the source all waves arrive with the same height 
as the initial wave. A steady state is reached 
with the passage of the wave front. 
If, however, the phase velocity C is a function 
of the wave length, a fact which is expressed by 
describing the medium as dispersive, this ideal 
simplicity no longer exists. In general, C and V 
are no longer equal. Radio signals in hollow 
guides, seismic waves in the interior of the earth, 
and surface waves in deep water, are examples 
of waves traveling through dispersive media. In 
these only a portion of the energy travels along 
with the wave form, the remaining portion being 
left behind. Focusing attention on a single wave 
in deep water, one should expect this wave to 
gain the portion of the energy left behind by the 
preceding one and, in turn, leave a portion of its 
energy for the wave following it. Dealing with a 
group of, say, 10 waves, the first wave, which 
loses energy to the wave behind and gains none 
from the front, soon becomes very small, but the 
last wave, which leaves energy behind, will form 
a new wave behind it and become the second from 
the last. 
This is exactly what has been observed and 
reported by numerous observers. In the Ad- 
miralty Navigation Manual (1939), Vol. III, page 
389, it is stated that: 
if motion of the first wave of the group is followed, it will 
be found that this motion dies out and that the wave next 
behind takes the lead. If, on the other hand, the last 
wave of the group is watched, another wave will be seen 
to appear behind it. The new waves constantly rise in 
the rear as rapidly and as constantly as those in the front 
die out, so that the general appearance of a group of waves 
remains unchanged. ‘The group as a whole has a definite 
velocity of propagation, which has been found to be 
half of that of the individual waves comprising the 
group * * * 
Kriimmel (1911, page 95), also states that: 
* * * the wave, which at any instant is in front, flat- 
tens so much as it travels over the surface, that it becomes 
invisible after traveling 2-4 meters, whereupon the next 
one becomes the first, again goes through the appearance 
of flattening, ete. 
The same experierice in tanks is reported in the 
Technical Report No. 2 of the Beach Erosion 
Board (1942). 
In wave tanks, wave groups may be generated by operating 
the wave machine through only a few strokes * * * 
the observer can follow a particular wave crest only a finite 
distance before it disappears. Close observation reveals 
that the wave group maintains its identity, that individual 
waves pass through the group, rising out of comparatively 
calm water at the rear, reaching a maximum at the center, 
and then disappearing at the front of the group. 
When a long group reaches an observer stationed 
at a given distance from the source, the initial 
waves will be very low, but the wave height will 
increase with time. It will reach a maximum as 
the center of the group passes the observer and 
will then decrease. If waves of constant height 
are continuously generated at the source the ob- 
server will find that, after a transient stage of wave 
growth, a steady state condition involving con- 
stant wave height will be approached. The prob- 
lem is to find how long it takes to reach, for example, 
50 or 90 percent of this constant height. The prob- 
lem can also be stated: to find the rate at which an 
appreciable portion of the energy of the disturbance 
is propagated through the area of calm. 
It will be shown that there exists an actual 
“‘wave front” which advances with the velocity C 
of the initial wave but the magnitude of the dis- 
turbance so propagated is negligible compared 
with that of the main group which travels at 
lower speed. 
Group Velocity and Energy Flow 
The slow rate at which a disturbance is trans- 
mitted through an area of calm, as compared to 
the rate of travel of the individual waves, is gen- 
erally explained by stating that the disturbance 
travels with the group velocity V’. An expression 
for group velocity is derived by considering the 
combined effect of two trains of waves of equal 
wave height whose lengths differ by a small 
amount dL. The resulting interference pattern 
will travel with a velocity equal to 
dC 
V=0-L7 (11a) 
Since O?~LZ, it follows: 
NG) 
Was (1b) 
(Beach Erosion Board, 1942, p. 32; Cornish, 1934, 
p. 137; Krimmel, 1911, p. 95.) 
This derivation has led to much confusion 
because, unless further qualified, it would indicate 
that group velocity is important only under very 
special conditions as, for example, in the case of 
two wave trains of equal height and slightly dif- 
ferent lengths. But equation (11a) is much more 
general and can also be derived by making use of 
methods of summation or integration if the 
simultaneous existence of an infinite number of 
wave trains involving a frequency spectrum is 
