O44 — 



(Ya-l)Jo(k2,a) 



v _ IPo a C a 2 C p 



3 Ca^C pa \ 

 COK a / 



[ 1 + (dlk:) ( 



ki a j '(k 1a a) \i 

 co 2 a 2 p 0a / V j (k 1a a) )\ 



where equation II-68 has been used for k 2f 2 . 



The solution of the boundary conditions for B w is 



(111-15) 



where 



B w _ 0,1) + g 2 W 

 A S,,U + S 21 W 



(111-16) 



U — 044(022033 O23O32) + S 34 (S 2 3S4 



022S43) 



1-17) 



and 



W — b 4 4(0 13 032 0,2033) + 034(0,2043 0,3042) 



(111-18) 



This solution is easily arrived at after several pages of substitutions. 



Equation 111-16 could be solved numerically with a computer by choosing values for the various physical 

 parameters and calculating the S./s and a,'s. However, this will not be done because the purpose of this 

 report is to obtain simplified equations for co and o which can be solved without resorting to use of a 

 computer. The simplified solution is based on the ranges of the physical properties and the limits of the 

 variables given in Chapter II. It is obtained by assuming that | k, a a I is large and that k 2w b, k 2| b, andk 2a aare 

 small and accepting any errors which are less than ten percent. Order of magnitude comparisons are then 

 made between various terms and any term which is always less than ten percent of another term is 

 neglected. This simplification process is shown in Appendix B. The result of this process is: 



% = - (1 + A + iA) (1 + 2 ^ w 3 

 A \ p 0w co 2 a 3 



+ i 



t 



3p ,c w (y a - 1 ; 

 Po w w 2 a 2 



Jb£fc)'( 1 



/ c wPo, \ r /3p 0a c a 2 \ / 



i, a)ap 0w ) iwm^)\ 



3p 0a c a 2 



2s 



w 2 a3p 0f , 



2s x 



3p 0a c a 2a^ 



11^2 



3p 0| 2c t 2a2 



)]!' 



1-19) 



where 



and 



\ p 0f 2 c,^a3 ; v 



1 1 Po,c,2 

 9p 0w c w 2 



) 



= / 2(o^b3 x r / p 0( c,2 x /_p 2l c !i + H \ /Ji\ ] 



V 3p 0f c, 2 a3 ) L Uo„Cw 2 f Ipo w c w 2 4 )\&) J 



(III-20) 



1-21) 



Equation 111-19 can now be substituted into 11-104 to yield a. However, before doing this, equation 111-19 

 will be manipulated so that the final results can be easily compared to those of other researchers. Thus, the 

 resonant frequency is determined by setting the imaginary part of the denominator of equation 111-19 equal 

 to zero and the real terms of the denominator represent the damping [18]. Therefore, letting 



20 



