For the center wave (m=450\) the agreement 
is exact, but for all other waves there are found 
small differences. The approximation is satis- 
factory. 
Table 2 
Degree of Approximation Involved in Equa- 
tion (27) 
(1) (2) (3) (4) (5) (6) 
OR wR 19R oR WR 
He & ou om 29m on + 29m 
300 | 1.0000 | —.0000 | +0.00x10-#] —0.00%10-4| —0. 0010-4 
400 | .9996 | —.0014] +0.44X10-4] —0. 4710-4] —0. 0310-4 
420 | .9788 | —.0508 | +16.36x10-4| —16. 9310-4] —0. 57%<10-4 
430 | .9147 | —.1561 | +50.84Xx10-4] —52.03x10-4) —1. 1910-4 
435 | 18485 | —.2347 | +76.89X10-4| —78. 2310-1] —1. 3410-4 
440 | .7500 | —.3123 | +102. 87X10-4| —104. 1010-4] —1. 2310-4 
445 | .6430 | —.3730 | +123. 57x 10-4 | —124. 3310-4] —0. 7610-4 
447 | | 5020 | —. 3882 | +128.90X10-# | —129. 4010-4] —0.50%10-4 
449 | .5398 | —. 3970 | +132. 1110-4] —132. 3310-4] —0. 2210-4 
450 | .5033 | —. 3088 | +132. 8610-4 | —132. 9310-4 | —0.07%10-4 
45034]. 5000 | —. 3089 | +132. 9810-4] —132.98%10-# | —0. 0010-4 
451 | .4967 | —.3988 | +132. 8610-4 | —132. 9310-4} —0. 0710-4 
+0. 0010-4 —0. 0010-4 | —0.00X10-4 
Significance of the Solution 
Table 2 also illustrates an important point: 
within a very short distance the ratio R decreases 
from very nearly 100 percent to a minute percent- 
age. In figure 3, & has been plotted against m, 
the distance in wave lengths from the generating 
area. The scale to the left gives percent of max- 
imum wave energy, the scale to the right gives 
percent of maximum wave height; the height 
ratio equals the square root of the energy ratio. 
The center wave is halfway between the wave 
furthest advanced and the one at the very rear. 
Its energy is exactly one-half the maximum energy, 
its height 70.7 percent of the maximum height. 
In this example the total waves present number 
900, hence the center wave is 450.5 wave lengths 
from the generating area. Let the “region of 
sharp decrease in wave height,” the shaded portion 
of the figure, be defined as the region within which 
the wave heights decrease from 90 to 10 percent 
of their maximum value. These two limits cor- 
respond to m=485, and m=435, and the area of 
sharp wave decrease is only 50 wave lengths wide. 
The “region of sharp decrease in wave height’ 
has at any instant traveled only half as far as the 
leading wave; its velocity is half the velocity of 
REGION OF 
SHARP INCREASE 
IN WAVE HEIGHT 
i 
CHANGE OF WAVE HEIGHT 
WITH DISTANCE FROM 
GENERATING AREA 
AAAMTANAUANNN IN 
KON 
\ 
WAVE FURTHEST 
ADVANCED 
WN 
) 
w 
wa 
~ 
° 
= 
wi 
z 
wi 
w 
> 
< 
Ef 
2 
=) 
= 
x 
< 
= 
uw 
° 
w 
RK 
100 200 360 400 $00 600 700 6800 
M IN WAVE LENGTHS 
Figure 3.—Change of wave height with distance from source re- 
gion, assuming that uniform waves produced at the source region 
advance through water originally at rest. Deep water waves. 
the leading wave. Therein lies the answer to a 
controversial question: how fast does a disturbance 
travel from the storm area, at wave velocity or at 
half the wave velocity? The answer is that each 
wave travels with its own wave velocity and arrives 
at a distance D from the storm area at a time 
t’= D/C, where C denotes the mean wave velocity. 
However, these early waves are extremely low. 
From a practical point of view appreciable waves 
will appear at D only slightly before 2t’, the time 
of arrival of the “center wave.” Hence for pur- 
poses of forecasting, the velocity of the disturbance 
should be taken at one-half the wave velocity. 
ENERGY TRANSFER FROM WIND TO WAVES 
Energy Transfer by Normal Pressure 
The average rate at which energy is transferred 
to a wave by normal pressure equals 
1(2 
Ry=7 : Po2W old (28a) 
where 
Wo= —kaC cos k(x— Ct) (28b) 
is the vertical component of the particle velocity 
at the surface, and p,,:the normal tension acting 
on the sea surface. 
When a wind blows, the tension p,,, which is 
directed opposite to the pressure exerted on the 
sea surface, can, according to Jeffreys (Jeffreys, 
1925 and Lamb, 1932, p. 625), be taken as the 
sum of two negative terms, one representing the 
constant atmospheric pressure and the other, the 
wind pressure against different portions of the 
wave: 
Pez=—p—Ap 
The wind pressure can be considered composed 
of a series of harmonic terms of wave lengths L, 
a 
