throughout the entire range is apparent, but in 
accordance with the assumption that the character 
of the sea surface is determined by the highest 
waves present, 8 remains constant after reaching 
a value of 6, (64). Comparison with observa- 
tions, which are indicated by points, will be made 
later. 
Wave height will be represented by the nen- 
dimensional parameter gH/U*. According to 
(52): 
NEE, ; t 
a= 2x08 =i( Fr or (G) 
because 5=/(@) (fig. 5) and B=f(gx/U”) (fig. 6) or 
B=f(gt/U) (fig. 7). The wave heights are plotted 
nondimensionally in figures 6 and 7. Comparison 
with observations will be made later. 
(72) 
General Case 
So far, the growth of wind waves has been 
examined in two special cases; the growth with 
fetch, assuming constant wind of unlimited dura- 
tion, and the growth in time, assuming a constant 
wind blowing over an unlimited fetch. Under 
actual conditions both fetch and duration are 
limited, and for any given situation the wave 
height and age from the fetch graph (fig. 6) will 
differ, in general, from height and age from the 
duration graph (fig. 7). The smaller of the two 
values is considered valid for the following 
reasons. 
If a wind of constant speed has blown for many 
hours over a small lake the height of the waves 
depends entirely upon the distance from the up- 
wind shore. If, on the other hand, a wind has 
blown for just a few hours over a fetch of several 
thousand miles the limitation of the fetch can be 
of no consequence and the wave height must de- 
pend upon the duration only. Therefore, for any 
given wind velocity the wave height is determined 
by either the fetch (fig. 6) or the duration (fig. 7), 
depending upon which of the two factors imposes 
the greater limitation to the full development of 
the waves. This can also be stated by saying 
that for any given fetch there exists a “‘minimum” 
duration, ta, for which the fetch and duration 
graphs give the same wave height and age. If 
the duration‘is less than tyi,, the waves are de- 
termined from the duration graph; if the duration 
is greater than tm, the waves are determined 
from the fetch graph. 
The minimum duration can be expressed by 
means of the nondimensional parameter ty,,U/z 
777333 O - 48 - 4 
19 
which is found by reading off corresponding values 
of gz/U* and gt/U for various values of 6 from 
figures 6 and 7, and dividing: 
tminU (gt | gx 
er =(¢/ t), 
As a numerical example, assume that a 20 
m/sec wind blows in an offshore direction. When 
the wind first starts to blow, :=0 and H=O over 
the entire fetch (fig. 8A). Five hours later, ac- 
cording to the duration graph, the wave height 
would equal 4 meters as shown by the straight 
line in figure 8B, but in the immediate vicinity of 
the coast the waves will be lower, and at the very 
beginning of the fetch, at s=0, the wave height 
remains zero. The point marked “P” is placed 
at the distance from the coast at which a sieady 
state has been established. To the left, upwind 
from point P, the significant wave height depends 
upon the fetch only. To the right of P the sig- 
nificant wave height is constant at any given time 
and its value depends only on the duration of the 
wind. 
As the wind continues to blow, the point P 
travels downwind. For t=5 hours P lies 60 km. 
from the beginning of the fetch (fig. 8B). For 
t=24 hours P lies 500 km. from the beginning of 
the fetch (fig. 8C), and after 130 hours when P 
has advanced 4,500 km. the waves have attained 
the maximum height and the wave height is 
everywhere determined by the fetch graph alone 
(fig. 8D). A similar reasoning applies to the 
wave velocity or the wave age. 
Let z and ¢ be fetch and duration for a chosen 
value of 8, and x-+dz, t+dt, correspond to 6+d8. 
Then the rate at which the influence of the limit- 
ing fetch advances, that is, the velocity of the 
point P, equals 
(73) 
dx__ dg/dt_C 
dt dB/dx 2 
according to equations (55a, b). Hence the region 
in which the limiting fetch is dominant expands 
at group velocity. The same conclusion was 
reached by Commander Suthons (verbal commun- 
ication). 
When developing the theory of the growth of 
significant waves, the wind velocity in the gener- 
ating area has been assumed to be constant in 
time and space. The limited experience gained so 
far indicates that satisfactory forecasts can be 
made on this assumption, especially for wind sys- 
tems separated by well-defined fronts. It might 
(74) 
