LONGITUDINAL MODES IN CYLINDERS AND SLABS 959 



W = Be* *, is a solution (where A and B are constants measuring the am- 

 plitude), if either 



(0 kl = -^ -y' and y A, = hB,, 



2 



or (ii) ^2 = — — 7^ and ^2^2 = —7^2- 



M 



When solutions of both types are so superposed as to make U odd and W 

 even 



U = iAi sin hx -{- iA2 sin ^2^, (1) 



'Y kv 



W = Ai - COS kix — A2 — cos ^2:*^. (2) 



ki 7 



The normal and tangential stresses on planes perpendicular to x are 

 du 



X.= (2. + X)?^ + x(g4-g), X. = .(£ + g) 



/aw aa>\ 



and the requirement that they vanish 2X x — dtza leads to the boundary 

 conditions 



-4i(X7^ + (2/x + X)^i) cos ^la + 2^2M^i^2 cos ^2^ = 0, 



2^117^ sin kia -\- ^2(7^ — ^2) sin kza = 0, 



the vanishing of whose eliminant with regard to A\ and ^2 is the secular 

 equation. Although in principle that equation establishes a relation be- 

 tween 7 and CO, it is more conveniently examined when expressed in terms 

 of a = kia and /8 = kia, which are quadratically related to co and 7 by (i) 

 and (ii). In those terms it becomes 



(Xi82 + (2m + \)o?Y cos a sin j8 



(3) 

 -f 4(m + X)Q:/3(Mi8'^ - (2m + \W) sin a cos iS = 0. 



The physically interesting quantities can be expressed in terms of a ard 

 /8, with the aid of (i) and (ii). Thus 



P"^ = 4roV (^ - "'>> ^»d (4) 



a^M + X) 



2 MiS' - (2m + \)a /.x 



'>' = 2/ _L A\ • ^^^ 



