Here we will restrict our attention to a wave type for which
for simplicity.
We will suppose the wave train has a slowly varying wave length
L as a function of x and t
The rate of change of L following an individual wave is
Since 
depends on k or 
, then
will also be a slowly varying function of x.
An alternate representation of 
following the wave
is then given by the rate of stretching associated with
successive wave crests having different phase speeds:
to first order in L. Combining (1.15) and (1.16) gives
which implies that an observer who travels at speed
always sees the same wve length beneath him or her,
but not the same wave.
However, since 
and 
it is readily shown that
(1.18) reduces to 
.
Thus we have the following alternative interpretation of the
group speed: