162 BELL SYSTEM TECHNICAL JOURNAL 



By virtue of equation (1), equation (3) is linear in ^ only if dpjdp 

 = Kp-"^ or dpjdv = — K, where v = 1 /p = specific volume and X" is a 

 constant. This condition is not satisfied during any ordinary varia- 

 tions of state of a gas, but is approximately satisfied when the variations 

 are very small. For isothermal changes we have pv = poVo and for 

 adiabatic changes : 



pjpo = {v.Ivy = (p/po)^ (4) 



where 7 is the ratio of the specific heats and pa is the undisturbed 

 atmospheric pressure. In either case, for very small variations, the 

 pv curve is practically identical with the tangent to the curve, hence 

 dpjdv is practically constant (Fig. la). 



From equations (1), (3), (4) we obtain the exact equation of adiabatic 

 plane wave motion in a non-viscous fluid : 



d2^/d/2 = c\\ + d^/dx)-y-'id^^/dx^), (5) 



where c^ = ypolpo- This equation is given by Rayleigh.^- ^ Rocard * 

 was first to call attention to the generation of harmonics in the air 

 within an exponential horn. His theoretical solution is based on a 

 plane wave equation in which the term d^^/dt^ was replaced by 



a/2 '^ dt dx\dt / ^^ 



In support of this substitution Rocard cites Riemann's ^ treatment of 

 the problem. However, Riemann's analysis is based on the Eulerian 

 form of the hydrodynamical equations whereas equation (5) is derived 

 from the Lagrangian equations. (For a comparison of these systems 

 of equations see Lamb.^) In the Lagrangian notation d^/dt and 

 d^^/dt^ are the exact values of the velocity and acceleration, respectively, 

 of the particle whose displacement from its equilibrium position {x) is |. 

 It is to be noted that in equation (2) the term po, or undisturbed 

 density, does not represent an approximation. 



A rigorous solution of (5) for the displacement ^ as an explicit 

 function of x and / has not been obtained. As a first approximation to 

 equation (5) we take 



dt' dx'' ^^ ^ ^ dx dx" ^^ 



' Lamb, "Uynaniical Theory of Sound," 2nd Ed., p. 182. 



* Y. Rocard, "Sur la propagation des ondes sonores d'amplitude finie," Comptes 

 rendus, 196, 161 (1933). 



* Riemann, " Ueberdie Fortpflanzungebener Luftwellen von endlicher Schwingung- 

 sweitte," GoUingen Abhandlungen, No. 8, 1860. 



"Lamb, "Hydrodynamics," 6th Ed., Chapter 1. 



