Consider a combination of two plane waves with slightly
different 
and 
but having the same A:
Using trigonometric identities this can be rewritten as
The carrier wave has the phase speed 
as
before.
The modulation may however have a different propagational speed.
Since the modulation is a measure of the wave amplitude or
intensity, its speed of propagation characterizes the rate at
which the wave energy or group moves.
Let the value of 
following a given phase of
the modulation be denoted by 
, the group
velocity; then
In general will be a function of (i.e., of
, 
, 
) for a given type of wave, the
function 
being the dispersion relation characterizing the wave type.
Thus
considering 
, 
, 
suitably small, but otherwise arbitrary.
Thus (1.11) can be written
Now, since , , are arbitrary
and
In the special case where depends only on the magnitude of
the wave number 
, then 
and 
. For such a wave mode and 
have the same direction but not necessarily the same speed,
unless is directly proportional to k (as in a
non-dispersive compressional wave).
In more general wave types, and may differ
both in direction and magnitude (an example being a Rossby wave).