^and VS= (3^. 3^^). (10) 



are equivalent tq -co and k, respectively, but that other variations of S 

 are of larger scale and correspond to the waves' refraction. In this 

 notation, the vectors k and x are horizontal vectors with components in 

 Xt and Xo directions, and following a common convention, a Greek suffix 

 represents the two components, e.g., VS = 8S/8x . 



If the propagation of a wave is to be followed, or predicted, by 

 refraction theory, then k and co must be defined and vary smoothly along 

 the wave's path. This means that for any part of the wave field 

 accessible to refraction theory, S, oj , and k must be smooth functions. 

 Mathematically, these are required to be dif f erent iable. Then, for 

 consistency, the partial derivatives of S must be independent of the 

 order of differentiation, i.e.. 



2 2 



3 s as 



3t8x 3x 9t 



a a 



(11) 



and 



2 2 



3 S 3 S 



3x 3x 3x„3x 



(12) 



In terms of oj and k, these "consistency conditions" become 



+ Vw = (13) 



3k 

 3t 



and 



V X k = (14) 



Equation (13) can be interpreted as the "conservation of waves" or 

 "conservation of wave number." The description consistency condition is 

 preferred since it helps to emphasize the underlying assumption that the 

 phase, S, is a smooth function. 



27 



