C„ = C„- = 3o/9k (19) 



is the group velocity of the wave relative to the current. 



The structure of equations (17) and (18) is quite simple. The only 

 derivatives of the unknowns k and oj appear in the bracketed operator 

 on the left-hand side of both equations. The form of this operator 

 indicates that the characteristics of the equation are given by the 

 space-time paths or "rays," 



dx/dt = u + C (20) 



Thus, these rays are in the direction of the total group velocity, u + 



The major difference from the case of zero current is that the ray 

 direction, in general, differs from the direction of the wave number 

 vector. That is, rays are not orthogonal to wave crests except in the 

 special case of parallel wave and current directions. Thus, unlike 

 depth refraction, the orthogonals to the crest (i.e., lines parallel to 

 k) do not show the direction of wave travel, but only indicate the local 

 orientation of the wave crest. The difference between the orthogonal, 

 k, direction and the ray direction can be seen in Figure 5 by the 

 geometrical formulation of the vector sum in equation (20) . This dif- 

 ference for a particular case is illustrated in Section II, 5. 



The mathematical structure of these refraction equations (17) and 

 (18) is unchanged between zero and nonzero current velocity fields. 

 Thus given values of k along some line, not coincident with a ray, 

 equation (17) can be integrated simultaneously with equation (20) to 

 give rays originating from each point of the original line. For exam- 

 ple, the initial line could be a wave maker creating waves in a labora- 

 tory basin, or waves incident from deep water on a coastal region. 



The only published example directly using these equations in space- 

 time is that of Barber (1949), who traced a single ray across the 

 Continental Shelf to the southwest of Britain in order to explain 

 observed fluctuations in the periods of oceanic swell. The variation of 

 tidal currents in space and time appeared adequate to explain the 

 variation although there was not detailed agreement. With the 

 computational advances now available, such calculations should be tried 

 again. 



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