taking into account that =H/L=2rH/gT’, one 
obtains: 
4r 
Dera dp rte |. 
eee =) | (93) 
to eae 
peer 60 
vy tTta 
1 il op Sr-ta 
Gib. Qa op Be? 2) (9) 
Even in this form the equations are not satis- 
factory because it is impossible to eliminate all 
the unknowns. The parameters 6; and 5p-=f(6r) 
appear on the right-hand side of the equations 
which relate decay distance, travel time, and wind 
velocity to the observed quantities Hp and Tp, 
and the unknown quantity By. 
In order to use equations (93) to (95) it is 
necessary to assume values of By. This can be 
done by assuming a relationship between wind 
velocity and duration based on the common 
experience that high winds are usually of short 
duration while weaker winds may blow for a 
long time. Such a relationship determines Bp 
because when ¢ and U are known the parameter 
gt/U can be computed and @; read from figure 7. 
In figure 13 two specific relationships are shown 
in the inset, and in the upper and the lower parts 
of the figure the corresponding values of D, tp, 
and U are represented as functions of the height 
and period of the swell, Hp and T>. 
A comparison of values read off from the two 
parts of figure 13 reveals that long period, high 
swell can occur only if the wind duration in the 
generating area has been long. Thus, a wind of 
velocity 20 m/sec and duration 24 hours (curve 
A, inset) gives a swell of period 16 seconds and 
height 2 feet at a distance of 2,800 km. (upper 
part of fig. 13), but if the duration were only 12 
hours (curve B, inset) the swell at a distance of 
2,800 km. would have a period of 14.8 seconds 
and a height of only 0.9 feet. It is also found 
that for given values of Hp and T> conclusions 
as to distance of decay and travel time can be 
drawn with greater certainty than conclusions as 
to wind velocity in the generating area. 
The Energy Front 
The preceding discussion deals with steady 
state conditions in the area of decay. At any 
given locality in the area of decay significant 
29 
waves will attain their maximum steady state 
height after waves have been emitted from the area 
of generation for a long time. Wave heights will 
be lower at an earlier time, when the waves ad- 
vance into an area of decay which is relatively 
undisturbed. The time required to reach, for 
example, 50 or 90 percent of the wave height 
corresponding to steady state conditions must be 
found from a study of the transient state. 
Consider the fundamental equations (13) or 
(48a) as applied to the simple case of waves in 
vacuum, traveling with constant wave velocity in 
a norviscous fluid: 
OH , 0 
ny Syn on a 0 
(18) 
This hypothetical case is dealt with in an earlier 
section of this paper (p. 6). Equation (27) 
which is a solution of (13) gives the wave height 
at any given time and distance from the generating 
area with good approximation. According to (27) 
the wave height increases from a minute percent- 
age to very nearly 100 percent of its maximum 
value within a very short distance (fig. 3). The 
“‘region of sharp increase in wave height” has at 
any instant traveled only half as far as the leading 
wave; its velocity is half the velocity of the leading 
wave. For purposes of forecasting, therefore, the 
rate at which the disturbance advances in the area 
of calm should be taken at one-half the wave 
velocity, since waves further advanced than the 
center wave are very low. 
The following numerical example will serve as 
illustration. Let waves of 10-m. height and 12- 
sec. period be generated in a storm area 4,000 km. 
from the point of observation. According to 
figure 12 the wave height at 4,000 km. will have 
decreased to 270 cm., the wave period will equal 
19.5 sec., and the “‘center wave” will arrive about 
90 hours after the first waves left the generating 
area. According to equation (27) a height equal 
to 10 percent of the steady-state wave height, or 
27 cm., will be attained 295 wave periods, or 96 
minutes, before the arrival of the central wave; 
a height of 242 cm., 90 percent of the steady-state 
wave height, will be attained 112 periods or 36 
minutes after the arrival of the “center wave.” 
Therefore, the wave height increases from 16 to 
90 percent of its maximum value in 96+36 min- 
utes, or in about 2% hours, while it took 90 hours 
for the “‘center wave’’ to arrive. 
So far, a hypothetical case has been treated for 
which a sudden increase from zero to full wave 
