 cm and x = 200 cm. They are obtained from the 

 original curve above by reducing the amplitudes of 

 the component waves by multiplication with the 

 factor e'^" and by shifting the component waves in 

 the positive time axis by the amount ziv. The data 



FIG. 15. Distortion of a pulse-wave composed of three har- 

 monics at different points from the origin. The data used were 

 ist harmonic: v — 860 cm/sec (3 = 0.0013; ^rid harmonic: 

 V = 8gi cm/sec /3 = 0.0030; 3rd harmonic: v = 897 cm/sec 

 /3 = 0.0052. 



used for the damping constants and the velocities of 

 the components are given in the legend to figure 15 

 and are those which were obtained with the thick- 

 walled tube (see above). As can be seen from the 

 figure, distortion is already appreciable at z = 100 

 cm, and is of course enhanced at z = 200 cm. 



If we try in a naive way to determine the speed of 

 propagation from the time lag of corresponding 

 points, for example AA' AA" or BB' BB" , we obtain, 

 respectively, the values 923, 1030, 750, and 880 

 cm/sec, which are appreciably different from the 

 velocities v\ , v-y , and V3 of the component waves. 



For the given example, the main causes of dis- 

 tortion are the difTerent damping constants of the 

 components. This is also the case in blood vessels 

 where the dispersion (frequency dependence of Edyn) 

 is much like that of rubber. Higher dispersion would, 

 of course, enhance the distortion. 



Even though these distortions tend to invalidate 



