CHAPTER 7 



Propagation of pulse waves in 

 visco-elastic tubings 



VICTOR HARDUNG 



Physiologisches Institul der Unhrrsitat Freiburg, Freiburg, Switzerland 



CHAPTER CONTENTS 



Fundamental Notions and Equations for the Pulse Wave 



Dynamic Elasticity and Viscosity 



Influence of Internal Wall Friction on Damping and Speed of 

 Propagation 



Viscosity of Filling Liquid and Its Influence on Damping and 

 Speed of Propagation 



Dispersion, Phase, and Group Velocity, and the Importance 

 of Harmonic Analysis 



Reflection 



Hydrodynamic Considerations 



Electrical Analog of the Elastic Tube and its Limits of Applica- 

 tion 



APPENDIX I 



The Concept of Mechanical Impedance 



APPENDIX -2 



Derivation of the Complex Reflection Coefficient R 



EXACT UNDERSTANDING of the physical phenomena 

 which come into play in the arterial system as a 

 consequence of heart action has been developing 

 rather slowly. This is due in part to the very compli- 

 cated geometric and physical structure of that system. 

 Also, it may be due in part to the tendency of many 

 investigators to look upon these phenomena as a 

 single entity, without having a clear picture of the 

 different individual physical elements which are the 

 basis of the response as a whole. 



The author's personal research published in this chapter 

 has been supported by grants from the Swiss National Founda- 

 tion for Scientific Research and the Office for Creation of 

 Employment Possibilities. 



Such an individual element is the propagation of a 

 pressure wave or pulse wave in a tube with visco- 

 elastic walls filled with viscous fluid. This present 

 chapter deals mainly with this phenomenon. It does 

 in no way claim completeness. Indeed, at this time 

 completeness would not be possible, because even 

 in this narrow field many questions still remain 

 unanswered. 



The purpose of the present article is merely to 

 give to the investigator in hemodynamics a clear and, 

 insofar as possible, a simple and illustrative description 

 of all the physical phenomena which may be of 

 importance for pulse-wave propagation in the arterial 

 system. 



In order to see how these physical principles can 

 be applied to actual hemodynamics, the reader should 

 consult the very clear monograph. Blood Flow in 

 Arteries by D. A. McDonald (lo), which appeared 

 recently. 



We will not overburden our treatise with a detailed 

 description of the historical development. The in- 

 terested reader will find a good account in a paper 

 by P. Lambossy (7). 



The mathematical development in this chapter 

 does not claim to compete with the more detailed, 

 and in many ways more accurate, hydrodynamic 

 theories of Korteweg (6), Frank (3), Morgan & 

 Kiely (11), Lambossy (8), and Womersley (24). 

 We have merely tried to give a semi-empirical and 

 phenomenological treatment of the kind most u-sed 

 in engineering. Where needed, the treatment includes 

 results of experiments and the afore-mentioned 

 hydrodynamic theories. 



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