of propagation against the current they are turned in the direction of 

 the fastest current (which slow the waves most). See curves for 160 

 and 340° in Figure 6. 



Figure 6 shows the rays calculated for a shear flow with linear 

 variation, V(x)j. The rays (full lines) are shown for every 20° of 

 initial directilin from a single point on the line of zero current. 

 Lines parallel to k are also shown (broken lines), these are orthogonal 

 to wave crests and can be used to deduce what a particular wave field 

 might look like. The two sets of lines differ because of the current. 



Certain properties are simply deduced from Snell's law (eq. 26), 

 which shows that the wavelength is proportional to cos in this case, 

 where G is the angle between u and k. Thus, when waves propagate 

 directly with or against the current, 9=0 or it, they have their 

 maximum or minimum wavelength and relative group velocity, but as they 

 turn to become more nearly perpendicular to the current, -> '^tt, the 

 wavelength shrinks toward zero, and so does their relative group 

 velocity, C . As C -»■ 0, a greater part of the wave propagation is 

 simply due to the current. However, as the wavelength gets smaller, 

 wave steepness increases, the waves break, and the limit = %Tr is not 

 attained. 



A coastal engineer is accustomed to having an initial region in 

 which waves are uniform. A ray diagram for such a case can be deduced 

 from Figure 6 by choosing a single initial direction along a streamline 

 and repeating the corresponding ray many times by parallel translation 

 up (or down) the diagram (see Fig. 7 for a closely related example). 



When wave direction (k) becomes parallel to the current, shown 

 for initial directions 20°," 40°, 200°, and 220° in Figure 6, they are 

 reflected toward weaker currents. The reflection line is called a 

 caustic. Simple refraction theory (ray theory) predicts singular (i.e., 

 infinite) wave amplitudes at such lines, but a better approximation 

 to the full linear theory gives a finite amplitude (Section II, 11). 

 Examples of the rays for two caustics are shown in Figure 7, which is 

 a sketch of waves on a flow in a channel with reflecting walls. The 

 upstream propagating waves are trapped at the center of the channel, the 

 downstream waves near the edge. Only the latter case is comparable with 

 Figure 6 since the waves in midchannel cannot reach the zero current at 

 the edge. 



A recent study (Hayes, 1980) models waves propagating across the 

 Gulf Stream near Florida in this way. In agreement with the model, 

 measurements show that the current shelters the coast from certain waves 

 by reflection at a caustic. 



35 



