We shall find it convenient to write the density, pressure and velocity in the 

 combined incident sound and turbulence field in the following form: 



p = po + Pt + Ps , V = Po + Pt + Ps, U = u + v, (36) 



where p and p are the uniform density and pressure in the completely undisturbed 

 medium, and p + pt , p + p t , u are exactly the values of the density, pressure and 

 velocity in the turbulent field in the absence of the incident sound wave. From what 

 has already been said, |»| <^ \u\ although \p s \ and \p s \ may be comparable with |p t | 

 and \p t \. Moreover, since the Mach numbers of the incident sound wave alone (M s ) 

 and of the turbulence alone (M t ) are both small, we can safely assume that 



I Pt+ Ps I « PO , I Pt + Ps I « Po • (37) 



We shall ignore again the effects of molecular conductivity, so that the motions are 

 adiabatic and 



/ Pt + ps Y / Pt V 

 (1 + ) -( i + -J 



\ Po / \ Po / 



= 1 1 + 



Po 



rs* 



yps V ( pt + p s \ 2 / p t 

 — + i7(7 - l) 



Po L\ Po / \ Po / J 



(38) 



These are the approximations that we can now use in the equations of motion. 



The equations for the quantities p, p, U relating to the combined field can be 

 written as 



dp d( P Ud 



— + = 0, (39) 



dt dxi 



d(pUi) dipUiUj) dp 



+ + — = 0, (40) 



dt dxj dxi 



without approximation, apart from the neglect of the effects of viscosity. A combined 

 form of these equations is 



6 2 p d 2 (pUiUj) 



— - V 2 p = 0. (41) 



dt 2 dxidxj 



This equation is satisfied when p, p, U are given by (36), and also when 



P = Po + Pt, P = pU + p t , U=u, (42) 



so that 



a 2 p s a 2 



V 2 p s [p s (ui + Vi)(y,j + Vj) + (p + p t ){2uiVj + ViVj)] = 0, (43) 



dt 2 dxidxj 



again without approximation. p s and p s are the increments in density and pressure in 

 the combined field due to the incident sound wave, and they are unlikely to be smaller 

 than the corresponding increments in the absence of the turbulence. Thus p s /po an( ^ 

 Ps/Po are at l east °f order M s . Examination of the magnitudes of the different terms 

 of this equation in the light of the above permissible approximations, together with 

 a little (uncontentious) guesswork about the magnitudes of the spatial derivatives of 

 u, then shows that to a first approximation the equation reduces to 



p s 1 d 2 (p s /p ) 



V 2 = 0. (44) 



Po c 2 dt 2 



420 



