668 BELL SYSTEM TECHNICAL JOURNAL 



nitely long wire can sustain vibrations of any wave-length, or vibrations 

 of two or any number of wave-lengths simultaneously, with any inter- 

 relation whatever among their several amplitudes and phases. On this 

 fact rests our freedom to impose any initial conditions whatsoever 

 on such a wire (subject to the usual restrictions of continuity and 

 finiteness). For, if it be demanded that at / = the displacement y 

 shall vary along the wire according to any totally arbitrary function 

 f{x), and the transverse speed y according to any totally arbitrary 

 function g{x), then we have only to expand these functions / and g 

 into Fourier series, or if need be, Fourier integrals; and each term in 

 such an expansion corresponds to such a solution as (109), with a 

 specific value of m and such specific values of C and D as the initial 

 conditions require; and the configuration of the wire forever before 

 and after is described by the sum of all these solutions. In such a case 

 we should not see an unchanging distortion of the wire slipping 

 steadily along its length with a constant speed, nor a stationary pattern 

 of nodes and loops. All the obvious features of wave-motions would 

 be blotted out; and yet the infinitely complicated and variable figure 

 of the string would be equivalent, in the last analysis, to a multitude 

 of sinusoidal wave-trains perpetually gliding to and fro with the 

 same uniform speed. 



As soon, however, as we impose boundary-conditions , the vibrations 

 which the string can execute are severely restricted. 



As a simple and familiar example of boundary-conditions, I will 

 assume that the string is clamped at the points x = and x = L, and 

 concern myself only with the finite length of string, L, comprised 

 between these two fixed extremities. 



As a preparation for future developments, it is advisable to restate 

 the underlying difTerential equation, and solve it ab initio. We have: 



y = tib", (113) 



in which it stands for the speed of propagation of a sine-wave along 

 an infinite wire. We essay a tentative solution, in the form of a 

 product of a function of / only by a function of x only: 



y = g{t)-f{x). (114) 



The differential equation subjects the functions g and/ to the condition : 



ril = IhH = - m\ (115) 



for, since the first member of this triple equation does not depend on 

 /, and the second does not depend on x, each of the two must be inde- 



