between wave height and duration of the form 
Hn, 
HT 
(81) 
where H,, is the maximum wave height for any 
given wind velocity, and v a coefficient which 
must be determined from observations. Although 
a close comparison with the relations given in this 
report is not possible, the chief features of (81) are 
in agreement with the present theory. 
6. Wave height and wind velocity.—According to 
equation (72) the wave height equals 
H= Use" 
and depends, therefore, not only upon wind 
velocity but also upon fetch and duration, since 
6 and £6 are functions of these variables. The 
maximum wave height is found by setting B=8,, 
6=5,, (p. 18) and therefore depends upon the wind 
velocity only: 
_0.26 
Hn 
9 
UV? (82) 
Equation (82) is a dimensionally correct 
expression and is in fair agreement with the 
equation 
Ha=22U? (83) 
g 
proposed by Rossby and Montgomery (1935) 
from entirely different considerations. 
Linear relations between wind velocity and the 
general (not the maximum) wave height have 
been proposed by Cornish (1934) and Zimmerman 
(Patton and Marmer, 1932). These relations, 
together with equation (82) (marked “ o’’) and 
(83), are shown in figure 10. Several tines of 
equal fetch and equal duration have been com- 
puted according to the nondimensional relations 
of this paper and are also shown in figure 10. 
The points represent the 128 observations of 
table I, appendix 2, which were plotted on figure 
5. Where observations were too crowded for 
individual plotting they have been combined and 
are shown by large circles with the number of 
observations indicated in the center. 
Equation (82) gives a fair indication of the 
maximum wave heights but the two linear rela- 
tionships, which relate wave height to wind 
velocity only, do not fit well. Some measure of 
success of these linear relationships may be 
explained by the fact that the authors have been 
concerned mainly with the highest waves observed 
WAVE HEIGHT AGAINST WIND VELOCITY 
COMPARISON BETWEEN THEORY AND OBSERVATION 
LEGEND 
THEORY 
UNLIMITED FETCH B 
UNLIMITED DURATION 
+ — LiMiTEO BY FETCH 
++ —= LIMITED BY DURATION 
1200 
EMPIRICAL RELATIONS 
— —_ TiMMERMANN 
———— CORNISH 
— — — _ 205587 & MONTGOMERY 
WAVE HEIGHT, H, IN CM 
fe) 
1000 1500 2000 2500 
WIND VELOCITY, U, IN CM/SEC 
3000 
Figure 10.—Wave height as function of wind velocity only. 
at different wind velocities. These wave heights 
will be too high at low wind velocities when it is 
likely that waves are present which were previously 
generated by stronger winds or which come from 
nearby areas. On the other hand, observed 
wave heights will be too low at very high wind 
. velocities, where the limitations imposed by the 
fetch and the duration may not have permitted a 
full development of the wave. 
7. Wave velocity and wind velocity—This re- 
lationship is shown in figure 11, the legend of which 
corresponds closely to that of figure 10. The line 
marked infinity follows from the definition of 
Bu=Cn/U (p. 17). The same 128 sets of observa- 
tions have again been entered. Three empirical 
laws, shown by the thin dashed curves, give the 
empirical relationship between velocity of the 
largest waves and the wind velocity. According to 
Zimmermann (Patton and Marmer, 1932), C= 
2.35 U?*, and hence C exceeds U for wind veloci- 
ties above 1331 cm/sec, but according to Cornish 
(1934), C=0.8 U, and to Schott (1893), C= 
0.76 U, that is, the wave velocity is less than the 
wind velocity. Observations in figure 8, however, 
show that the wave velocity frequently exceeds 
that of the wind. 
