2L, 3L,.... , but of these terms only the one 
in phase with w, does a net amount of work. 
Jeffreys assumes this term to be proportional to 
the product of the density of the air, the square 
of the wind velocity, and the slope of the surface: 
C)? Os C)*ka cos k(a—Ct) 
(29) 
and he calls the coefficient of proportionality, s, 
the ‘‘sheltering coefficient.’”’ It might be more 
descriptive to use the term ‘“‘streamlining coeffi- 
cient’’ because the coefficient is a measure of the 
form resistance offered by the wave. 
From (28a), (28b), and (29) it follows: 
Ap=sp' (U 
igs 
Re 58" (U—C)'aC for C<U_ (30a) 
Equation (30a) holds for waves traveling more 
slowly than the wind. Should the wave velocity 
exceed that of the wind, and the relative wind 
velocity be directed against the directionsof wave 
motion, then Ap is 180° out of phase with the slope 
of the wave, and the sign in equations (29) and 
(30a) must be reversed. Hence 
ieee a (U—C)k?eC for C>U  (80b) 
Jeffreys assumes Fy to be the only important 
source of wave energy. On this assumption the 
energy of a wave can increase only if Ry exceeds 
Ru, the rate at which energy is dissipated by 
viscosity, or if, according to (7) and (30), using 
(3a), 
sp’ (U—O)*C>4ug 
This is Jeffreys’ criterion for the growth of deep 
water surface waves. According to (31a), waves 
cannot grow if, approximately, 
Sy TEE 
C>U 4 ap 
and the wave velocity cannot exceed the wind 
velocity during the stage of growth. Jeffreys 
evaluated s from equation (31a) by noting that 
for a given wind velocity the term on the left- 
hand side is at a maximum when 
1 
(31a) 
(31b) 
The least wind which can maintain waves is, 
therefore, 
_3(4#9)3 
Una =3( £9 (32b) 
10: 
Observations by Jeffreys gave Umin approxi- 
mately 110 cm/sec. With this value and with 
p=0.018, g=980, and p’=1.25 107-3, one obtains 
SS O)e2ile 
The corresponding wave velocity is Umin/3, or 
about 35 cm/sec, the corresponding wave length is 
8 cm, and the wave period 
T min= 0.22 seconds (32c) 
Observations by others agree as to the order of 
magnitude of the least wind velocity and the 
smallest wave length. 
The mechanism described by Jeffreys appears, 
therefore, to give a satisfactory explanation for 
the initial formation of waves. However, it is 
not possible to apply Jeffreys’ concept and his 
numerical value of s when studying the growth of 
waves after their initial formation because the 
observed increase in wave height indicates that the 
transfer of wind energy is only about one-tenth of 
that demand by Jeffreys. This conclusion is sub- 
stantiated by experiments conducted with small 
wooden models of waves placed in a wind tunnel 
(Stanton, 1937). The pressure distribution along 
the wave profile was measured and the amplitude 
of the component in phase with w, (28b) deter- 
mined from a harmonic analysis of the pressure 
distribution. The sheltering coefficient can be 
evaluated from these measurements, using equa- 
tion (29), 
ApL 
Ue 
The results are summarized in table 3. 
Table 3 
Determination of Sheltering Coefficient From 
Experiments by Sir Thomas Stanton (1937) 
= mak 
F - Wave | Wave Wind 
Te oy ie tunnel length | height | velocity 8 
he (em) | (cm) | (cm/sec) 
- 055 325 0. 036 
- 55 470 - 047 
peat 330 - 068 
11 580 . 090 
0.75 1, 400 - 006 
Zoe c S| Eee eae - 049 
Although the measurements were subject to 
large experimental errors as indicated by the wide 
variation in s, the average shows clearly that 
Jeffreys’ value of 0.27 is*too high. 
