I20 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



10 



IS 



20 



FIG. 13. Measured phase and group velocity as functions of 

 frequency. At point (i) 1/ dv/dv = 4.4- 14 = 62. v = v -{- p 

 dv/di/ = 460 + 62 = 522 against a measured value of v = 

 564. At point (2) I. dv/dv = 14.4-31 = 45. i' = 530 + 45 = 

 575 against a measured value of 586 cm/sec. 



5. DISPERSION, PHASE, AND GROUP VELOCITY, .A.ND 

 THE IMPORTANCE OF HARMONIC ANALYSIS 



We speak of dispersion when the velocity of 

 propagation depends on frequency. This dispersion 

 produces some phenomena which we shall now dis- 

 cuss. If we produce a wave train of finite length or 

 finite duration, as sketched in figure 12, the speed of 

 propagation of the head or tail of the wave train 

 will difi"er from that of the hills and troughs of the 

 wave. The former is called group velocity and the 

 latter phase velocity. The picture is somewhat like 

 that of a caterpillar crawling on a leaf. The two 

 velocities are connected by the equation 



V + V 



dv 



dp 



V = V — \ — 

 d\ 



(5.0 



velocity and X the wavelength. The graphs of figure 

 13 show how these two velocities depend on the 

 frequency. The data have been taken from Miiller's 

 paper (12, table 3). If we calculate v dvjdv from the 

 slope of the jj-curve, we obtain a difference v — v 

 which is smaller than the difference measured di- 

 rectly, but at least demonstrates the sense of the de- 

 viation. For the derivation of equation 5. i the reader is 

 referred to any physics textbook. 



We now turn to waves which are still assumed to 

 be periodic but no longer sinusoidal in shape. We 

 think, for example, of a pulse-pressure curve recorded 

 with a suitable manometer. It is always possible to 

 represent such a periodic function by an infinite 

 series of sin-functions and cos-function. 



f(t) = flo -H zZ '''■■cos (2T!-nl/T) + 2I ^1 sin iivnt/T) 

 1 1 



or 



(5-2) 



y(x) = oo -t- ^a„co& (n-x) + 2lh,riin (nx) 

 1 1 



The coefficients ao , a,, and 6„ are called the Fourier 

 coefficients of the given function f{t) or y{x). The 

 integer numbers n may take all the values from i to 

 infinity. The coefficient ao is simply the mean value 

 of the function to be considered, T being the funda- 

 mental period. Instead of using sin-functions and 

 cos-functions, we can also represent f{t) or y{x) by 

 means of sin-functions alone. Instead of equation 

 5.2 we have, then. 



/(') 



-I- Y.<:.-sm[{2imt/T) + ,j,„] 



(5-3) 



where 



Cn = {a,? + b„ 



and tan \j/n 



a„/b„ 



well known in physics, where v denotes the group 



Because tan ^ = tan (ip ± nm), we use the additional 

 relationship sin ^,1 = a,Jc„ to determine in which 

 quadrant the angle xp is to be found. The determina- 

 tion of the coefficients a„ and b„ can be made mathe- 

 matically, with numerical approximation methods or, 

 best of all, with a harmonic analyzer). 



We shall illustrate this with an example. Figure 14 



* A very practical type of analyzer, which can be made in 

 every laboratory with cheap radio parts, has been described 

 by Rymer & Butler (17) and in an enlarged and improved 

 form by Taylor (20). The author has used such an instrument 

 with success, and it furnishes the larger coefficients of the first 

 three harmonics with no larger error than i per cent, if the 

 period is divided into 30 intervals. The whole analysis takes 

 less than a half hour, when the 30 corresponding ordinates have 

 been evaluated before. 



