INFINITE PARABOLIC HORN 87 



element of volume must be equal to the time rate of growth of mass. 



dp' ^ d{Sp'u) 



-^ S^x = - Ax ' 5.2i 



dt dx 



or 



S^ + = 5.25 



w dx 



From equations 1.19 and 1.6 



- P0 = c^p' 5.26 



From equation 1.11 - 



u = — 5.27 



dx 



Substituting equations 5.26 and 5.27 in 5.25 the result may be written as 



d<b d , d^(f) 



dx dx dx^ 



Equation 5.28 is the wave equation for the axial motion in a tube of 

 varying section. 



5.15. Infinite Parabolic Horn ^^ — The equation expressing the cross- 

 sectional area as a function of the distance along the axis is 



S = SoX 5.29 



The general horn equation for the parabolic horn is 



c^ d<b av 



^_^2_^0 5.30 



X dx dx^ 



The velocity potential, pressure and volume current are, 



<i> = A[Mkx) - jY,{kx)y'^' 5.31 



p = - jo^pA[J,{kx) - jY,{kx)]e^''' 5.32 



U = ASk[ - Jo'ikx) +jYo'{kx)]e'"' 5.33 



The real and imaginary components of the acoustic impedance at the 



21 Olson and Wolff, Jour. Acous. Soc. Amer., Vol. 1, No. 3, p. 410, 1930. 



