WIND CURRENTS AND WIND WAVES 137 



Stokes' results lead to the conclusion that at increasing amplitude the 

 wave form deviates more and more from the trochoid. Studies of the 

 stability of waves by Michell show that the wave must become unstable 

 if the angle formed by the crest is less than 120 degrees. In this case, 

 the ratio of height to length is 1 :7 (see fig. 35). The velocity of progress 

 of these waves is no longer independent of the height, but can be written 



= [4(— (OT 



In the case of the extreme Michell wave the velocity of progress is about 

 1.14 times greater than that of waves of small amplitude. 



Accurate measurements of actual wave profiles would be very useful 

 for examining the correctness of the above-mentioned theoretical con- 

 clusions. Such measurements have been based on photogrammetric 



Fig. 35. True dimensions of steepest possible wave, according to Stokes and Michell. 



pictures, but the observed profiles show little similarity to the theoretical 

 curves. 



The wave theory also leads to certain conclusions concerning the 

 character of motion of individual water particles. In a wave that has the 

 form of a trochoid, the single water particles will describe circles whose 

 radii decrease with increasing depth: 



ae '''^ (VII, 28) 



where a is the amplitude of the wave (a = 3^//), L is the wave length, 

 and z is the depth below the undisturbed water surface. Each water 

 particle describes a circle with radius r in the time T that represents the 

 period of the wave. The velocities of the individual water particles are, 

 then, 



. = ^ae"''2. (VII, 29) 



These formulas are valid only when the amplitude of the wave is small 

 relative to the length, but they can nevertheless be used for an approxi- 

 mate computation of the greatest velocities that may be encountered in 

 waves. The first columns of table 16 show the periods and the corre- 

 sponding wave length and velocities of progress and the assumed values 

 of the heights of waves. It should be observed that the height equals 

 twice the amplitude. The heights entered in the table reach approxi- 



