304 BELL SYSTEM TECHNICAL JOURNAL 



face drawn in the medium completely around the point; second, that 

 the function which is propagated in waves conforms to the so-called 

 ' ' wave-equa tion . " 



We have found that these definitions are compatible with one 

 another, the latter being included under the former. 



We have applied them to the case of a function which at any par- 

 ticular point of the medium varies as a sine-function of time, thus: 



/ = 7^(x, y, z) -sin ^; ^ = nt-h{x, y, z), 



and have found: 



(a) that provided the functions F and 8 conform to certain stipula- 

 tions, the function / will satisfy the wave-equation; 



(b) that (p itself is propagated by wave-fronts; although there is 

 nothing periodic or vibratory about (p, each surface over which (p 

 possesses any constant value wanders onward through space, changing, 

 it may be, in shape as well as position; 



(c) that the speed with which the wave-fronts of the phase-function 

 <p travel is the speed at which the contributions, out of which the value 

 of f/F at any point and moment is built up, travel to that point from 

 any environing surface. 



A Test of Kirchhoff's Theorem 



Having formed the conceptions of plane waves and sine-vibrations 

 and frequency and wave-length, we now can practice on Kirchhoff's 

 theorem by applying it to a problem of which the answer is predeter- 

 mined, so preparing ourselves for other problems of which the answers 

 can be discovered only by means of the theorem. 



Imagine plane monochromatic waves, of frequency v = lirti, 

 wave-length X = 27r/m, and phase-velocity c = vX — n/m, travelling 

 in the positive :x;-direction in an endless procession through an infinite 

 medium. They are described by the function: 



5 = cos {nt — mx), (61) 



which is a solution of the wave-equation (6). 



The value Sa of the function 5 at the origin at the moment / is by 

 hypothesis 



Sq = cos nt (62) 



and according to Kirchoff's theorem it is given by the following 

 equation: 



4.50 = J'^^l^cos (n,0|^7 -^^] . (63) 



