86 Generation, Growth and Propagation of Waves 



for the surface is u = n 2 d 2 c. One obtains for the transfer of energy by 

 tangential stress 



R T = k 2 7i 2 Q 2 d 2 cU 2 (U > 500 cm/sec) . (IV. 31) 



If we take this transfer of energy also into account, the energy of waves 

 can increase only if R N +R Ti the rate at which energy is added by both normal 

 and tangential stresses of the wind, exceeds R^, the rate at which energy 

 is dissipated by viscosity, or when 



±sq 2 (U- cfc + 2k 2 g 2 U 2 c > 4]ug , (IV.32) 



in which + refers to c < U. This equation replaces the Jeffreys criterion 

 (IV. 27) for the growth of waves. This cancels the conclusion drawn from 

 the Jeffreys criterion that the waves cannot attain a velocity exceeding the 

 wind velocity. According to (IV.32), waves can go on growing even after 

 their velocity exceeds that of the wind, which is in agreement with observations. 

 This is then an action of the tangential stress which accelerates the motion 

 of the particles in the wave crest and slows it down in the wave trough, which 

 equals an increase of energy, just when the wave moves faster than the wind. 

 Since it must be assumed that the wave velocity increases the longer the 

 wave travels, the ratio (3 = c/U will indicate the state of development of 

 the wave and can appropriately be considered a parameter which describes 

 the age of the wave. 



Equation (IV.32) is valid only for U > 500 cm/sec and, therefore, cannot 

 be applied to the problem of the first formation of waves which takes place, 

 when U is approximately 100 cm/sec. At wind velocities less than 5C»0 cm/sec 

 the sea surface is hydrodynamically smooth according to Rossby (1936), 

 and the relation between the stress and the wind velocity differs from that 

 expressed in (IV. 30). 



The effect of molecular viscosity is always small compared to the wind 

 effect; thus, Sverdrup and Munk show that: 



for U= 500 cm/sec /S=0-1, and c= 50 cm/sec respectively, 

 for U = 1000 cm/sec /? =01, and c = 100 cm/sec respectively, 



RJ(R T +R N ) becomes -0-296 and -0036 respectively. Consequently, for 

 all very small values of /? and for moderate and large values of U, R^ is 

 small compared to R T -\-R N and can be neglected when dealing with the 

 growth of waves. 

 (g) New Considerations and Measurements 



The arrangements given in the preceding sections on influences of wind 

 pressure upon the water and on the boundary friction between air and water 

 have recently been improved. W. Wust (1949) has applied these arrangements 



