BARS 115 



Equating this to the difference of the forces on the two ends, 

 we have 





If the section be uniform, this reduces to 



^-^ (3) 



~** 



where c* = E/p ........................... (4) 



It will be noticed that in this investigation it is not 

 necessary to assume the substance of the bar to be isotropic, 

 provided the proper value of the Young's modulus be taken*. 



The result is also unaffected if the bar, or wire, be subject 

 to a permanent longitudinal tension, since by Hooke's law 

 the stress due to the extension (1) may be superposed on the 

 permanent tension, so long as the limits of perfect elasticity 

 are not transgressed. 



As in 23 the general solution of (3) is 



g = f(ct-x) + F(ct+x), ............... (5) 



representing two wave-systems travelling unchanged in opposite 

 directions with the velocity c given by (4). In terms of the 

 length-modulus, we have by 42 (14) 



c = V(<?); ........................ (6) 



this is the velocity due to a fall from rest through a height J L. 

 Some numerical values of c are given in the last column of the 

 table on p. 119. 



The application to particular problems may be treated very 

 briefly. The various cases that arise present themselves in 

 a more interesting form when we come to the vibrations of 

 columns of air. 



In the case of a rod or wire fixed at both ends, we have 

 f = for x = and x = I (say) ; and the mathematical theory 



* In an "aelotropic" or crystalline solid the values of E will be different for 

 bars cut in different directions from the substance. 



82 



