WIND CURRENTS AND WIND WAVES 139 



somewhat in the direction of progress, meaning that an actual transport 

 of water takes place in this direction even in the absence of wind. 



The irregular appearance of the surface is not accounted for by 

 the theories that have been mentioned so far, but a somewhat better 

 understanding of the pattern of waves is obtained when one takes into 

 account the phenomena of interference. Suppose that two waves travel 

 in the same direction but with a slightly different velocity. The waves 

 at the surface can then be represented by the equations 



^1 = ai sin (kiX — dt) and ^2 = ^2 sin {k2X — (Jit), (VII, 30) 



where ^ is the vertical displacement of the sea surface, where k = 27r/L, 

 and where a = 2'k/T. The actual appearance of the surface is obtained 

 by adding the displacements due to the two individual waves. If the 

 amplitudes are equal, one obtains 



^ = 2a cos [H(ki — K2)x 



- M(o-i - G^)t] sin [}i{K^ + K2)x - ma, + <T2)t]. (VII, 31) 



If the wave lengths differ by a small amount only, this new equation 

 represents a wave whose length is the average of the two waves of which 

 it is composed but whose amplitude varies between zero and 2a. At the 

 locality where the two waves are in phase, the amplitudes are added, 

 and a wave appears of twice the amplitude of the two original waves, but, 

 where the waves are in opposite phases, the amplitudes cancel. The free 

 surface takes the appearance of a sequence of wave groups separated by 

 regions with practically no waves. A simple pattern of interference is 

 involved, and, owing to this interference, the two individual waves are 

 no longer conspicuous, but are replaced by a series of wave trains that 

 appear to progress with a definite velocity: 



c = "-1^^. (VII, 32) 



Kl — K2 



If the wave lengths are only slightly different, this velocity is very nearly 

 equal to J^c, where c now represents the average velocity of the two 

 interfering waves. Thus, the wave train progresses with a velocity that 

 is only one half that of the single waves, which therefore advance through 

 the wave trains. 



The upper curve in fig. 36 is a reproduction of a record of waves 

 obtained at the end of the Scripps Institution of Oceanography pier and 

 represents a good example of interference of waves of nearly the same 

 period length but of different amplitudes. In the middle portion of the 

 figure are shown two sine curves, one of period 9.6 sec and amplitude 

 0.75 m, and one of period 8.7 sec and amplitude 0.32 m. The heavy 

 curve represents the wave pattern that would result from interference 



