PHOTONS AND ELECTRONS 7 



to Hiiygens' mind, but it is essential to our present conception of mono- 

 chromatic light, and to the explanation of the many and variegated 

 phenomena of diffraction. 



When the undulatory theory is formulated in mathematical language, ^ 

 one finds that the progress of a wave front — in the sense of the foregoing 

 implication — across a refracting surface may be predicted by Huygens' 

 construction. The wavelets of Fig. 1 acquire new attributes, first 

 imposed on them by Fresnel. Huygens in effect assumed that the 

 amplitude of each wavelet is zero except at the point where it touches the 

 envelope common to all. In the undulatory theory, the amplitude of 

 a wavelet varies continuously from a maximum value at the point just 

 defined to zero at the point diametrically opposite (the wavelet being 

 a full sphere, not a hemisphere as indicated in Fig. 1). This modification 

 is necessary to account for diffraction, but does not affect the case of 

 refraction of an extended wave front. We may continue to accept equa- 

 tion (3), and the equation obtained by comparing (3) and (4): 



^ = -, (5) 



p V 



where now v and v' stand for the wave speed of the train of monochromatic 

 waves. 



The final step is taken by remembering that in the two equations 

 connecting wave speed, frequency, and wave-length, one for each side of 

 the bounding surface, 



V = p\ v' = v'\' 



the frequencies v and v' must be equal ; for it is of the nature of vibration, 

 that if two or more continuous media are in continuous oscillation, the 

 frequency (or frequencies) of that oscillation must be the same throughout 

 the entire system, as otherwise the media could not remain in contact. 

 Accordingly we write in place of {5) : 



'P' ^ (R\ 



which amounts to writing 



V = 



const. 



which is the same as equation {1) except for lacking any intimation of the 

 value of the multiplying constant. 



^ Cf., for instance, an article of mine in the Bell System Technical Journal, 7: 

 281-320, 1928. 



