The dispersion relation 
for short surface waves
in which both surface tension and gravity play significant roles
as restoring forces is given by (1.23a) of which the preceeding
two examples are special cases.
The phase speed and group speeds for this general case are
A minimum phase speed occurs at k = 
and a mimimum group speed at a smaller k.
These minimum values are 
and 
for 
and 
, respectively.
The existence of a minimum implies that for a given
wave ray 
there can be two
different k for a common ray, except for that the ray
corresponding to the minimum (which must represent
the trailing edge of the wave group).
For given 
greater than the minimum, one
root for k corresponds to wves behaving like gravity waves,
the other like gravity waves.
For this general case, one must employ the implicit method of
generating phase contours n in the x, t-plane;
specifically x and t are computed from (1.33) and (1.34)
for given n by allowing k to vary from small to large
values across the value at which is minimum.
This produces the two families of intersecting phase lines
shown in Fig. 3.
The closely spaced lines (small wave length) represent
nominally capillary waves; those with the larger wave length
are the gravity modes.
There exists one unique phase line in this plot which is exactly
a straight line and has a slope 
corresponding to the minimum
phase speed.
This phase line is straight because =
when is a minimum (see section 1.2, Ch. 2 of these
notes, Eq. 1.18 in particular).
For this case the trailing edge of the group, assuming that the
energy density is not zero, generates both wave modes which
advance faster than 
.