CONTEMPORARY ADVANCES IN PHYSICS 303 



which is found anywhere repeats itself at intervals lir/m all along the 

 x-direction ; exactly as, at a given point, any value of sin cp which is 

 found at any moment repeats itself at intervals Ix/n all through time. 

 Owing to the coincidence aforesaid, any value of / which is found any- 

 where also repeats itself at spacings 27r/m along the direction of x. 

 This constant spacing serves as the elementary definition of wave- 

 length ; and in this special case the elementary agrees with the general 

 definition, for 



m = \dipldx\ = \v<p\ = 27r/X. (54) 



But there is an almost equally simple case in which the spacing between 

 wave-fronts and the spacing between surfaces of constant / are not the 

 same. I refer to the case of spherical waves of sound or the circular 

 ripples on a water-surface, in which we have: 



/ = 7^ sin (nt — mr), F = constant/r. (55) 



In this case f/F does not conform to the wave-equation, but the func- 

 tion / does. Nevertheless it is the phase, and not the displacement, 

 which advances steadily outward (or inward) in a sequence of steadily 

 diverging (or converging) spherical wave-fronts which expand or 

 contract with the constant phase-speed c. For any two of these 

 spherical wave-fronts differing in radius by l-wlm, the values of 4> are 

 the same. The surfaces of constant/ are also spheres, but they expand 

 or contract with variable speed, and for any two which differ in radius 

 by 2x/w the values of / are different. This shows that one must not 

 be misled by experience of plane waves into defining "wave-length" as 

 "distance between points where at the same moment the displacement 

 is the same" but must hold fast to the phase as the central fact of any 

 wave-motion. 



If the phase-function </> does not vary in space, we have the case of 

 stationary waves. The coefficient of cos in the general expression 

 for v!/ (footnote on p. 339) now vanishes automatically; the coefficient 

 of sin reduces to the term v^-^, and this must be equated to — w^F/c', 

 which if n and c are preassigned leads to an alternative form of the 

 wave-equation 



V-i^ + y^ ^ = (56) 



very common in acoustics and in wave-mechanics. 



In summary: 



We have considered two definitions of wave-motion: first, that the 

 state of affairs at any point and moment in the medium is controlled 

 by the state of affairs at earlier moments all over any continuous sur- 



