WD-(i-4 + -)-rO-f + - )• (B ' 16 » 



h.'(z)--f (i-^-.. ■) + "5( 1+ ^---- )• < B ' 17 > 



VW-"T OH? ♦-)-§('♦*-■ ) (B-18, 



In order to limit errors in these functions to under ten percent, it must be assumed that 



k 2w 2 b 2 < 2 x 10-' , ( B-19) 



k 2f 2 b 2 < 2 x 10-' (B-20) 



and 



k 2a 2 a 2 < 6 x 10' 1 . (B-21) 



This implies that in fish flesh and water 



cob < 7 x 10 4 cm/sec (B-22) 



and in air 



coa < 2.4 x 10 4 cm/sec . (B-23) 



Then with these limitations and the acceptance of at most a ten percent error, equations B- 10 through B-18 

 can be written as: 



(B-24) 



(B-25) 



(B-26) 



(B-27) 

 (B-28) 

 (B-29) 

 (B-30) 

 (B-31) 



(B-32) 



It is necessary to keep both the largest real and largest imaginary term in the spherical Hankel function 

 expansions, regardless of their comparative magnitudes. 



Equations B-24 through B-32 will now be utilized to simplify equations III-2 through 111-15. In addition, 

 equations II-59, II-60, II-68, and II-69, will be substituted for k, a , k 2a , k 2f , and k 2w . These substitutions yield, 

 after neglecting any term which causes an error of less than ten percent, the following expressions for the 

 Si/s: 



Sl1 = <£±_ - ic * u ' (B-33) 



11 3p 0w c w 2 p 0w a)b 2 



42 



