Savitsky 



of the wave-wake system is shown in Fig. 22. 



For progressive deep water gravity waves in still water, the 

 phase velocity of the wave, Cq, is given by: 



Co^=g/ko (3) 



where g = acceleration of gravity, Xq = wave length in still water, 

 kg = wave number = Zir/\ , C^ = wave velocity relative to still water. 

 After the waves have run from still water into a current, the kine- 

 matical condition that must be satisfied is that the wave period, T, 

 remains constant while the wave length, X, velocity C, and height 

 H change. Given a current velocity V , the constancy of wave 

 period is expressed as: 



a = ^ = k(C + VJ = ko(Co + V„^) (4) 



where the subscript, o, refers to the still water conditions. 

 Thus: 



For the present case Vy,^ = so that: 



^0 Co Co 



and 



C =-^(Co +Vc|+4V^Co) (5) 



which is the wave speed relative to the water for waves progressing 

 in the saxne direction as the current. 



More generally, for waves whose crest line is at an angle 

 relative to the x-axis: 



C = i (Co + VCJ + 4V^Co cos a) (6) 



where a is the angle between a wave ray and the x-axis (Fig. 22). 



The wave velocity relative to the bottom C is the vector sum 

 of the wave speed relative to the water and the local current. 



424 



