length traversed by the incident wave-front, that is, the logarithmic decrement of the 

 attenuation by scattering, is then 



7 = / cr(I) dl 



= - / $(k - d) dl (12) 



where the integration is over all directions of the unit vector I. 



The special case of sound of very high frequency is worth noting, in view of 

 its common use in under-water instruments and devices. If k is considerably larger 

 than the reciprocal of the length scale of the inhomogeneities of the medium, $(k — kI) 

 will be small for all directions of I except those close to the direction of k- Most of 

 the energy of the scattered wave is thus thrown forwards, at small values of the scatter- 

 ing angle 0. To the first order in the angle of the cone containing most of the scattered 

 energy, 



7 = — / *(m) dm (13) 



2k* J 



where m is a vector in the plane perpendicular to v. and the (two-dimensional) integra- 

 tion is over all values of m. If we choose the jq-axis to be parallel to k, this can be 

 written as 



it r+ x r+ x 



7 — / / <&(p,m2,mz) dmzdmz 

 1 



Q(yi + ti,y*,yi)Q(yi,y2,y*) d£i , ( 14 ^ 



showing the dependence of y, in this case, on a kind of integral scale of the quantity Q. 

 To proceed further we must know more about the right-hand side of the equation 

 that was adopted at the beginning of this discussion, and we therefore turn now to a 

 consideration of the equations that describe the propagation of sound waves. There 

 are two important cases, first a medium of variable physical properties which is sta- 

 tionary except for the disturbance caused by the incident wave, and second a medium 

 of uniform physical properties in turbulent motion. These two cases often occur 

 together in practice, but separate consideration is desirable so that we can see which 

 is likely to be the more important effect. 



Stationary medium of variable properties 



Provided we can neglect the effects of molecular viscosity, which usually plays 

 the minor role of a small damping agent in these problems, the exact equations describ- 

 ing any kind of fluid motion are 



(15) 



where v, p and p are the local velocity, density and pressure. Since the changes in the 



415 



