SCATTERING 



481 



has some variable value c + Ac, while in the sur- 

 rounding water the sound velocity has a constant 

 value c. Suppose also that a plane sound wave, of in- 

 tensity h, and wavelength \, is progressing through 

 the medium in the x direction. The intensity I, of the 

 sound scattered from S may be different in different 

 directions, but at long ranges will fall off as the in- 

 verse square of the radial distance r from the center 

 of the region S. Since /, must be directly proportional 

 to h, 



I -^I 



(1) 



where A; is a constant. A more detailed discussion of 

 this equation is given in Section 19.1 of this volume, 

 describing in general the reflection, or scattering, of 

 sound from objects or scattering regions in the sea. 

 The target strength T as usually defined is simply 

 10 log k. 



The quantity k, which depends on the direction of 

 the scattered sound under consideration, must be re- 

 lated to the values of Ac, the sound velocity fluctua- 

 tion, at different points in the region S. Only the 

 energy scattered directly backward need be con- 

 sidered here, since this corresponds to the situation 

 of practical interest. It may also be assumed that 

 the scattering is sufficiently small that the sound 

 level at all points in S is practically equal to its value 

 in the incident sound wave in the absence of scat- 

 tering. This assumption tends to overestimate k; if 

 the scattering is large the sound level will decrease 

 as the wave penetrates the region S, because energy 

 is lost by scattering in the portion of the region S 

 already passed through. 



Since the scattering is produced by the relative 

 change in sound velocity, it is reasonable to assume 

 (and, in fact, it can be shown) that the pressure dp, 

 of the sound scattered from each volume element 

 dxdydz in S is proportional to the value of Ac/c for 

 each element. In adding up all the sound from differ- 

 ent elements, the differences in phase must be con- 

 sidered. Since sound must travel to the scattering 

 element and then back along the x axis, the difference 

 in phase between two elements separated by a dis- 

 tance X along the x axis will be ^ttx/\. Thus to find 

 the pressure of the scattered sound, Ac/c must be 

 multiplied by cos (47ra;/X -|- 27r/<), where / is the 

 frequency of the sound, and integrated over the en- 

 tire scattering region S. The scattered sound in- 

 tensity is then proportional to the square of this 

 integral. In this way it may be shown that the 



quantity k in equation (I) is given by the formula 

 A; = [^ jJJ- cos (47r - -|- 27r/<j dxdyd-J^ ' ■ (2) 

 By writing 



/ 4-7rx \ X 



( f- 2irft j = cos 4ir — COS 27r/i 



cos 



— sin 47r — sin 2Tft , (3) 



A 



the integral in equation (2) becomes the sum of two 

 integrals. Now square this sum, and average over the 

 time t, using the relations 



(4) 



cos^ 27r/« = sin2 2Tft = | 

 and cos 2vft sin 2irft = , 



where the bars denote an average over the time t 

 Then the quantity k, which measures the scattered 

 sound intensity, becomes 





(5) 



As pointed out above, the target strength of the 

 scattering region is 10 log k. 



When the volume of the scattering region is small 

 compared with the wavelength, the trigonometric 

 functions in equation (5) are constant; since the 

 sum of their squares is unity, 



^ = ^(///7^^''^'^V 



(6) 



When Ac/c is constant throughout the region, this 

 equation reduces to 



where V is the volume of the region. Equation (6) is 

 the so-called Rayleigh scattering law, which predicts 

 only a small amount of scattered sound. On the other 

 hand, when c is constant over a region large compared 

 with the wavelength, k is again sma-ll; as a result of 

 the oscillation of the sine and cosine factors in equa- 

 tion (5) each integral adds up to only a small value. 



29.3.1 Effect of Temperature 

 Microstructure 



Equation (5) may be used to compute the sound 

 scattered by a mass of water in which the tempera- 



