The series number n gives the total number of 
waves present and equals the time in periods since 
the first wave entered the area of calm; the wave 
number m gives the position of the wave measured 
from the plunger and equals the distance from the 
plunger expressed in wave lengths. In any series, 
n, the deviation of the energy from the value #/2 is 
symmetrical about the center wave. Relative to 
the center wave all waves nearer the plunger show 
an excess of energy and all waves beyond the 
center wave show a deficit. For any two waves at 
equal distances from the center wave the excess 
equals the deficiency. In every series, n, the 
energy first decreases slowly with increasing dis- 
tance from the plunger, but in the vicinity of the 
center wave it decreases rapidly. Thus, there 
develops an “‘energy front’? which advances with 
the speed of the central part of the wave system, 
that is, with half the wave velocity. 
Table 1 
Distribution of Wave Heights:in a Short Train 
of Waves 
Series Wave number, m Total 
num- energy 
ber : of 
n 1 2 3 4 5 6 7 group 
1 1 — 1/2E 
2 3/4 1/279 CORO S | Pena eis |e | were wx | Feaey 2/2 
3 7 4/9 tis | G01) 97am | Roe een | ig Oe | cote aa 3/2 
4 1/15/16 | 11/16 5/16 1/16 E79 | Renee | Eas | ees 4/2 
5 | 31/32 | 26/32 | 16/32 GEya |W Erye |e eS 5/2 
6 |63/64 | 57/64 | 42/64 | 22/64 |7/64 | 1/64 |_-___- 6/2 
According to the last line in table 1 a définite 
pattern develops after a few strokes: the wave 
closest to the plunger has an energy E (2"—1)/2” 
which approaches the full amount E, the center 
wave has an energy E/2, and the wave which has 
traveled the greatest distance has very little 
energy (H/2"). 
Approximate Solution to Equation (15) 
Let "k,,="E,,E, where "E,, denotes the energy 
of the mth wave in a group of n waves. Then 
{r=2—-—m 1 
"Rn 22 
2" 26 ri(n—r)! (22) 
gives the value of any term in table 1. This table 
and, therefore, equation (22) are in agreement with 
observations. It remains to be shown that (22) 
satisfies the differential equation (15). 
Let t=nT, z=mL (23) 
where, as usual, 7 and LZ denote wave period and 
length. Then 
ok ,10R 
which can be written as the following difference 
equation 
CRS Ci oil) Sao 
(n+1)—(n) 
m—1 
3 (m)—Gm—1)~° 
or 
a) tad eT oe-(y (25) 
us 1) SO) Sip 
The general expression for "Rf, given by (22), 
satisfies equation (25). The proof, based on the 
method of mathematical induction, is given in 
appendix 1. 
Equation (22) is not practicable since we are 
dealing with such large distances that m and n 
have values up to 10*.. The process of summation 
would be very cumbersome even if tables of bi- 
nominal coefficients were available. From an 
analogy with the probability theory it is possible, 
however, to find approximate values of *#,, directly, 
no matter how large m andnhappentobe. Since 
the binominal coefficnt *C, is defined as 
yah ay OH 
CAG 
equation (22) can be written 
af T=2—-™ 
or rT=0 
The, binominal distribution can be closely 
approximated by the normal frequency distribu- 
tion curve because the binominal summation 
corresponds nearly to the area under the normal 
curve which is found in tables of the probability 
integral. From an analogy with the probability 
problem we can write at once 
"k= *C, (26) 
To find the ee of approximation the approxi- 
mate solution (27) is substituted directly in the 
differential equation (24). 
OR ORdu _ +( =) 2m+n— *) 
oe ns? 
10R_10R du 
20m 20udm — 
be “5 at and their differences for n= 900 
on Oudn 
7) 1 ) 
— — 2 — 
(= oe ( n 
10R 
are deed in columns 4-6 of table 2 
Values of 
