indicating that the potential energy is transmitted 
with wave velocity. The kinetic energy is obtained 
from the horizontal and vertical components of 
particle velocity: 
Be dba CEE 3 
r= 50 | ve 1 u\dz 
where f is the depth. For deep water waves, 
w= Tide sin k(x —( t) 
C 
(19) 
w= — SH cos k(z—Ct) and 
1 2 
In contrast to the potential energy, which is a 
periodic function and advances in phase with the 
deformation of the surface, the kinetic energy is 
evenly distributed along the entire wave and is 
independent of the position or the velocity with 
which the surface deformation advances. (See 
fig. 2.) 
DEEP WATER WAVES 
> 
uJ 
= 
uJ 
i 
a 
2) 
wn 
Figure 2.—Variation of surface eleyation, potential energy, Ep, and 
kinetic energy, Ex, along one wave length. 
From (18) we find the mean potential energy 
over a wave length, Hp, 
1 E ; 
Ep= {@9elD = Ex=5 (21) 
and equation (16) is satisfied, since 
| OE 
FATE Mn Ona Gy 
The following interpretation can now be given 
to observations of wave motion in deep waiter. 
Again quoting the Technical Report No. 2 of the 
Beach Erosion Board (1942): 
As the first wave in the group advances one wave length, 
its form induces corresponding velocities in the previously 
undisturbed water and the kinetic energy corresponding 
to these velocities must be drawn from the energy flowing 
ahead with the form. If there is equipartition of energy 
in the wave, half of the potential energy which advanced 
with the wave must be given over to the kinetic form and 
the wave loses height. Advancing another wave length 
another half of the potential energy is used to supply 
kinetic energy to the undisturbed liquid. The process 
continues until the first wave is too small to identify. 
The second, third, and subsequent waves move into water 
already disturbed and the rate at which they lose height 
is less than for the first wave. At the rear of the group, 
the potential energy might be imagined as moving ahead, 
leaving a flat surface and half of the total energy behind 
as kinetic energy. But the velocity pattern is such that 
flow converges toward one section thus developing a crest 
and diverges from another section forming a trough. Thus 
the kinetic energy is converted into potential and a wave 
develops in the rear of the group. 
This concept can be interpreted in a quantita- 
tive manner, by taking the following example from 
R. Gatewood (Gaillard 1935, p. 34). Suppose 
that in a very long trough containing water 
originally at rest, a plunger at one end is suddenly 
set into harmonic motion and starts generating 
waves by periodically imparting an energy E/2 
to the water. After a time interval of n periods 
there are n waves present. Let m be the posi- 
tion of a particular wave in this group such that 
m=1 refers to the wave which has just been 
generated by the plunger, m=(n-+1)/2 to the 
center wave, and m=n to the wave furthest 
advanced. Let the waves travel with constant 
velocity OC, and neglect friction. 
After the first complete stroke one wave will 
be present and its energy is 1/2H. One period later 
this wave has advanced one wave length but has 
left one-half of its energy or 1/4# behind. It now 
occupies a previously undisturbed area to which 
it has brought energy 1/4H#. In the meantime, 
a second wave has been generated, occupying the 
position next to the plunger where 1/4 was left _ 
behind by the first wave. The energy of this 
second wave equals 1/4#+1/2H=3/4E. Repeated 
applications of this reasoning lead to the results 
shown in table 1. 
