assumed (Havelock, 1914, p. 5). If the group 
velocity V’ is defined in such a manner that the 
wave length L does not vary in the neighborhood of 
a geometrical point traveling with velocity V’, 
dL ol, ob 
dt ot ' o& 
then equation (11a) is valid for any limited initial 
disturbance provided the wave crests maintain their 
identity (Lamb, 1932, p. 381). This initial dis- 
turbance can be expressed in terms of a Fourier 
integral and the resulting interference pattern 
derived as a function of time and distance. 
V’=0 (12) 
As a simple case, assume again that waves of 
constant height and length are emitted at the 
source and that their wave length remains constant 
thereafter. It might appear that under these as- 
sumptions only one single wave length is present 
and that group velocity is not involved. Since, 
however, the wave train is of finite size the wave 
length must either equal a constant value L’ 
(within the train), or be zero (outside tHe train) 
and this distribution can be represented by a 
spectrum of wave lengths in a Fourier integral. 
For long trains the various wave lengths in this 
spectrum cluster closely about L’ but the use of 
one single wave length is justified only in the case 
of an infinitely long train. 
A study of the propagation of a disturbance into 
an area of calm could be based on these considera- 
tions but we prefer to use a method based on 
consideration of energy, which appears to be 
simpler and in better accord with the point of 
view from which this paper is prepared. 
From a comparison of equations (11b) and (8b) 
follows 
Vi== Vi 
This identification of the group velocity with the 
mean rate of transmission of energy, here shown 
for deep water waves, can be extended to shallow 
water waves, indeed to all kinds of waves (Have- 
lock, 1914, p. 55 and p. 61). However, the physi- 
cal significance does not appear obvious, as evident 
from explanations attempted by Lamb (1932, 
p. 383) and Rayleigh (1877, p. 21). 
To the present problem it is of particular 
importance to know whether equation (8b) shall 
be interpreted to mean that 
(1) all the energy advances with group velocity, 
or 
(2) half the energy advances with wave velocity. 
To answer this question consider the flow of 
energy through a parallelepiped of unit width, 
length dz, and extending to a depth below which 
wave motion is negligible (fig. 4, p. 14). The 
time rate of change of energy within the paral- 
lelepiped must equal —d0(V£)/dz, the net inflow 
in the direction of the x-axis; and, therefore: 
of, 2 
oF +2 (VE)=0 (13) 
Tn the first case (13) becomes 
dE, CdE_ 
with the solution 
(14 ) 
E =f(2-§!) 
which gives no information as to the manner in 
which £ varies with z. 
In the second case pies (13) becomes 
=0 (15) 
leading, as will g fees to a solution of the 
transient state which is consistent with necessary 
physical boundary conditions, 
Transmission of Energy by Wave Motion 
The difference in form between equations (14a) 
and (15) has been given physical significance: 
equation (15) has been interpreted to mean that 
the ratio “‘group velocity to wave velocity” 
denotes the fraction of energy, #’, which advances 
with wave velocity.* 
1D AY 
EO to) 
This physical significance follows from a consider- 
ation of the distribution of potential and kinetic 
energy along a wave. Consider a sinusoidal wave, 
for which the surface elevation is given by 
n= 5H sin k(e—Ct) (17) 
The potential energy is computed from the 
elevation or depression relative to the still water 
surface as: 
TUE i "ede= hapEP sin? k(w—Ct) (18) 
Substituting in equation (8a) we find for the rate 
at which energy is transmitted 
VE=CE> 
*This interpretation is, according to Rossby (1945), not generally valid, but 
it is applicable to the type of waves examined here. 
