REGULAR COMBINATION OF ACOUSTIC ELEMENTS 261 



Substituting the value of A and B in (5) and (7), we have 



piioiS sinh ax 



V = Vi cosh ax — 



Poya 



= pi cosh ax — Vi ■" „ sinh ax. 



(9) 



{Poya)l(icci) is, by analogy with the electric line, the characteristic 

 impedance ■* per square centimeter of the tube. It is the ratio of 

 pi/^i for an infinitely long tube. For since cosh ax = ^(e"^ + e~"=^) 

 while sinh a = K^""^ ~ ^~"^), then when x approaches infinity, and 

 dissipation exists in the tube, cosh ax approaches sinh ax, and both ap- 

 proach infinity. Hence the ratio of Pi/Vi equals Poja/iooS. The 

 propagation constant a has the physical significance that e~°^ equals 

 the ratio of V to Vi or p to pi, when we are dealing with an infinitely 

 long tube, as can be seen by substituting pi/ Vi = Poja/io^S in (9) and 

 solving for the above ratios. The real part of a, i.e. a, determines the 

 rate at which the linear or volume velocity, or pressure, decreases with 

 distance, while the imaginary part b determines the phase of pressure 

 or velocity with respect to the initial values, and hence is known as the 

 phase constant and gives the phase rotation per unit length of pipe. 

 Now since the velocity of propagation C is 



we have by equation (4) 



^ ~ b' 



C = C 



'^wp \ 



IR 7 



2 S ^ leap 



\ 



The attenuation constant and the velocity reduce to the familiar 

 Helmholtz formulae, for circular sections.^ 

 We write (9) as 



V = Vj_ cosh ax — ^-^ sinh ax, 



viz . ^'°^ 



p = p\ cosh ax ^sinh ax, 



where Z^ represents the specific characteristic impedance P^yalio}. 



^ The analogy between pressure and electromotive force, volume velocity and cur- 

 rent, and impedance to ratio of pressure and volume velocity was first pointed out by 

 Webster*. Another system in which force and e.m.f., and linear velocity and current 

 are related, is very convenient when we are dealing with combinations of mechanical 

 elements such as masses and elasticities and no account has to be taken of the area. 

 In the first system, the total impedance is Zi, (per sq. cm.) divided by S whereas in 

 the second system it is Zi,S. We follow the first system expressing, however, the 

 impedance in terms oi the impedance per square centimeter, which is the same on 

 either systems of units. 



* See Lamb, "Dynamical Theory of Sound," p. 193, or Rayleigh, "Theory of 

 Sound," Vol. II, p. 319. 



