4. Refraction of Waves by Currents — Theory. 



Refraction methods are reasonably well established for water waves 

 over water of variable depth and in many other physical applications, 

 especially in acoustics and optics. However, two significant 

 differences occur for water waves on a current. First, the current 

 carries the waves, so that wave energy propagates with the sum, u + C , 

 where C is the group velocity relative to the current, u. Since this 

 sum is usually not perpendicular to wave crests, energy is usually not 

 transmitted in the direction of wave motion. Second, wave energy is not 

 conserved in the absence of frictional dissipation since energy is 

 transferred between the waves and the currents. 



Refraction theory in general has advanced considerably within the 

 last 20 years, particularly through recognition of the concept of wave 

 action. Wave action is important for waves on currents since it, unlike 

 wave energy, is conserved in the absence of wave generation or 

 dissipation. For linear waves, wave action equals (wave energy 

 density)/(wave frequency relative to the current). Much of the recent 

 theory has arisen from the study of nonlinear waves, but the presenta- 

 tion in this review is based on linear theory unless explicitly stated. 

 Some linear results also hold for nonlinear waves, e.g., the Doppler 

 relation (eq. 3), but others are modified. Refraction theory has the 

 primary assumption of locally plane waves, i.e., at any point waves can 

 be recognized as a train of plane waves on a local scale (on time and 

 length scales corresponding to at least a few wave periods or wave- 

 lengths). This restricts consideration to large-scale currents (defined 

 in Section II, 1) and to large-scale variations in the wave field (but 

 see the discussion of caustics in Section II, 11). For linear waves, 

 separation of waves into two or more superposed wave trains is 

 permissible . 



Wherever the primary assumption of locally plane waves holds, the 

 phase of a single progressive wave train can be identified. That is, a 

 description of the form 



a cos (k • X - ojt + 6) (8) 



is possible for most wave properties such as surface elevation. The 

 phase, S, is 



S(x,t) = k • X - (a)t + 6 (9) 



On a large scale, k and (jj , and 6 are also functions of position, x, and 

 time, t. This means it is possible to take the function S(x,t) and say 

 that the partial derivatives. 



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