LONGITUDINAL MODES IN CYLINDERS AND SLABS 967 



to become proportional to sin — and cos — respectively, and jS approaches 



a a 



the value nw where n is the branch number in order of increasing cut-off 

 frequency, with « = ascribed to the lowest branch. Thus the feeling, de- 

 rived from more familiar cases of wave motion, that the order in which 

 the branches arrange themselves should be correlated with the number of 

 nodes they display retains an asymptotic vaUdity here, in respect of each 

 displacement component. 

 Nodes of absolute displacement will occur only at special frequencies. 



OCX 8x 



With the notation a' = — , /8' = — , the conditions for their occurrence can 

 a a 



be written 



Oui8'2 - (2iu + X)a'2) sin ^' cos a' + (JX + X)a'/3' cos |8' sin a' = 0, 

 sin 2a' sin 2^' ff j8 



sin 2a sin 2^ ' 

 taken together with (3). 



^' < ^, «' < a, 



3. The Cylinder 



Procedures analogous to those of the preceding section, and presented by 

 Love®, lead to Pochhammer*s secular equation, which in the pressnt 

 notation* is 



(XjS2 + (2m + \)o?yj,{a)jm 



+ 40z + X)c^(Mi82 - (2/x + \W)Ji{a)Jo(fi) (9) 



+ 20z + X)(2m + X)a(a2 - 0^)J,(a)J,(fi) = 0, 



where (4), (5) and (6) still hold, with a signifying the radius of the rod. The 

 analogy between (3) and the first two terms of (9) is striking. Again the 

 roots jS = and ^ = a can be neglected, and the equation when divided 

 by |8 becomes even in a and /3, and a plot of p!^ against a^ becomes appro- 

 priate, with the restrictions (7) as to regions of significance. The following 

 paragraphs are lettered to correspond with their analogues of the preceding 

 section, 



(a) Setting fi^ = (2 /x -f X)^^ in (9) reveals the cut-offs at Ji(fi) = 

 and at (2/x .+ X)a/o(«) = 2fiJi{a). 



(b) Setting o: = in (9), it can be seen that the roots intersect the line 



* In comparing this treatment with that of Hudson^, interpret his symbols 



. (MX 2u 



