PULSE WAVES IN VISCO-ELASTIC TUBINGS 



133 



APPENDIX I 



THE CONCEPT OF MECHANICAL IMPEDANCE 



The concept of impedance was first used by electrical 

 engineers in the treatment of problems involving alternat- 

 ing current. Impedance denotes the ratio of tension to 

 current in alternating circuits. Because an alternating cur- 

 rent or tension is defined by amplitude and phase, imped- 

 ance can be most suitably represented by a complex quan- 

 tity. Within the last few decades, due to the emergence of 

 broadcasting and sound-recording technics, acoustics has 

 become a new field of engineering and analogies with the 

 theory of electrical currents and lines have been established 

 (18). Also, because the physics of pulse-wave transmission 

 is essentially homologous with the acoustics of very low 

 frequencies, the concept of impedance can be explained 

 using a simple example, similar to the well-known Helm- 

 holtz Resonator in acoustics. 



Let us consider a rubber balloon filled with an incom- 

 pressible liquid and tied to a rigid neck as represented by 

 figure I A. Some excess pressure p may be applied to the 

 neck. The volume of the balloon shall be V. If a column of 

 X cm of liquid from the neck with the cross section () is 

 pressed into the balloon, by the excess pressure^, its volume 

 will be increased by the amount v = Qx. A restoring force 

 of —pCi will thus be created which balances the pressure p. 

 We have thus 



Q., 



(i) 



where k = pV/v denotes the volume elasticity. The amount 

 of V is assumed to be small with respect to V. The restoring 

 force is then 



-PQ.= - "^^ 



(2) 



If we change now from a constant excess pressure p to an 

 alternating pressure/)' = poe'"' we have to consider a second 

 force needed to accelerate a certain mass m of liquid. This 

 mass corresponds approximately to the mass of liquid 

 m = QJ contained in the neck, where p stands for the 

 density of the liquid, and / is the height of the liquid column 

 in the neck. This inertial force is then Q^-l -p-d^x/dt". Be- 

 sides this we need another force Rdx/dt to overcome the 

 friction in the neck, where dx/dt is the mean velocity in the 

 neck. The dynamic equilibrium can then be expressed by 

 the equation 



</'.v dx 



at' at 



0- 



P'Q. 



(3) 



A solution of this equation will be .v = a: oe' '"'+'''. With 

 this we obtain from equation 3 the complex equation 



( -«2(3-/ ^]^R -Fk^J 



Q.-1 +1<^R +Ky\ei'<' = p^Q_ (4) 



Since we are interested in the ratio of pressure to flow we 

 write ;' = Q^-dx/dt, and with dx/dt = JLOxae''''e>'' = 

 J •;V"'«''' we obtain from equation 4 





or for the desired rate 



<■" = Po 



(5) 



ir = '^ = 



P« 



■e-''' = R+jlci-lp-o^-'-^] (6) 



This equation looks much like the analogous equation 



W, = R-f;L.L-4;) (7) 



for the electrical circuit represented by figure 2A, where W 

 is the impedance of that circuit and ; is the current flowing 

 through the circuit due to the electromotive force « of a 

 certain external power supply. R is the ohmic resistance, 

 L the self-induction, and C the capacity of the circuit. Any 



L 



C 



-mmiw- R 



u 



FIG. I.'K 



FIG. 2A 



