302 BELL SYSTEM TECHNICAL JOURNAL 



nary particle happens to be moving with just the speed defined by the 

 equation 



U = (d<p/dt)/^(p = C, (50) 



the coefficient of dt in equation (49) vanishes; that is, the particle as 

 it moves along keeps up with the preassigned value of (f, but this is the 

 same thing as saying that c is the speed of the wave-front. 



The phase-function <p therefore possesses a quality which in itself 

 suggests one aspect of a wave-motion. It is not periodic, neither does 

 it conform to the wave-equation ; but each of the surfaces over which 

 (p has any constant value is perpetually travelling. Each of them 

 may be changing continually in size, it may even be changing in shape; 

 but each retains its identity, and if it is completely known for any 

 given instant, its past and future history are determined completely; 

 for each of the area-elements of such a surface is moving at the speed 

 c and in the direction normal to itself, and from the position of each 

 area-element at the moment t we can determine the position of the 

 area-element into which it evolves at the moment / + dt, and repeat 

 this process of prediction or retrospect ad infinitum. I will speak of 

 this state of affairs as propagation by wave-fronts. 



Having determined by (48) the relation between frequency and 

 phase-speed, we now can give both the definition and the formula for 

 the wave-length. The wave-length X is by definition the quotient of 

 phase-speed by frequency: 



(«/27r)\ = C, (51) 



and in this special case, the formula for it is: 



X = 2ir/\v<p\. (52) 



The reason for giving this quantity a name, and such a name as "wave- 

 length," arises from the best-known and too-exclusively-known special 

 case, that of "plane waves" commonly so called — meaning not only 

 that the wave-fronts are plane, but also that the amplitude is constant 

 over each. Such waves travelling along any direction, say that of 

 X, are described by the expression : 



/ = Fsin {nt — mx), F = constant. (53) 



The wave-fronts — that is to say, the surfaces over which (p = {nt — mx) 

 is constant at any moment — are planes normal to the axis of x. These 

 planes are likewise the surfaces over which the displacement/ is con- 

 stant at any moment, and it is tempting to define the wave-fronts as 

 the loci of constant displacement; but this is a coincidence which should 

 be regarded as an accident. At a given moment, any value of sin ip 



