Vibrations of Bars of Uniform Cross-Section. 127 



n = VKv' 



V = ■£, denotes the velocity of waves of 

 a transverse vibration, 



y i= \/ ^? denotes the velocity of 



P waves of dilatation, 



and V 2 = \f ~i denotes the velocity of waves 



P of distortion. J 



It is easily verified that equations (3) are satisfied by the 

 expressions 



u = cos ax (M sinh 7713/ + N sinh ny) cos pt, 1 



v = sin ax I M - cosh my 4- N - cosh ??,y J cos p£, 



and the conditions that the boundaries (y=+ c ) are free 

 from traction give us 



[(\ + 2/ji)ni 2 — Xa. 2 ] M sinh mc +■ 2^a 2 Nf sinh nc = 0, 1 

 and 2?nnM cosh mc + (a 2 + n 2 ) N cosh nc = 0. J 



Hence, by eliminating the ratio M/N, we obtain the 

 u frequency equation " in the form * 



4=/uLoc 2 mntnnhnc — (a 2 + ?i 2 j [(\ + 2fjb)m 2 — X* 2 ] tanh 



mc : 



whence, denoting the length of the waves by 7, and puttino- 

 V/Vx =/, V/V 2 = h, we have 



4 \/(l-/' 2 Xl-A 2 ")tanh(?p Vl^A 2 ) 



= (2-A 2 ) 2 tanh (?p Vl^l (5) 



If V be given, the corresponding value of the ratio l/2c 



* This "frequency equation" was found by Prof. P. Ehrenfest and 

 myself in collaboration. The solution given in this paper is my own. 



