128 Prof. S. P. Timoshenko on the Transverse 



can be calculated from this equation. Some results are 

 given in the table below *. 



h = 



0-5 



0-7 



0-9 



09165 



0-9192 



l/2c = 



4-8 



2-5 



0-75 



0-47 



0-30 



§ 3. In the case of long waves, the velocity V is inversely 

 proportional to the length Z, but as I diminishes it can be 

 seen, from equation (5j, to approach a limit which is lower 

 than V 2 » an d can be found from the relation 



4 V(l-/ 2 )(1-A 2 ) = (2-A 2 ) 2 . 



This limit is the velocity of the " Rayleigh waves' - ' f. 



If we put A, = //,, we obtain, in the limit, V = 0*9194 Y 2 . 



As the wave-length / increases, the arguments of the 



hyperbolic functions in equation (5) decrease, and we can 



employ the ascending series. I£ we limit our attention to 



the first three terms, we may write equation (5) in the 



form 



a(*cy + b{uc) 2 -\-d = 0, . . (6) 



where 



« = A[4(i-A 2 ) 3 -(2-mw 2 ) 2 ], i 



5 = -KJ(l-A 2 ) 2 -(2-/, 2 ) 2 (l-/ 2 )], • ■ (7) 

 d = -A 4 . J 



When etc is very small, we can obtain a first approximation 

 by neglecting the first term in (6) and putting 



(«)» = ~j (8> 



This approximation will be 



. ^-i^w <*> 



whence 



and 



y 2 _ ±( aC y A X + ^ no) 



V - 3 W p (x + 2fiy ' ' ' l W) 



2 2V2 4.2 4#(A + //,) n . 



^ = aY = uc MxT2^y ' ' • (11) 



* In these calculations, a has been taken to be 0'25. 

 t A. E. H. Love, op. cit. § 214. 



