Generation, Growth and Propagation of Waves 83 



(/) The Action of Normal Pressure Forces and of the Tangential Stress of the 

 Wind on the Waves 



All waves, once generated, suffer a loss in intensity coupled with decreased 

 wave height through viscosity and through air resistance. To generate and 

 maintain waves on the surface, it is necessary to supply energy from the 

 outside, mainly by wind blowing over the water surface. The question as 

 to how waves are generated and grow can be brought back to what the 

 action of the wind is on the waves. Jeffreys (1925, p. 189; 1926, p. 241) 

 has made the first test of a theory of wave generation which goes beyond 

 the general description given by Lord Kelvin of the influence of an unsym- 

 metrical distribution of pressure on the wave formation. According to 

 Jeffreys (1925) the principal supplier of energy R N is the wind pressure 

 A , which he assumes to be proportional to the product of the square of 

 the relative wind velocity (U— r) 2 , the density of the air q 2 , and the slope 

 of the surface in the direction of the wind drj/dx. We thus have 



A P =SQIU-Cf d -^, 



wherein s is an unknown proportionality constant called by Jeffreys 

 "sheltering coefficient". As drj/dt is the elevation of the surface in unit time, 

 the pressure Ap performs in unit time over the area dxdy the work 



se 2 (U-cff x f t dxdy. 



For an area of the width b and of the length X we then obtain with the help 

 of (IV. 18) an equation for energy transferred to a wave by normal pressure. 



R N = l( S Q 2 )(U-cf>c 2 A 2 cbX . (IV. 25) 



(for c < U) . 



This equation (IV. 25) is for waves with a velocity smaller than the wind 

 velocity. If the latter exceeds the wave velocity and if the relative wind blows 

 in the direction opposite to the wave motion, a negative sign must be inserted 

 into the equation for Ap, which means that Ap is 180° out of phase with 

 the slope, and we obtain 



r n = -i( S Q 2 )(U-c) 2 x 2 A 2 c-bl (fore > U) . (IV. 25b) 



The loss of energy through viscosity is given by the equation (IV. 19). 

 Jeffreys assumes that a wave can only grow when the energy it receives R N is 

 greater than the energy dissipated by viscosity Rfi. Jeffreys' criterion for the 

 growth of deep water or surface waves is then 



W=#c*^k. (IV. 26) 



C SQ 2 



