obtained from equation (5) where O is given by the positive root of 

 equation (4). In Figure 2 only positive values of wave number k have 

 chosen, i.e., the positive direction is by definition that of wave prop- 

 ation. Positive current, u, therefore means a following current (waves 

 moving downstream), and negative u means an opposing current (waves 

 moving upstream) . 



Solutions to equation (6) are most easily understood by first 

 considering the no-current case, which corresponds to the dashline in 

 Figure 2, parallel to the k-axis. Only one solution is found, 

 corresponding to solution point E from equation (4). The wavelength 

 here is that found in any conventional wave table. When there is a 

 current, the line for lo - ku splits, the two branches corresponding to 

 waves going with or against the current. Then four solutions are 

 possible located graphically at points A, B, C, and D on Figure 2. 

 Points B and D correspond to waves and currents in the same direction (u 

 positive); points A and C correspond to opposite directions (u 

 negative) . 



In coastal engineering practice, solution points A and B are usually 

 the only ones of interest. This is easily seen by following a wave of 

 constant depth from a no-current environment through a gradual change 

 into a current; simple continuity reasoning shows that either solution 

 point A or B will be met with. It further appears from the figure that 

 everything else being equal, a following current increases wavelength 

 (k is diminished), and an opposing current has the opposite effect. 

 More discussion may be found in Jonsson, Skougaard, and Wang (1970), who 

 also present tables for a direct determination of wavelengths for an 

 arbitrary angle between current and wave direction. These tables can 

 also be found in Jonsson (1978b). A general procedure, including 

 nonlinear terms, has been given by Hedges (1978). 



A complete discussion, including solution points C and D, is given 

 by Peregrine (1976, pp. 22-23). Solutions C and D correspond to shorter 

 waves than A and B, and they have no corresponding solution in the no- 

 current case. Solution point D corresponds to waves propagating with 

 the current, and solution point C to waves propagating against it. For 

 cases A and B, energy propagates in the wave direction, while for cases 

 C and D energy is swept downstream by the current. Alternatively, for 

 case A, only, energy propagates against the current. 



In stronger currents the two solution points A and C draw closer 

 together until they are coincident; for still stronger currents there 

 are only the B and D solutions. Two coincident solutions (A = C) occur 

 when the current velocity is equal and opposite to the waves' group 

 velocity, C = da/dk, relative to the current, that is, total group 

 velocity, u + C„, is zero. In such a case, the energy of the waves is 

 held stationary against the current (their phase velocity indicates 

 upstream travel). They are then "stopped" by the current. For a more 

 complete discussion, see Peregrine (pp. 22-23, 1976). 



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