666 BELL SYSTEM TECHNICAL JOURNAL 



or using dots to symbolize dififerentiation with respect to time, and 

 dashes to represent differentiation with respect to space: 



ij = -^iTp-y". (103) 



This equation, a linear combination of a second derivative with 

 respect to time and a second derivative with respect to space, is the 

 first and simplest of our wave-equations. 



It is called a wave-equation, because it may represent — it does not 

 necessarily represent, but it may — a shape or a figure or a distortion 

 of the string (whichever one may choose to call it) which travels con- 

 tinually and indefinitely along the string with a constant speed. 



To illustrate this possibility, let us suppose that at the time / = 

 the string is distorted into a sinusoidal curve described by the equation : 



y — A sin mx at / = (104) 



and that its points are moving parallel to the v-axis with speeds 

 described by the equation : 



y = nA cos mx at / = 0. (105) 



At any other moment /, the configuration of the string is described by 

 the equations: 



3^ = ^ sin (nt + mx), y = nA cos («/ + mx), (106) 



for these satisfy the differential equation which underlies the whole 

 theory, and they satisfy also the "initial conditions" specified by 

 (104) and (105). They satisfy these equations, that is to say, provided 

 that a certain relation is fulfilled among the constants n and m, and 

 the quantities T and p which describe the physical nature of the 

 stretched string; this relation being: 



n/m = ■ylrfp- (107) 



If this relation is fulfilled, the condition of the string throughout all 

 time is described by the equations (106). 



Examining these equations, we perceive that they signify that the 

 values of displacement and speed, which at the time / = existed at 

 any point Xq on the string, are at any other time t to be found at the 

 point Xi = xq — {n/m)t. These values are moving steadily along the 

 string; the whole configuration of the string, its sinusoidal shape and 

 its transverse velocities, is slipping steadily lengthwise in the direction 

 of decreasing x — the shape of the string is being transmitted as a 



