Consider a progressive, plane surface wave group of form
where
i.e.,
with 
, 
0 such that the local phase speed 
is everywhere positive.
The dynamics of the system assert that and k are
related:
In particular (as we will show later):
and
where 
.
As shown previously, k (and hence ) are conserved
following at group speed,
, i.e.,
Thus 
is also conserved, which implies that a
contour of k in the x, t-plane is a straight line:
with 
at t = 0.
S at given k and t may be obtained by integrating (1.21)
with t along 
:
where 
at t = 0.
Functions 
and 
are not
independent; (1.20a) requires that
Equations (1.25) and (1.26) combined yield implicitly the field
of 
and hence of 
.
In practice, for the most general dispersion relation
, and for general initial states ,
satisfying (1.27), contours of S = constant may
be generated by solving (1.25) and (1.26) for x and t as
functions of S and k.
In order to quantify the behavior of the amplitude function M,
we appeal to considerations of energy conservation.
Let 
denote the local energy density, which we consider
proportional to 
.
For the x-directed local energy flux we take 
,
and hence energy conservation requires
or
for the plane waves under consideration. The above relation can be rewritten in the form
But from (1.25)
so (1.29) takes the form
along 
= 
, i.e. for k constant.
Integration of (1.30) then yields
where 
= 
with
.
Thus the time dependence of M depends critically
on 
.