The last example of a dispersive wave group concerns the behavior
of long gravity waves near the leading edge of an
earthquake-induced wave train, the source being treated as a
centered line source.
The dispersion relation for this case is given by (1.23b).
For 
( 
),
.
Hence (for constant D), a maximum 
and 
exist having the value 
.
For general 
but not zero, 
and hence
are transcendental functions of k, but the phase lines
n in the x, t-plane can be generated by the implicit method.
Resulting calculations lead to the illustration in Fig. 4, in which
x is in units of D and t is in units of
.
Near the leading edge, the wave length grows proportional to
in agreement with an analysis by Kajiura (1963)
based on quite a different approach.
These illustrations demonstrte the robustness of the simple kinematic method. (The domain of validity of this approach is that for which k is a slowly varying function of x and t, i.e., for a well resolved modulation.)
In order to show the behavior of wave length near the front,
we examine the behavior of 
near the front, where
kD is regarded as small (or L/D large).
Expanding the 
function in (1.23) via a power
series gives
or
so
while
both of which approach 
as 
.
From (1.44d) we get
Now the parametric relations for 
are
so near the front
where
At the front 
so k and hence S = 0,
the front being of constant phase.
For that position behind the front where S = -2 
,
for given t.
At that position
where 
.
Hence from (1.45a) with S = -2 and x = 
Thus the wave length of the first wave near the front is given by
Sample values are:
(Note that these satisfy the constraint L/D 
1.)
which can be verified by referring to Fig. 4.