HANDBOOK OF PHYSIOLOGY ^^ CIRCULATION I 



or with Q, = Trr- to 



d-p E-a d-p 

 at- " 2rp 3^2 



(!.l8) 



This equation is known in physics as the wave equa- 

 tion. It describes the propagation of any periodic or 

 nonperiodic disturbance in the c-direction. Solution 

 of the equation in an analogous way for current 

 rather than pressure leads to the same equation 



d-i _ E-a d^i 

 a/2 ~ 2rp dz' 



(1. 19) 



for the current. If we take for p or ;' any arbitrarily 

 chosen function /(< — z/v) of the expression {t — z/v), 

 we see at once, by carrying out the differentiations, 

 that it represents a solution of 1.18 or 1.19 if 



• ' m" 



(1.20) 



which is the well-known "Moens-Korteweg equation." 

 A special solution of 1.18 is, of course, 



pU) = Po sin u(t =F z/v) 



where the minus sign corresponds to a wave running 

 in the positive, the plus sign to a wave running in 

 the negative .^-direction. 



In regard to the derivation of the wave equation, 

 some remarks must be made concerning equation 

 1. 1 5. This equation holds strictly only for static 

 conditions. Under dynamic conditions the variable 

 part of internal pressure must not only balance the 

 increasing elastic-restoring force but also the inertial 

 resistance of the mass of a part of the liquid and of 

 the wall, due to the acceleration of that mass M. If 

 this modification of equation 1.15 is made, one ob- 

 tains instead of equation r.20 the following: 



(E-a M V* 



V2rp 47rp J 



(1.21) 



The effective mass M cannot be calculated from the 

 present theory. Frank (3) found for the additional 

 term under the root coV/S. The introduction of 

 that term means that the speed of propagation would 

 drop to zero when a certain frequency limit is reached. 

 It can easily be calculated that this limiting frequency 

 is so high, and that the correction factor so small for 

 the cases to be considered, that it need not be ac- 

 counted for. 



For some purposes, the use of the modulus of 

 volume elasticity may be indicated. To obtain a 



relationship between p and V, we consider a section 

 of tube of length /. This has the volume V = wr-l 

 and the differential dl' becomes dV = 2-T-r-l-dr 

 for a variation of radius dr. For the modulus k = 

 {dp/dV) ■ V w'e obtain therefore 



K = (dp/dV)-V = y^ridp/dr) ('1.22) 



Using equation 1.14 we get 



K = ]>^r-dp/dT = E-a/iir) (1-23) 



The Moens-Korteweg equation can then be written 

 in the form 



(-c/p)"' 



(1.24) 



which is used by Frank (3). Like the Moens-Korteweg 

 equation, it only holds for thin-walled tubes. 



Equation i .24 is well known in physics as the 

 equation for propagation of a sound wave in a 

 liquid with the compressibility i 'k. In liquids, i/k 

 is very small and the velocity of propagation rather 

 high (on the order of 10^ cm/sec). On the other hand, 

 in a liquid-filled rubber tube, or a blood vessel, i//c 

 depends for the most part, not upon the character- 

 istics of the liquid, but upon the distensibility of the 

 tube wall, which is rather great, the pulse-wave 

 velocities being therefore comparatively small (a 

 few hundred cm/sec). 



In the preceding description of the pulse-wave 

 phenomenon we have considered only the geometric 

 features of the conduit, the density of the liquid, and 

 the static modulus of elasticity for the tube walls. 

 E.xperiments with pulse waves in rubber tubes and 

 researches in hemodynamics show that the amplitudes 

 of such waves diminish with increasing distance from 

 the source (pump or heart), at least if no reflections 

 occur. This damping of the wave is more pronounced 

 for high frequencies than for low ones. Wave propa- 

 gation is therefore connected with a loss of energy. 

 Such a loss of energy can only be due to friction, 

 either in the liquid or in the tube wall, at least when 

 the tube is freely suspended in air (see equation 1.5). 



However, before we try to complete our theoretical 

 picture of wave propagation by taking into account 

 the observed damping, we must learn something 

 about the elastic behavior of the tube walls. We shall 

 see in the following section how these elastic properties 

 depend upon the manner in which the load is applied 

 to the material; that is to say, on the way the stretch 

 occurs in time. We shall see then that internal fric- 

 tion is closely connected with elasticity. 



