STKINGS 61 



In the alternative method we calculate the work done in 

 stretching the string against the tension P. The increase in 

 length of an element &e is 



approximately, so that 



F=4P/y'=<fo. ..................... (6) 



The formulae (5), (6) lead to identical results when applied to 

 the whole disturbed extent of the string. For by a partial 

 integration we have 



-Syy"dx = -[yy r l+Sy'*-dx, ............... (7) 



where the first term refers to the limits. It vanishes at the 

 extremities of the disturbed portion, since y is there = 0. 



23. Waves on an Unlimited String. 



The solution of 22 (2) is 



y=f(ct-x) + F(ct + x) ................ (1) 



where the functions /, F are arbitrary. It is easily verified 

 by differentiation that this formula does in fact satisfy the 

 differential equation, and we shall see presently that by means 

 of the two arbitrary functions which it contains we are able to 

 represent the effect of any given initial distribution of displace- 

 ment (y) and velocity (y\ It was published by d'Alembert* 

 in 1747. 



The two terms in (1) admit of simple interpretations. 

 Taking the first term alone, we see that so far as this is 

 concerned the value of y is unaltered when x and ct are 

 increased by equal amounts ; the displacement therefore which 

 exists at the instant t at the point x is found at a later instant 

 t 4- r in the position x 4- CT. Hence the equation 



y=f(ct-x) ..................... (2) 



represents a wave-form travelling unchanged with the velocity 

 c in the direction of ^-positive. The equation 



y = F(ct + x) ..................... (3) 



represents in like manner a wave travelling with the same 

 velocity in the direction of ^--negative. And it appears that 



* J. le Kond d'Alembert (171783), encyclopaedist and mathematician ; he 

 made important contributions to dynamics and hydrodynamics. 



