114 DYNAMICAL THEORY OF SOUND 



The potential energy (W) per unit volume of a strained 

 isotropic substance may be found by calculating the work done 

 by the stresses on the faces of a unit cube, on the hypothesis 

 that the strains increase from zero to their final values keeping 

 their mutual ratios unchanged. The average stresses are then 

 one-half the final stresses. 



Thus in the case of a uniform dilatation A we have 



In the case of a pure shear 77, 



Tr=4^ = J^ ................... (16) 



In the extension of a bar, with freedom of lateral con- 

 traction, 



W=J ft e, = itf,' ......... . ......... (17) i 



In the general case we have 



, + 2 + 3 ) 2 + p (ef + 6 2 2 + 6 3 2 ) 



+ J /* {(e 2 - 6 3 ) 2 + (e 3 - 6l ) + (e t - 6 2 ) 2 ). . . .(18) 



This shews that in order that the potential energy may be 

 a minimum in the unstrained state K and p must be positive. 

 It is otherwise obvious from the meaning of the symbols that 

 if either of these were negative the unstrained state would be 

 unstable. 



43. Longitudinal Vibrations of Bars. 



We take the axis of x along the bar, and denote by x + f 

 the position at time t of that cross-section whose undisturbed 

 position is a?, so that f denotes the displacement. An element of 

 length is then altered from Sac to S(#+ f), or (1 + ')Sa;, where 

 the accent denotes differentiation with respect to x. Equating 

 this to (1 + e) 8x, we have 



.-!* ............................ CD 



dx 



The tension across the sectional area (&>) is therefore Eeco. 

 The acceleration of momentum of the mass included between 

 the two cross-sections corresponding to x and x + Sx is p&Sx . %. 



