Generation, Growth and Propagation of Waves 75 



pressure between B and T vanishes. Then there is equilibrium in all forces 

 present and the wall can be removed without changing the situation. We 

 have a water current with a stationary wave surface. For an observer who 

 is moving with the inner velocity of current to the right, we have surface 

 waves travelling to the left with a constant velocity, whereas the deeper water 

 is motionless. The process of stationary wave motion can be based in principle 

 on the fact that the statical and dynamical pressure differences within the 

 water masses are in equilibrium. See Einstein (1916, p. 509) and Cornish 

 (1934, p. 139, Notes of H. Jeffreys). 



We can see that, in this case, the air above the wavy water surface must 

 have the same velocity as the velocity of propagation of the waves (to the 

 left), in order that there be equilibrium. If this is not the case, the wave motion 

 is no longer in equilibrium. If the wind velocity is greater than the wave 

 velocity, then the negative pressure generated in O in Fig. 39 by the air 



to to 



Fig. 39. 



movement in a wave crest is greater than the excess pressure caused in U 

 by the water motion. Then there exists an excess pressure in the wave crests 

 on the side of the water, which tries to lift the water surface. The wave must 

 increase in height. If, on the contrary, the wind velocity is smaller than the 

 wave velocity, the wave must lose some of its amplitude, so that it approaches 

 again a stationary, stable condition. 



These fundamental considerations show that, each time there is the slightest 

 disturbance of the water surface by air in motion, the smooth sea surface 

 represents an unstable situation and the wavy water surface a stable situation 

 and that to each wind velocity at the sea surface corresponds a fixed, stationary 

 wave system. Helmholtz(1889, p. 761, 1890, p. 853) was the first to conclude, 

 from general viewpoints that the generation of waves is related to an unstable 

 condition of the smooth sea surface. It can be proven that a wavy surface 

 is in a stable, stationary condition, when in a system of two superposed layers 

 of water and air in motion the difference of the potential and kinetic energy 

 of the system is at a minimum. The determination of the wave form belonging 

 to this system is very difficult. Wien (1894, p. 509) has elaborated the thoughts 

 expressed by Helmholtz and has derived a number of theoretical wave profiles 

 which satisfy these conditions. The results obtained by him, however, are 



