APPENDIX B 



SIMPLIFIED EXPRESSION FOR (B w /A) 



Simplification of (B w /A) as given in equation 111-16 is based on the ranges of physical properties and the 

 limits of the variables given in Chapter II. 



The first step in the simplification of equation 111-16 is the simplification of the spherical Bessel, 

 Neumann, and Hankel functions in the S, 's. In the k, a terms: 



sin(k, a) 

 JoO^a) = k 1a a (B- 1 ) 



and 



i 'fk *\ ■- cos < k ^ a ) sin(k 1a a) 



Jo (kl * a) k^ \^jtT ■ (B-2) 



Now, if 



sin(k 1a a) = sin[(1+i) xa] , (B-3) 



then 



sin(k 1a a) = sin(xa)cosh(xa) - i sinh(xa)cos(xa) (B-4) 



Similarly, 



cos(k 1a a) = cos(xa)cosh(xa) - i sin(xa)sinh(xa) . (B-5) 



An examination of the parameters involved shows that xa > 10, so that, 



cosh(xa) » sinh(xa) x —■ . ( B " 6 ) 



and 



sin(k 1a a) z. i cos(k 1a a) . (B . 7) 



Thus, 



io(k 1a a) = [ C ° S( ^ a) (B-8) 



and Ka 



Jo'(ki a a) = -j (k 1a a) [i + ^-y | . 



(B-9) 



One of the assumptions in the model is that the shell is small compared to the wavelength of the incident 

 compressional wave. This means that k 2w b, k 2f b, and k 2a a are small. The definition of "small" will now be 

 determined by examining the expansions for spherical functions of small argument: 



Jo(z) = (l --!+..), (B-10) 



j '(z) = -|-(l -f Q + ....), (B-11) 



j "( Z ) = -^(l-^ + ..-.). „ (B " 12) 



n.W- -4- (1 -| + ....> (B " 13) 



n '(z) =^(1 +-f - .... ), (B-14) 



n »(z)= -J, (1 + *-.-. .. ), (B . 15) 



41 



