io8 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



I. FUNDAMENTAL NOTIONS AND EQUATIONS FOR 

 THE PULSE WAVE 



In order to obtain a clear and illustrative picture 

 of the pulse-wave phenomenon, we will start with a 

 simplified and approximate mathematical descrip- 

 tion, which we shall improve and complete stepwise 

 as we penetrate deeper into the matter. 



For this purpose let us first consider wave propa- 

 gation in an infinitely long tube, the walls of which are 

 completely elastic; that is to say, the wall material 

 does not show any elastic hysteresis or internal fric- 

 tion, and its stress-strain relationship obeys the law 

 of Hooke. In addition, let us consider the whole 

 problem as essentially a linear one, and suppose the 

 pressure and velocity of the liquid to be constant 

 over the entire cross section of the tube. 



If we produce, in a straight elastic tube of infinite 

 length, a sinusoidal pressure variation at a given point 

 .; = o, a pressure wave will start from this point in 

 both directions. Let the pressure at this point be 



piz = o, () = P(i sin oit 



(I. I) 



where u denotes the angular frequency 2t', v being 

 the frequency in cycles per second. At any other 

 point z on the tube we observe then, assuming damp- 

 ing to be negligible, a pressure variation of the form 



P {z, i) 



PfiS\n 01 [ t — - j 



(1.2) 



In order to understand the physical meaning of that 

 equation, let us assume some arbitrary value for/; and 

 then follow that value oi p as it progresses along the 

 tube; that is to say, we shall look for the mathematical 

 expression indicating that p is constant. Obviously 

 p remains constant if / — z/v remains constant. If 



t increases, as from / to /', we must proceed along 

 the tube for a distance z' = vl' in order to find the 

 same pressure again; v signifying the speed of propa- 

 gation. 



As a consequence of pressure variations along the 

 tube, the particles of the liquid will likewise undergo 

 some displacements from their equilibrium positions, 

 which will also vary according to a sinusoidal func- 

 tion 



iiz. I) = fo sin 



<'-0 



+ 4' 



(1.3) 



where i/- accounts for a phase shift which might occur 

 between pressure and displacement. The particle 

 velocity u = d^/dt is therefore described by the 

 formula 



f W COS 



w( / — - I + vl- = «o cos O) I / — - j 



+ 4' 



(■.4) 



As a consequence of friction, due to the viscosity 

 of the liquid, the amplitude will generally decrease 

 with increasing distance from the source. To account 

 for this damping of the wave, let us first consider the 

 simplest case, assuming an exponential decrease of 

 the amplitude 



p{z,t) = P„e- 



sin u I / 1 



(..5) 



where /3 denotes the damping constant. Similar for- 

 mulas can be used for f and u as well. 



It is the purpose of the pulse-wave theory to de- 

 termine the ciuantities v, fi, and xp from the geometric 

 and physical properties of the tube and its filling. 



In order to derive the fundametital equations, let 

 us consider a tube of radius r and cross-sectional area 

 Q, = Trr'-. When the wave travels along the tube, the 

 cross-sectional radius r varies with the coordinate z 



SYMBOLS USED IN THE TEXT 



a = wall thickness 



1 = flow (or current in electrical ana- 

 Iog)_ 



i = V-. 



k = ajr ratio of wall thickness to tube 

 radius 



/ = length 



m, n = integral numbers 



p = pressure 



r = tube radius 



t = time 



a = particle velocity (alternating elec- 

 trical tension in electrical analog) 



V = phase velocity 



V = group velocity 

 X, y, z = coordinates 

 C = t)a/(2 rp) 



C = capacity 



E = Young's modulus of elasticity 



G — conductivity 



L = self-induction 



/ = complex amplitude of flow witli 



respective indices 

 P = complex amplitude of pressure 



with respective indices 

 Q^ = cross section of tube 

 /?, if], R-i = resistance constants 

 R = reflection coefficient 

 R = ohmic resistance 



T 



V -■ 

 Z 



time of period 



volume 



surge impedance 



a = tjijv or a 



V 

 (3 = damping constant 

 7 = -\- ja complex propagation 



constant 

 ifxp = phase angles 

 ■q = viscosity of tube wall 

 r}i = viscosity of liquid 

 x = frequency 



f = displacement in Z-direction 

 p = density of fluid 

 a = Poisson"s ratio 

 u = angular frequency = iir-y 



