IV -5 



We may see tiiat this corresponds to a single scattering approximation 

 by following a different procedure yielding the same result. If we introduce the 

 harmonic time dependence of p (IV- 9) into the wave equation (IV- 7), we can 

 make the approximation: 



V^p + k^p = - (2 eu + e^ ^^) k^p~ -2 e uk^p (IV- 13) 



This approximation involves only the assumption of weak inhomogeneities, i.e. , 

 e < < 1. We may integrate (IV-13) to give an integral equation for the pressure 

 distribution: 



2-^3 f ikr 



p(x) = p(o)(x) - -^^ I d|u(§)p(£)-^ . (IV-14) 



This solution is obtained by the following reasoning. When l-i = 0, the desired 

 homogeneous solution to (IV-13) is the original incoming wave p(°'. The inhomo- 

 geneous (or scattered) portion of the solution is obtained by regarding the right 

 side of (IV-13) as a source function emitting spherical waves of the appropriate 

 strength. The distinction between (IV- 12) and (IV-14) is that (IV-14) contains the 

 unknown function p under the scattering integral rather than the known incident 

 field p(o). As usual, the solution for (IV-14) may be obtained by successive 

 iteration starting with an initial guess. If we use p^°) as the initial approximation, 

 one iteration of (IV-14) will then yield (IV-12) as the single scattered pressure. 

 It is clear that the single scattering approximation will be valid only as long as 

 the twice- scattered power is small compared to the once- scattered power. In fact, 

 one would expect the single scattering approximation to hold only when the single 

 scattered power is small compared to the incident wave. For the actual conditions 

 encountered, we shall show later that the single scattering approximation is adequate 

 as long as the radius of the scattering region (R) satisfies: 



where the radius of the scattering region is to be measured in kilometers and 

 f denotes the frequency of the sound in kilocycles. In most practical cases, we 

 do not exceed this region. For sufficiently high frequencies, e.g., frequencies 

 above the 10 kilocycle range, we may be interested in propagation through regions 

 of inhomogeneities greater than those permitted by (IV- 15). For example, accord- 

 ing to the above, the single scattering approximation for sonar at a frequency of 

 40 kc is certainly invalid at ranges in excess of 7 km. We shall therefore want to 

 develop an approximation scheme which is more uniformly valid, in the sense that 



artliur B.HittbJnt. 



S-7001-0307 



