and equation 11-64 can be written as 



WA » " -FsTfce + k 3< 2A * = ° ■ C 1 " 75 ) 



The plane wave impinging on the shell will be represented by 



P wi = Ae*a«* , (11-76) 



where P W| is the perturbation pressure of the wave in water, and A is the pressure amplitude. P W| can be 

 expanded in spherical waves as 



P w| = A^ (2 C + 1)i\(k 2w r) P c (cos 6) , (H' 7 ?) 



5=0 



where 2 is the mode number, ^(k^r) is a spherical Bessel function and P tt (cos 9) is a Legendre polynomial. 

 The incidence of the wave upon the shell gives rise to five additional waves, which are represented by the 

 solutions of equations II-62, II-63, II-65, II-66, and II-75. These equations can be solved by standard 

 separation of variables techniques. The solutions are: 



*i = ^ D ac j c (k 1a r) P c (cos6) , (||78) 



2=0 



OO 



^ =Z B aE j c (k 2a r)P e (cos9) , (II-79) 



P, = £ [B, g j g (k 2f r) + E, e n 2 (k 2f r)] P g (cos0) , (II-80) 



c=o 



OO 



Pw s =X B wft h c (k 2w r) P e (cos 9) , (|| . 81) 



c=o v ' 



OO 



A <p f = J [F. £ j £ (k 3f r) + G, c n c (k 3) r)] P' c (cos9) , (M-82) 



and 



e=i 



where B, D, E, F, and G represent the amplitudes of the waves, q ^k 2l r) is a spherical Neumann function, 

 and h s (k 2vv r) is a spherical Hankel function of the first kind. P Ws indicates the scattered compressional wave in 

 water, so that 



P = P + P (H-83) 



r w r wj ' r w s 



One of the assumptions of this model is that only the fundamental mode contributes to the scattering. Thus 

 only the 2=0 mode is considered. Examination of equation M-82 shows that for 2=0, 



A,, = . (H-84) 

 The remaining equations are now written for 2 = 0: 



Pw = Aj (k 2w r) + B w h (k 2vv r) , (II-85) 



Pf = B f j (k 2f r) + E f n (k 2f r) , (H_ 86 ) 



«Pi = DJ (k 1a r) , ( l|. 8 7) 



14 



