These are disturbances for which the ray paths of k project
backwards in time to a common point in the x, t-plane,
which represents a virtual source point at which a
finite total energy has been injected with ordered phases
and a given wave number spectrum 
.
Let the virtual source be taken as x = 0, t = -a, then
the required 
for a centered disturbance is
such that (1.25) becomes
Now (1.27) and (1.32) require that
Integration by parts gives
recognizing that 
.
Thus (1.26) takes the form
where b is an arbitrary constant of integration. Finally, from (1.32)
and hence (1.31) takes the form
Thus
which implies that, except for a scaling factor 
,
the spectral distribution of energy versus wave number (or
frequency) is conserved for a centered disturbance.
In the following examples we will shift the time origin to the
virtual source, which means we replace 
by t.
Equation (1.35) the becomes 
where 
now represents
the value of M vs. k at t = a.