II- 11 



where L is the range (the distance from the scatterer at which we observe the 



field), and t is the scatterer' s volume (proportional to R ). The constant of 



proportionality depends a bit on the shape of the scatterer but is generally of the 



order of magnitude of unity. One could almost arrive at (II- 4) by the following 



very simple dimensional argument. The linearity of (II- 4) in 1 v 1 is the result 



of using 1 V 1 as the parameter of smallness and ignoring all terms of second or 



iiigher order. The scattered sound is in the nature of a directional spherical 



/ 1 



/ wave, and the field should, therefore, decay as -■ It is not unreasonable that 



the total scattering should be proportional to the volume of the scatterer. The 

 only remaining parameter which can enter the expression is the wave number k 

 characterizing the incident wave, and if the resulting expression (II-4) is to be 

 dimensionless (as the ratio of two pressures must be) the wave number must 

 enter quadratically. 



On the basis of this result for a single scatterer, we may derive, in a 

 very heuristic fashion, the principal conclusions of Section IV-D for a large col- 

 lection of scattering patches. In the following, we shall present these intuitive 

 arguments, aimed at determining the order of magnitude of the ratio of scattered 

 to incident power. It should be clearly understood that these arguments are to be 

 regarded only as an after -die -fact explanation of results obtained by somewhat 

 more rigorous methods in the body of this report . 



Consider first the case of low frequency sound, i.e. , sound whose wave- 

 length is much greater than the radius of the typical scatterer (kR < < 1). In this 

 case, the scattered pressure from a single inhomogeneity will be omnidirectional, 

 and the ratio of the scattered pressure to the incident pressure is given by (II- 4) 

 in every direction. Suppose we were to insonify by a plane wave a half space of 

 an inhomogeneous medium packed densely with inhomogeneities all of radius R, 

 as shown in Figure II-3. We wish to determine the ratio of the scattered pressure 

 to the incident pressure a distance L inside the inhomogeneous medium. To this 

 end we divide the inhomogeneous medium into slabs, each slab one scatterer thick 

 (i.e. , a slab thickness of 2R). Consider now the scattered pressure due to the 

 scattering from a single slab. Clearly this pressure is independent of the distance 

 behind the slab at which it is observed. It can, therefore, depend only on the 

 properties of the individual scatterer ( | v 1 and R) and on the wave number of the 

 incident sound k. Since each of the inhomogeneities of the slab has a scattering 

 strength proportional to | v | the whole slab should have a scattering strength which 

 is also proportional to | v | . Similarly, since the ratio of scattered to incident 

 pressure for the single scatterer is proportional to the square of the frequency of 

 the incoming sound, one would expect the same behavior for the slab. As a result, 

 one is led at once by a dimensional argument to the expression 



P- 



k^R^ (11-5) 



Arthur B.littlcJnt. 



S-7001-0307 



