84 Generation, Growth and Propagation of Waves 



If we neglect the influence of the surface tension, then equation (IV. 9) 

 is applicable to c and the equation (IV. 26) becomes 



(U- cfc ;> 4 M fo . ^Z£l . (IV. 27) 



■S £?2 £l + £?2 



The smallest wind velocity (7 which is capable of generating and maintain- 

 ing waves is the one which gives a maximum in the left hand term of equa- 

 tion (IV. 27). It occurs when c = \U and, in this case, 



ULn = 27 -^ £ . ^ZJt (IV . 28) 



■y & ei 4- Q2 



and with (IV. 9) the corresponding wave length follows from 



3 _ 2 g el gi— g 2 



^min — 22" 1 



V'l go Pi T- g2 



It is not possible to test Jeffreys' suppositions by experiments, because 

 the value of the sheltering coefficient s is unknown. Jeffreys can only deduct 

 from his observations of the first waves generated by the wind (A = 8-10 cm 

 at a minimum wind velocity of 104-1 10 cm/sec, see p. 67) that s has plausible 

 values between 318 and 0-229, which he considers to confirm his theory. 

 For U^ = HO cm/sec and (i = 0018, g = 980 and g 2 = 1 25 x 10~ 8 one 

 obtains s = 0-27. The corresponding wave velocity it/ min is approximately 

 35 cm/sec and the wave period T mia = 0-22 sec, which would be in agreement 

 with observations. 



Motzfeld, in his laboratory experiments in a wind tunnel, has submitted 

 Jeffreys' theory to a test and was able to prove that the air resistance of the 

 waves in Jeffreys' hypothesis in respect to the pressure distribution leads to 

 values c d of a pressure resistance coefficient which do not agree as to the order of 

 magnitude with those derived from the laboratory tests. Therefore, he does 

 not find a satisfactory solution of the problem in the hypothesis of Jeffreys. 

 In Jeffreys' postulate the pressure coefficient has a value c d = sA 2 % 2 , which 

 is proportional to the square of the ratio A\l = d, the average slope of the 

 wave. Motzfeld, elaborating on Jeffreys' postulate, puts c d — sA"k n , in which 

 s and n are at first unknown, but which can be derived empirically from 

 laboratory experiments. He thus finds for the value of the coefficients in 

 the two first model tests n = 1-5 and s = 0014. The value of s, consequently, 

 is about 20 times smaller than the value claimed by Jeffreys. Stanton (1937, 

 p. 283) has obtained a similar result with his tests on small wooden models 

 of waves placed in the wind tunnel. The pressure distribution was measured 

 along the wave profile and allowed the numerical determination of the 

 sheltering coefficient s. It is of the order of magnitude of 005, i.e. about 

 5 to 10 times smaller than found by Jeffreys. It follows that the normal 



