Introducing the definitions of Rc, Rr, and Ry into 
(57b) gives 
dB 
ae — AG Wim Bimin bh + (1—8)?] (58a) 
dB 19-2 __ Q)2 
di =AgU Br [1 = (1— 8)? (58b) 
and by comparison with (55) leads to the following 
analytical relationship between 6 and B: 
ene. (a8) 
6)’ (59) 
Agreement between equation (59) and the 
empirical data in figure 5 for all but the youngest 
waves justifies the assumptions made regarding 
the “energy split-up.”” It would have been pos- 
sible to establish an equation involving 6 and 8 
by applying certain technique of curve fitting, 
but in view of the scatter of the observations it 
seemed preferable to find the relationship from 
physical assumptions and to use the observations 
to evaluate the constants. 
Equation (59) can be written: 
ding _1+a—2(r+o)B+(r+a)p’_ 2 
2 
SS 2rp?(2—B) gy OS6S1 
(60a) 
GIO fie Gate) (Bel) 2 2 
dp 2rgll+(@—1))) + 8B’ 1<8 (60b) 
whose integrals are 
Opie Lone 1—b_ite 
i 4r iB In(2— Bg) + 
5} 20r—1 1 
ot In 2 0<sX1 (61a) 
ino = —2 EET ing —*E* Inf + (8—1)"1+ 
1 T 
se 
AS tan-"(6—1), 1<8 (61b) 
where 6, refers to the wave steepness when 
Bl 
According to (61a) 60 for B=0. This is not 
in agreement with experience because very young 
waves are known to have considerable steepness 
and even the smallest waves generated by wind 
gusts have finite velocities and heights (equation 
32c). For that reason it will be assumed that 
ding _ (62) 
ap B=<p" 
Ind=1nd,+ ms (63) 
where 6) and m are constants to be determined by 
the conditions that 6 and d6/d8 must be continuous 
at B=’. Since 6 cannot be zero, the logarithmic 
curve should not be drawn all the way to the 
y-axis but the initial development of the wave is 
so rapid that the exact form of the logarithmic 
relationship is of very little consequence to the 
later development of the wave. 
Evaluation of Numerical Constants 
The logarithmic relationship (62) is assumed 
applicable from B=0, 5=6 to B=6’, 6=6’. Assum- 
ing continuity in slope at §’, equations (60a) and 
(62) give 
ele (tea) Bats (eae) 88 _ 2 
while if 6 is to be continuous 
Ini, =Ins, —* E21 F EES nop) + 
3 20r—1 1 
sat oornt In ame" (64b) 
according to (61a) and (68). 
The steepness reaches a maximum, to be denoted 
by 6’’, for B=£’’, which is found by setting 
diné/d8=0 in (60a): 
etal: Pil 
ioe Vx 
The corresponding value of 6’’ is given by (61a) 
for B=6’’. At B=1, 6=6, by definition. 
The wave height reaches a maximum, H,,, for 
B=Bm, where according to (52) 
(64c) 
dé 
dpuce a? 
and, from ce 
1—r = 
(64d) 
The corresponding value of the steepness, 6,,, can 
be determined from (61b) by substituting B=6n. 
In choosing numerical values for the constants, 
particular attention was paid to the point 6’’, B’’. 
The theoretical maximum value of 5 is ¥ (9b) but 
according to Kriimmel, British reports, and the 
data in figure 5, 6 does not exceed %o. The value 
of B’’ must be approximately 0.4 and that of 6, 
0.04. These considerations are sufficient to de- 
termine the chief constants, r, «, and 6. The 
fourth constant, 6’, was chosen to give 5) approxi- 
mately 0.05, but its value is only of secondary 
importance. 
