Generation, Growth and Propagation of Waves 85 



pressures exercised by the wind on the wave surface cannot by themselves 

 account for the generation and the growth of waves, especially as a trans- 

 mission of energy from the air to the water can only take place when the 

 wind velocity exceeds the wave velocity. 



Another hypothesis for the transfer of energy consists in that a tangential 

 stress is exercised on the water surface through the boundary friction between 

 water and air. A computation of Jeffreys showed that, when we take the 

 tangential stresses proportional to the square of the relative velocity between 

 water and air, a minimum wind velocity of 480 cm/sec would be required 

 to generate waves on a water surface, and these first waves should have 

 a length of 140 cm. This value, however, is contradictory to observations, 

 for which reason Jeffreys was of the opinion that the tangential stresses of 

 the wind can hardly be accepted as the initial cause of wave formation. The 

 average rate at which energy is transmitted to a wave by tangential stress 



equals 



x 



R T = i j ru dx, (IV. 29) 



b 



if « is the horizontal component of particle velocity in its orbit at the sea 

 surface and where x is the stress which the wind exerts on the sea surface. 

 According to Rossby (1936), the stress of the wind with wind velocities 

 of above 500 cm/sec is 



r = k-2 Q2 U 2 , (IV. 30) 



q 2 being the density of the air, U the wind velocity at a height of 8—10 m and 

 k 2 = 00026 the resistance coefficient (see, Physical Oceanography vol. 1). This 

 value k 2 is valid provided that the difference between the wave velocity and 

 the wind velocity is not too great. If this condition is not fulfilled, k 2 should 

 probably be increased. Introducing equation (IV. 30) in equation (IV. 29) 

 assuming r to be independent of x: 



r t = k 2 QoU 2 J u dx . (IV. 30a) 







For waves of small amplitude (see II. 9) 



u = — ;-H = g — sm(kx—ct) — ndcsink(x — ct) . 

 \ ox/ z =o a 



d = i//A = 2A\X is the average slope or steepness of the wave, and the 

 integral in equation (IV. 30a) vanishes. These seem to be the main reason 

 why Jeffreys assumed that the tangential stress of the wind is of no importance 

 for the generation of waves. Sverdrup and Munk (1947), however, draw 

 the attention to the fact that for Stokes's waves of finite amplitude (see 

 p. 26) accompanied by mass transport u = n 2 d 2 ce~ 27lz the value for 



