EXTRANEOUS FREQUENCIES 



161 



Theory of Propagation of Plane Waves of Finite Amplitude 



The derivation of the exact differential equation for sound wave 

 propagation in air involves the continuity equation, Newton's force 

 equation and the equation expressing the relation between pressure 

 and specific volume in a gas. Since there may be some question as to 

 the accurate definition of the density and force in the equation of 

 motion a somewhat detailed discussion of this subject will be given. 



Following Rayleigh,^ let y and y + {dy/dx)dx be the actual distances 

 at time t from the plane ^jc = to neighboring layers of air whose un- 

 disturbed positions are defined by x and x + dx, respectively, Fig. lb. 



Fig. lb. 



The displacement corresponding to y is thus ^ — y — x and the 

 equation of continuity of the fluid is 



p = po(dy/dx)-' = po(l + d^/dx)-\ 



(1) 



where p and po are the densities of the fluid in the disturbed and un- 

 disturbed states, respectively. If the effect of viscosity is neglected 

 the exact equation of motion of the element of mass p{dyldx) -dx is 



or 



d'^y dy J d^y , dp dy , 



Po{d^^/dt^) = - dp/dx, 



(2) 



p is the pressure at the point y (Fig. lb) which moves with the air 

 particle, not the pressure at a fixed point. Except for very large dis- 

 placements these pressures are nearly the same. From equations (1) 

 and (2) 



d^Udt^ = (dp/dp) -(1 + d^dxy^-id^^jdx''). (3) 



^ Lord Rayleigh, "Theory of Sound," 2nd Ed., Vol. II, p. 31. 



