80 Generation, Growth and Propagation of Waves 



represents the time in which the wave amplitude has been reduced to the 

 fraction e of its original value (e = 2-718), becomes r = \\2vk % or in terms 

 of the wave length (IV. 23) 



In the case of water 



(IV.23) 



if the wave length be expressed in cm. We can see that these capillary waves 

 are destroyed very rapidly by viscosity, whereas for a wave length of 1 m 

 the necessary time t would be about 2 h. This modulus of decay increases 

 very rapidly. A wave with a period of 8 sec would travel more than 2 years 

 before friction would reduce its height to 63 % of its original value, as a con- 

 sequence of friction; in that time it could circle the equator 10 times. The 

 internal molecular viscosity is therefore always negligible. However, we 

 observe a very rapid decrease of wave height of ocean waves and, therefore, 

 it becomes obvious that where we have turbulent flow, we have to substitute 

 the eddy viscosity for the molecular viscosity; according to the theory of 

 wind currents, this eddy viscosity is about a million times greater than the 

 molecular viscosity. It is not known whether the motion of the water on their 

 orbits is turbulent. However, we have to put the coefficient for eddy viscosity 

 for a wave motion many times smaller than for wind currents, because the 

 decrease of the angular velocity of the particles with increasing depth would 

 become too large, contrary to the observations. It is probable that the de- 

 crease of wave height is mainly caused by the air resistance encountered by 

 the travelling wave. 

 (e) Investigations of the Wind Fields above the Waves 



So far such systematical investigations, which are important in the study 

 of wave formation, have not been conducted. Whatever is known about 

 it is based on direct visual observations and occasional experiments. The 

 lower layer of air flowing over the water is slowed down by the water surface 

 which causes a vertical velocity gradient. The upper layer of air moving more 

 rapidly creates a sucking action upon the lower layer of air and also on the 

 water surface. The pressure exercised by the air on the water will be distributed 

 irregularly, depending on the turbulence of the current at the surface. The 

 consequence of this is an upward and downward motion of the water. Wind gusts 

 and internal turbulence in the air create variations in pressure in points lying 

 near each other in space, and these pressure differences cause the initial forma- 

 tion of waves. As shown some time ago by Schmidt (1934, p. 57) there are 

 variations in the vertical component of the air motion, hurling small air 

 bubbles like missiles, up and down. The water surface immediately follows 

 these variations of pressure and wave formation sets in. 



