Energy Transfer by Tangential Stress 
Jeffreys did not take into account a transfer of 
energy by tangential stress because he considered 
this process as negligible compared to the transfer 
by normal pressure, but the following considera- 
tions show that tangential stress cannot be 
neglected. 
The average rate at which energy is transmitted 
to the wave by tangential stress equals 
WL 
Bo=z | “rude (33) 
0 
where wu, denotes the horizontal component of 
particle velocity at the sea surface and where r is 
the stress which the wind exerts on the sea 
surface. 
At wind velocities above 500 cm/sec the stress 
of the wind equals (Rossby, 1936) 
T=Y'p" G2} 
where p’ is the density of the air, U is the wind 
velocity at a height of 8 to 10 m and 7’ is the 
resistance coefficient. 
Various types of observations have consistently 
led to the value 
y?=2.6 10-3 (34b) 
provided that the wave velocity does not differ too 
much from the wind velocity. If this condition is 
not fulfilled the value of y? is probably greater. 
Introducing (34a) in (83), assuming 7 to be in- 
dependent of x: 
(34a) 
Rr=e'U? [pute (35) 
For waves of small amplitude 
Up= 75C sin k(x— Ct) 
and the integral in (35) vanishes. This, appar- 
ently, led Jeffreys to assume that energy trans- 
mitted by tangential stress does not play an 
important part in the generation of waves. For 
Stokes’ waves of finite amplitude, which are 
accompanied by mass transport (10), the integral 
in (35) has the value 
Ww p= 7'5C (36) 
and, therefore, 
Rr=y'r'p'* CU? (for U>500 cm/sec) (37) 
It would be more correct to write (U—u’,)? 
for U? in equation (37), since stress is caused by 
the wind velocity relative to the water surface, 
but since wu’) is small compared to U, (37) gives a 
satisfactory approximation. 
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11 
The energy of waves can increase only if (Ry+ 
R,) the rate at which energy is added by both 
normal and tangential stresses of the wind exceeds 
Ru, the rate at which energy is dissipated by 
viscosity, or if, according to (7), (30), and (37), 
sp’ (U—C)?C+ 279’ WC >4ug (38) 
where + refers to CCU. 
Equation (38) now takes the place of (31), the 
Jetfreys criterion for the growth of waves. Accord- 
ing to the latter, waves cannot attain velocities 
exceeding the wind velocity, but equation (38) 
does not place this restriction upon the develop- 
ment. Take, for example, C=U and let U—500 
cm/sec, the lowest wind velocity for which (34) 
and hence (37) are valid. Then sp’ (U—C)? C=0, 
2y?p’ U?C=812, and 4ug=71; hence, according to 
(38), waves continue to grow even after their 
velocities exceed that of the wind, in agreement 
with observations. Since it must be assumed 
that the wave velocity increases the longer the 
waves travel, the ratio B=C/U will indicate the 
state of development of the wave and can appro- 
priately be considered a parameter which describes 
the age of the wave. 
Equation (38) is valid for U>500 cm/sec only 
and therefore cannot be applied to the problem 
of the first‘formation of waves, which takes place 
when U is about 100 cm/sec. At wind velocities 
less than about 500 cm/sec the sea surface is hydro- 
dynamically smooth (Rossby, 1936) and the rela- 
tion between the stress and the wind velocity 
differs from that expressed by' (34a). The problem 
of the initial formation of waves must therefore 
be approached in a different manner. It deserves 
further attention but lies outside the scope of this 
paper which deals with the growth of wind waves 
at wind velocities above 500 cm/sec. 
It is of interest to compare the accuracy of Rr 
and Ry, as defined by equations (37) and (80). 
Ry depends mainly upon the accuracy with which 
u’ and 7? (17b) are known. The expression for 
wu’ has been checked experimentally and found 
to be in good agreement with theory. The 
numerical value of the resistance coefficient y’ has 
been arrived at by several different methods but 
is, as already stated, applicable only to wind 
velocities exceeding 5 m/sec, for which the sea 
surface can be considered hydrodynamically 
rough. Otherwise it is independent of wind 
velocity. One might estimate roughly that R; can 
be obtained from equation (37) with an accuracy 
of +25 percent. The accuracy to which Ry can 
