REGULAR COMBINATION OF ACOUSTIC ELEMENTS 259 



only a determination of the propagation constant, that is, a determina- 

 tion of the attenuation and phase change per unit length, or as more 

 often stated, the attenuation and velocity characteristics. If we solve 

 the differential equations in the manner first employed by Heaviside in 

 the solution of the equation of the electric line, we obtain one more 

 parameter, namely, the characteristic impedance of the tube. 



The differential equation, given by Rayleigh,^ for the propagation of 

 plane waves of sound in a tube of uniform cross-section is 



where ^ denotes the displacement of the fluid at a distance x from one 

 end of the tube, 



/z = the coefficient of viscosity of the medium, 

 p = the density of the medium, 

 R — perimeter and 5' = cross-sectional area of pipe, 

 w = frequency of vibration times 2ir, 



C — \/— ^ = velocity of sound in medium, 



\ P 



7 = ratio of specific heats of medium. 

 This equation is valid for tube diameters and frequencies such that 



4 



PJ.S 

 2fiR 



and hence can be used for all frequencies of interest in connection with 

 acoustic filters. 



Kirchoff ^ extended the theory to take account of the losses due to 

 heat conduction in the medium. His results indicate that in order to 

 take account of this effect, the square root of the coefficient of viscosity 

 should be replaced by a quantity 7', given by 



7' = V/^ 4- ( V7 - -p jVu, 



V7 



where i; is the coefficient of heat conductivity of the medium. By the 



kinetic theory of gases v has the value 5/2 ju. 



The most useful solution for our present purpose is obtained by 



writing 



^ — gu^t^j^ (,Qg|^ ax -\- B sinh ax), (2) 



' Rayleigh, "Theory of Sound," Vol. II, p. 325. 



