state of a material element of fluid are adiabatic (in the absence of molecular transfer 

 effects), p and p are related by 



Dp Dp 



= C 2 = _ pC 2 V • V (16) 



Dt Dt 



where c is a parameter describing one of the physical properties of the material element 

 under consideration and is equal to the phase velocity that sound waves would have in 

 a uniform medium having the same physical properties as that element. Thus we have 

 two equations for v and p, the form of the equations showing that the only two 

 physical properties of the medium that are relevant are the local inertia per unit volume 

 p, and the local phase velocity of sound waves c. 



Our interest is in sound waves of small amplitude, and we therefore linearize in 

 the departures from the undisturbed state in which v and \jp are zero, obtaining 



(17) 



where p and c are now symbols denoting the local density and acoustic phase velocity 

 of the medium in the undisturbed state, and are independent of t according to the 

 assumption explained earlier. The equation for p alone is 



d*p ( Vp \ 



pC 2 V • J = 0, (18) 



dP \ p I 



while that for v is 



d*v 1 



V (pc2 V • v) = 0. (19) 



dt 2 P 



The velocity field v is not irrotational when p varies with position, but it is clearly 

 possible to define a quasi-potential such that 



1 

 v = - V <p, (20) 



p 



the equation for <p then being the same as that for p. If the length scale on which p 

 varies is large compared with the wave-length of the incident sound wave, the equation 

 for p may, for some purposes, be reduced approximately to 



d 2 p 



c 2 V 2 p = 0, (21) 



dt 2 



which is the ordinary wave-equation with variable phase velocity; however, the Fourier 

 components of the distribution of p that are most effective in scattering energy through 

 a large angle are just those with length-scale comparable with the wave-length, so that 

 we shall not make use of this approximation. 



If the variations of p and c in the medium are relatively small, we can write 



p = p + p\ , c = Co + ci (22) 



416 



