UNIT CREST —— DIRECT! 
wi 
SURFACE 
Se 
ON OF WAVE MOTION 
Figure 4.—Energy changes of an individual wave of length L traveling from left to right with velocity C. 
With this condition (45) takes the form 
dE , EdC 
Git O deen (47) 
which will be further examined. It is not imme- 
diately apparent that this equation can be applied 
to nonconservative systems, but when the con- 
stants in the, solution are evaluated for the sig- 
’ nificant waves the results check with observations. 
Integration of (47) gives the change with time 
of the significant waves at any locality in the 
storm area. Jnitially the significant waves will 
have originated in the immediate neighborhood 
of the locality considered. As the time increases, 
the waves reaching this locality will have travelled 
a longer time and originated at larger distances. 
In practice the distance from which waves can 
come is limited by the dimensions of the storm 
system or by a shoreline. This distance is called 
the fetch. 
The time necessary for the waves to travel from 
the beginning of the fetch to the locality in ques- 
tion is called the minimum duration, tny,. If the 
duration of the wind exceeds tm, the character 
of the significant waves that are present in the 
fetch remains constant in time: a steady state 
is established. 
To examine the steady state, consider a parallel- 
epiped fixed in space of unit width and length 6z, 
but otherwise similar to the one considered above. 
Since the parallelepiped is fixed in space, potential 
energy flows into the volume at the rear edge at 
14 
the rate CH/2 and leaves at the forward edge at 
the rate 
IO fe) 
0f+Z(0F i 
The local change in energy must equal the sum of 
the amounts which enter or leave the parallele- 
piped: 
OL fa) E™ 
fan éx= — (05 Jat (Rr+Ry) bx 
or, rearranging, 
oLE CoE, Ed 
Dio On OnOn 
=R,7+Ry (48a) 
This equation corresponds to (45) and applies to 
conservative waves.. In order to apply it to the 
significant waves which are present over a limited 
fetch after a steady state has been reached, we 
write 
of 
Sine 
and obtain 
Ode oY Ry tRy (48b) 
This equation is equivalent to (47) and the com- 
ments on the applicability of (47) apply. 
The solution of equation (48b) gives the height 
and velocity as function of fetch after a steady 
state has been reached, that is, fort=tam. It will 
