Another basic example is of waves propagating directly against or 

 with a current, i.e., take a current u(x)i, where i is a unit vector in 

 the X direction. This may occur in a channel of variable depth. More 

 details are in Peregrine (Section IID,'l976) for deep water and Jonsson, 

 Skougaard and Wang (1970) for finite depths. 



An effect here is the lengthening of the waves as the current 

 increases in the wave direction and a decrease in wavelength as the 

 current decreases. This corresponds, for a horizontal bed, with the 

 sign of the rate of strain which is simply du/dx in one dimension: 

 du/dx > is an extension; du/dx < is a compression. 



A different influence becomes prominent for sufficiently strong 

 adverse currents. As -u increases and L decreases, a point is reached 

 where the total group velocity u + C = and the waves are stopped. 

 Since the wave action flux is not zero in such a case, the wave action 

 and wave steepness become infinite in this simple refraction theory. 

 This is the basis of hydraulic and pneumatic breakwaters. The velocity 

 needed to stop deepwater waves is only h,CQ where C^ is the phase veloc- 

 ity of the waves on still water. When incident waves meet such a strong 

 current they break and lose their energy. Calculations aimed at clar- 

 ifying the effect of these breakwaters, including a representation of 

 the velocity variation with depth, have been made by Taylor (1955) and 

 Brevik (1976). See also Jonsson, Brink-Kjaer, and Thomas (1978). The 

 stopping point was recognized by Peregrine (1976) and by Stiassnie 

 (1977) as a form of caustic and is mentioned in Section II, 10 and 11. 



Further examples resembling these two, such as u(x)i + Vj where V 

 is constant, can also be examined analytically. These more general 

 examples are discussed in Peregrine (1976) and Peregrine and Smith 

 (1979). In all cases, caustics can arise. This implies that in any 

 general current field there may be areas of particularly steep waves. 

 Sufficiently short waves (i.e., where C is only a few times greater than 

 the maximum current speeds) will meet caustics or propagate into regions 

 where they have much reduced wavelength. In the field, such steep waves 

 often occur where tidal or freshwater flows are constricted. They are 

 common off headlands protruding into tidal currents and may also occur 

 off estuaries with strong tidal or freshwater flows and among islands 

 within an area of significant tidal range. 



The examples up to here involve wave-current interaction which makes 

 the surface waves steeper. The complement of these surface areas with 

 steeper waves is the areas of little or no wave activity. Areas of 

 reduced wave steepness are particularly likely with short-period waves 

 which are more easily stopped by adverse currents, or reflected by shear 

 currents, or dissipated when over steepened. Even following currents can 

 increase wave steepness to the breaking point (Jonsson and Skovgaard, 

 1978). Shear currents may filter wave spectra, dissipating or reflect- 

 ing certain components while transmitting others with substantial 

 increase or reduction in amplitude. This has been demonstrated in 



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