into equations (55) may make it possible to solve 
for H=H (z,t). 
2. Determine a relationship, 5=f (8), between 
the dependent variables from empirical evidence. 
The latter procedure has the disadvantage of 
being semiempirical, but it is the only one found 
possible so far. 
Observations of Wave Age and Wave Steepness 
It has been assumed that 6 is a function of 6 
only. Such a relationship has already been sug- 
gested by Kriimmel (1911, p. 82) who writes that 
“the ratio of wave length to wave height depends 
upon the stage of development of the waves; in 
case of a ‘young’ sea the ratio L/H (1/5) equals 
10 or may be even smaller, but this ratio increases 
for a more advanced stage. * * *” 
In the past many fruitless attempts have been 
made to relate wave steepness to wind velocity or 
other variables, but it has not been attempted to 
relate steepness, H/L, to wave age, C/U. Sucha 
relationship is suggested by dimensional consider- 
ations and, if existing, has the advantage of being 
independent of fetch and duration. The question 
can be examined by means of the 128 sets of ob- 
servations entered in table I of appendix II. The 
corresponding values of the two nondimensional 
parameters 6 and £ are plotted in figure 5, which 
clearly demonstrates that the two variables 
are related. 
The data have been collected from many 
different sources and from many localities, varying 
from a pond at Kensington Park, London, to the 
trade-wind belt of the North Altantic. Observa- 
tions listed under Gassenmayr, Officers United 
States Navy, Paris, and Schott were taken before 
the turn of the century. Many of these old obser- 
vations from small vessels are more reliable than 
recent observations. 
Wave data listed under Gibson, Berkeley, and 
Ehring represent values based upon a statistical 
analysis of instrumental records. Gibson recorded 
waves against a pole in Buzzards Bay by means 
of a moving-picture camera, and the wave charac- 
teristics entered in the table apply to the average 
of the one-third highest of 20 or 30 waves. The 
Berkeley observations were taken from a United 
States Navy Weather Patrol ship by means of an 
automatic pressure-recording instrument and the 
wave characteristics apply to the one-third highest 
of a small number of waves. The single point in 
figure 5, which is based on the observations dis- 
16 
cussed by Ehring, deserves particular attention. 
It gives the significant wave age and steepness 
derived from a statistical analysis of 579 waves 
from a 30-minute instrumental record in the North 
Sea. Observations marked U.S. 8S. Augusta were 
taken by a weather officer during the invasion of 
Normandy those marked H. M.S. Forrester during 
an Atlantic crossing in February 1944. Some 
casual observations at Dover prior to the invasion 
of the continent are also included. The observa- 
tions mentioned in this paragraph were obtained 
from unpublished reports. 
According to figure 5 the steepness reaches a max- 
imum value of about 0.10 as stated by Kriimmel 
and as has been mentioned in reports from Great 
Britain. The maximum steepness occurs for 8 
approximately 0.4, drops off somewhat for younger 
waves and diminishes rapidly for older waves. 
Derivation of an Equation for Wave Age and 
Steepness 
An empirical curve could be fitted to the 
empirical data shown in figure 5, and equations 
(55) and (54) could be integrated numerically. 
Instead, an analytical relationship between 5 and 
B will be chosen and the integration will be carried 
out analytically. The choice of the relationship 
can be guided by physical considerations. It is 
common experience that the wave period increases 
continuously* with z or ¢. Evidence will be given 
later but here it will be assumed that 
Re>0 (56) 
If it is assumed that the energy transmitted to 
the waves by tangential and normal pressures is 
divided in a certain fixed manner between the 
energy required to increase wave height (Ry) and 
wave velocity (fc), then either of equations (51) 
can be split up in the following manner: 
Ry=(1—)Rr£(1+2) By | Gae) 
Ro=rRr¥~ Rw (57b) 
where the signs are determined by (56). Addi- 
tion of equations (57a) and (57b) leads to (51). 
*Leonardo da Vinci was familiar with this when he wrote: “‘L’onda quanto 
piu si muove piu si abassa, e piu si dilata a piu fa veloce” (the further a wave 
moves, the more its height decreases and its length and velocity increase). 
