In this manner the values of the constants can 
be determined within fairly narrow limits. In 
choosing the exact values adjustments were made 
on the basis of other empirical relationships, 
particularly the ones dealing with the decay of 
waves, but by far the greatest emphasis has been 
placed on the relationship between wave steepness 
and wave age. The curves in figure 5 are based 
upon the following exact values of the numerical 
constants: 
ol | Value Source Par | Value Source 
r 0. 580 (ee . 407 | Equation 64c. 
@ 2. 500 4 - 0990} Equation 6la. 
™ 1.627 | Equation 64a. 61 - 038 
50 - 0537} Equation 64b. Brn 1. 369 Equation 64d. 
gf’ 350 Sm . 0219} Equation 61b. 
2095 Equation 61a. 
Wave heights decrease for B>8,, but younger 
and shorter waves which must also be present 
will be most significant. For that reason the sec- 
tion of the curve to the right of 8,, is dotted and 
will be neglected in practical forecasting. 
The solid line in figure 5 is assumed to represent 
the relationship between significant wave steep- 
ness and age. Other relationships, such as wave 
height and velocity as functions of fetch or dura- 
tion, and the decay of waves, follow directly and 
can be compared to observational evidence as 
check of the validity of our assumptions. 
Wave Velocity, Wave Height, Fetch, and 
Duration 
The wave velocity will be expressed in non- 
dimensional form by means of the wave age, 
B= C/U 
(a): 0<BSp’ 
Substituting (62) in (55): 
d 1 — 8)? 
B= 2AgU-*p-* ee (65a) 
dp _ Sync 
Fa AG Wim Bae anteoT em (65b) 
with the solutions 
a afe+S B+ KiB+5 (2Kk,—K,K;). 
ae + K+ ‘m(® ete) 
(RR le tan 8] (66b) 
gt 
OF 
where 
K\=5- +2=3.536 
K,=2.4+3-+=5. 673 
K,=1+4=1.400 
K,=a(2K,—K,K;— K.K;) = —2.446 
are combinations of known constents. 
(b): 6’<pS1 
using the upper signs of (58): 
48 _9 AgrU-*6-2(2—B) 
G—AgrU-p2—p) (67a, b) 
with the solutions 
guna, a §-3 + 
f-Z[ms, g 2 |—Ks 
f=-[@—p)—2n@—p)|—K, (68a, b) 
where 
K,=3.539 X10? 
K,=1.705X 105 (69a, b) 
are constants of integration, chosen to make solu- 
tions (66) and (68) continuous at B= 8’. 
(jel p 
Using the lower signs of (58): . 
1B 9 AgrU-*p-U1 + (8-1) 
dz 
10 = AgrU~6-[1+(B—1)"] (70a, b) 
with the solutions — 
2. 
Fe= sa; (inlio(e*—26+2))1+5+ 
26—2 tan“ (B— 1)—5|—K, (71a) 
f= [6-+In(s—26+2)]—K, (7b) 
where K, and K, are the constants in equations 
(69). 
A graph of 6 against the nondimensional fetch 
parameter gz/U? (equations 66a, 68a, 71a) is 
shown in figure 6; against the non-dimensional 
duration parameter gt/U (equations 66b, 68b, 71b) 
in figure 7. The continuity of the solutions 
