60 DYNAMICAL THEOEY OF SOUND 



neglected. Under these conditions the equation of motion 

 of an element Bx is 



(1) 



where i|r denotes the inclination of the tangent line to the 

 axis of x. The right-hand side is, in fact, the difference of 

 the tensions on the two ends of the element, when resolved in 

 the direction of y. In virtue of the assumption just made we 

 may write sin \|r = tan i/r = dy/dx, so that (1) becomes 



where c 2 =P/p ............................ (3) 



It is easily seen that the constant c has the dimensions of 

 a velocity. 



The kinetic energy of any portion of the string is given by 



T=lpjfdx ..................... (4) 



taken between the proper limits. The potential energy may 

 be calculated in two ways. In the first place we may imagine 

 the string to be brought from rest in its equilibrium position 

 to rest in any assigned form by means of lateral pressures 

 applied to it. For simplicity suppose that at any stage of the 

 process the ordinates all bear the same ratio (k) to their final 

 values y, so that the successive forms assumed by the string 

 differ only in amplitude. The force which must be applied to 

 an element Sx to balance the tensions on its ends is 



(P sin i|r) &c, 

 ox 



where sin -^ is now to be equated to kdy/dx', and the displace- 

 ment when k increases by &k is y 8k. The total work done on 

 this element is therefore 



where the accents indicate differentiations with respect to x. 

 The potential energy is accordingly 



v (5) 



