SECT. 4] 



SOUND SCATTERING BY MARINE ORGANISMS 



521 



staff of the University of California Division of War Research (NDRC, 1946a). 

 The echo-sounder is such an instrument ; its main lobe is directed vertically 

 downward. The directional radiating and receiving properties are considered 

 identical, that is, F{t, 6, </>) can be written F{t) b{d, (f)) and the integral equation 

 becomes : 



F{t) m{r, e, cf>)b^{d, 4>) 



m = 



dV. 



(23) 



f volume f ^ 



As noted earlier, observation with echo-sounders has shown that scatterers 

 are commonly horizontally stratified ; that is, w is a function of depth only. 

 Also, the transducer commonly has axial symmetry, b{d, (f)) = b{d). Thus (23) 

 becomes 



F{t) m{r cos e)b^e) 



m - 



volume f' 



r^ 



dV 



or 



I{t) = I f" r^" :^ m{r cos e)bHe) sin 6 dr dd d<f>. 

 'o Je=o J0=o r- 



Performing the ^-integration 



I{t) = 277 



Jo Jo 



" -^ m{r cos 9) b^d) sin d dr dd 



or substituting (22) 



^' c J.=oJo {t-rf 



-: (t — r) cos 6 



b^e) sin d dr dd. 



(24) 



(25) 



(26) 



(27) 



For a sinusoidal pulse beginning at t = and ending at t = to (i.e. a "pulse 

 length" to), we assume a form p—po sin ojr; 



^ po^sm^2n{fr-rlX) ^^^ 

 pc 



(28) 



where pQ is the instantaneous pressure in dynes/cm^ at range r= 1 m. Then 



'<'* = ^^Jo Jo JTW^ " 



- [t — r) cos 6 



62(61) sin d dr dd. (29) 



The usual choice of pulse length is such that to<^^ so that, for performing the 

 T-integration, we may take r = ctj'2. Then 



^ 477xl04 |o^^^ p Ic ^^^ \^^^ ^.^^ ^ ^^ 

 ct^ 2pc Jo \2 / 



Where m is regarded independent of position, 



^^^^ 477Xl04;P02 



^(0 = — — TT- rom 



or 



I{t) = 



ct^ 2pc 



77 X 104^90^ 



r bHd) 



Jo 



sin d dd 



-P 



^ Torn 62(^) sin d dd. 



(30) 



(3i; 



(32) 



