and 



4J 2 = B a j (k 2a r) , 



where the numerical subscripts have been eliminated from the coefficients for convenience. 

 The scattering cross section of the shell is 



a = $ s /li , 



where $ s is the average scattered power and I, the incident intensity. 



(11-88) 



— c 



$s : l-/.Pw s *U rws d 



where v is the area of an element of a sphere [49]. 



"V = nv , 



d-v = r 2 dx , 

 where n is the normal to the area and x is the solid angle. Also, 



J_ 

 2 



I = _ p * N 



so that it is necessary to determine u rws and u W| in terms of P Ws and P W| , respectively [49]. 

 In the fundamental mode u = - VQ, so that utilizing equation 11-31 for water, 



p w - Po w ar = ° • 



(II-89) 



(II-90) 



(11-91) 

 (II-92) 



(II-93) 



(II-94) 



Introducing the harmonic time dependence yields: 



Q = - 1 — P 



so that 



and 



i 



9P„ 



Urws cop 0w 9 r 



u = -— L ^p^ 



o) Pow az 



Thus, from equations 11-76 and 11-97, 



I - A2 



2p 0w c w • 



From equation 11-81, 



p w s = B w h (k 2w r), 



where 



h o(k 2w r) 



e ik 2w r 



ik 9 r 



(II-95) 



(II-96) 



(II-97) 



(II-98) 



(II-99) 



(11-100) 



15 



