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HANDBOOK OF PHYSIOLOGY ^^ CIRCULATION I 



measurements of pulse-wave velocities, the errors 

 might still he tolerable when dealing only with a 

 purely progressive unidirectional wave. In the pres- 

 ence of reflected regressive waves, on the other hand, 

 measurement of pulse-wave velocities with the above- 

 mentioned naive method becomes extremely prob- 

 lematical. Let us consider the extreme case of a stand- 

 ing wave in the strict physical sense, which is possible 

 only in the absence of damping. If we choose the 

 positions of the recording instruments in such a man- 

 ner that there is no node of vibration between them, 

 the recorded pressures have the same phase. In other 

 words, the corresponding points of the pressure 

 curves, which we suppose to be sinusoidal in this 

 case, correspond to the same time and the calculated 

 velocity would appear to be infinite. In actual cases 

 we obtain, of course, values which lie between the 

 true value and infinity. This difficulty was first 

 pointed out by Porje (15) and in an extensive paper 

 by M. G. Taylor (19). It will therefore be sufficient 

 to illustrate what we have said above with an ex- 

 ample. 



Upon a wave pie~^' sin (co< — ac) running in the 

 positive direction within a conduit, a second wave 

 p^e'*'^' sin {ict + az), running in the negative direction 

 and due to a reflection of some kind, may be super- 

 imposed. We have therefore 



p2 = /-If"*-' sin (oj/ - az) + p^e^^'sin (u< + az) ^5-4) 



If we substitute for this sum a pure sin-function with a 

 different constant a, v' = w/a' is then the "apparent" 

 velocity of propagation which we wish to find. We 

 write therefore 



p2 = A sin iwt — a'z) 



If we expand this expression and compare the coef- 

 ficients with those of 5.4, we obtain for a and .-1 the 

 equations 



tan (a'z) = tan {az)- (e~-^'-p\/pz ~ i)/(e-^'-pi/pi + i) 

 A = [F-cosHaz) + G"--s\nHaz)y" 



(5-5) 



with F = pie-i^' + poe+f'; G = -pie-i>' -\- p2e+^'. 

 For^i = p2 and /3 = o, that is, for the extreme case 

 of a standing wave, tan (a'z) will vanish, that means 

 that a'z = o or X. In the first case a' becomes zero 

 (unless .J = o or I)' = =c) and .4 = 2pi cos (az)- The 

 pressure function in the tube is therefore given by 



p: = 2pi cos (az) sin oit 



This is the expression for a standing wave with the 

 nodes Zk given by the condition cos (aZh) = o. 



For a numerical example we put pi/p2 = 10; that 

 is to say, we suppose that at z = o the reflected com- 

 ponent amounts to 10 per cent. With the same values 

 of |(3 = 0.0013 2'id a = 0.0073 fo"" the fundamental 

 angular frequency co = 6.28 (which we used for the 

 example in fig. 15) we obtain, at .; = 100 cm for 

 tan (a'z), the value 0.688 from (5.5) and therefore 

 a'z = arc tan 0.688 = 0.600; v' = w/a' becomes 

 therefore 6.28/0.006 = 1046 cm/sec instead of v = 

 860 cm/sec, which we had supposed to be the true 

 velocity. To be precise, we obtain this velocity v' = 

 1 046 cm /sec if we make the recording at two points 

 near c = 100 cm. If we were to make the recording 

 near z = o, we would obtain, under the same condi- 

 tions, tan (a'z) = (911) tan (az) or approximately 

 a'/a = 0.816, which leads to v' = f 0.816 = 860/ 

 0.816 = 1054 cm sec, which is not much different 

 from the first case. 



We see from this example that even a relatively 

 small amount of reflected wave can falsify the de- 

 termination of pulse-wave velocity made in the 

 usual manner. This is all the more annoying because 

 the presence of strongly reflected waves seems now 

 to be well established. Compare the papers of Hamil- 

 ton & Dow (2, 4), Porje (15), and Wetterer (23). 

 Even refinement of the method with the help of 

 harmonic analysis shows merely that we obtain dif- 

 ferent apparent velocities for the different harmonics, 

 but which of these lie nearest to the true value we 

 can only guess. 



6. REFLECTION 



Until now we have described only single progres- 

 sive waves in conduits free from reflection, with the 

 exception of the remarks in the last few paragraphs. 

 A single progressi\'e wave can, at least theoretically, 

 only exist in an infinitely long homogeneous conduit, 

 or in a conduit of finite length provided with some 

 device to absorb the wave at its termination. As 

 these conditions do not exist, we have to describe 

 any stationary state by assuming two wave trains, 

 one of which runs in the positi\e and the other in the 

 negative c-direction. We must therefore replace ex- 

 pression 3.10 for pressure by 



where 



p = p't.i^i-y! -I- p'VJ"'-h'-- 



y = +!« 



(6.1) 



