at a mean speed which is 27% of the phase speed (Longuet-Higgins, 1979). 

 These currents can be important in some applications involving transport 

 of heat, pollution, or sediments. The transfer of momentum from wave to 

 current motion, which occurs when waves break, usually leads to stronger 

 currents. 



One of the simplest effects of a current is in convecting a wave 

 field past a measuring instrument. For example, some fair-weather wave 

 measurements in the Bristol Channel showed a systematic variation of 

 significant period between 3 and 5 seconds. This was readily explained 

 once the measurements were plotted along with the tidal currents; a wave 

 field with a significant period of 4 seconds was being convected back 

 and forth by a tidal current of amplitude about 3 knots (1.5 meters per 

 second) . 



Once details are required of the wave field, the dispersion equation 

 is almost inevitably used since measurements are usually taken as a time 

 series, and information on wave number or wavelength is required. For 

 small-amplitude waves the dispersion relation is 



a^ = gk tanh kd (4) 



and use of the Doppler relation (eq. 3) leads to 



('J - k • u)2 = gk tanh kd (5) 



where d is the depth of water, and g the acceleration of gravity. 

 Consideration of equations (4) and (5), from the point of view of 

 solving for k, shows a significant difference. For given O and d, 

 equation (4) is readily solved numerically for k but gives no 

 information about the direction of k. Equation (5) includes both k and 

 k cos in it, where is the angle between u and k. The dispersion 

 equation (4) is anisotropic. 



Even if the angle between wave and current is known, there may be 

 either two, three, or four solutions for k. Even if waves and currents 

 run parallel, there may be one (if ^ = 0), two, three, or four 

 solutions for k, for given values of ij^, d, and current speed, |u|. For 

 the parallel case, solutions can be displayed graphically, as in Figure 

 2, by plotting each side of the reduced Doppler equation 



(JJ - ku = ±cr (6) 



17 



