[jat 



[^c,] = 



(1-cr) h - 



A. S. Iberall 



2ea/?2a:g 



(1-c^) Vj' 



[uj\(l -a) - 2eaS/32ga.aj] 



[awS(l-a) - 2ecr/3^ga(^a^] [jw(l-a) ■\/Ja7] 



[j (7- 1) ^2] [j(l-cr)/32a;a3] ,„..,,_.,,.,_ 

 [o- ( 1 - a) a^ ] 



(The brackets preserve the source of each factor.) Thence 



(1- J) 



7-1 

 \ + -— — /32a;2 



which when coupled with the quadratic equation for c^, leads to two equalities 

 from the real and imaginary parts. 



From the real part: 



k 



CO \J2co CO V 2ct) 



From the imaginary part: 



2 \/32^2 V^ 



/32aj2 



1 + 



7-1 /3= 



/32a.2 



^2a;2 



>' - 1 



yfa CO \f2co y/a co yj 2co 



7 - IV ^^ /i ^^ ~ 1 1 ^ 



7 - 1\ /32g2 

 ± (1 + I = , 



yfo^ I \J 2co 



— % /?; 



The secular equation thus finally leads to the pair (to the order of 1/v^ 

 terms) 



7 ~ V 



■d' 



1 + ^) + ( 1 + q + 



2^ 



1 . 2ULi , /I 



One may note that of the two roots possible for the fourth mode, only one 

 could satisfy the boundary conditions (namely, that it should nearly vanish 

 along a particular complex path). This root leads to a propagative system, the 

 elastic wave given by 



\ = a2 + §2 ^ ^2^2 _ 



722 



