If we consider periodic waves in time
then (2.10) and (2.11) yield (with 
):
where 
.
For f constant we obtain
A possible plane wave solution is
provided that 
where
These are the Poincare (or Sverdrup) solutions to the
LTEs. Note that as 
,
as in the previous analysis. However,
(3.4) has no upper limit on 
; this is a consequence
of the hydrostatic approximation. Indeed since the true
upper limit on for 
is 
then
(3.4) is clearly limited to
.
Possible cylindrical wave solutions of (3.2) may be obtained
by rendering 
in cylindrical coordinates
. The resulting solutions, which are finite for
all r, are
where 
is the Bessel function of the first kind of integral
order m, and 
and 
are related by (3.4).
This form of solution is appropriate for admissible free
modes in a circular bvasin of, say, radius b.
The radial velocity, from (3.1), is
and since 
must vanish at r = b we require that solution
(3.5) satisfy
while from (3.4)
For a given 
and azimuthal wave number
m, there are only certain admissible 
and hence
for the basin. Lamb (1945, Hydrodynamics,
Art. 210) shows that if
then the lowest root for will be less than
unity and hence is imaginary.
In this case the function 
should
be replaced by 
where 
is a modified
Bessel function, which is monotonic and an increasing function
of its argument. In fact, has the property
for 
.
For 
, Lamb shows that the lowest
frequency gravity wave in the basin has
and
which rotates counterclockwise in the northern hemisphere
(f > 0) and clockwise in the southern hemisphere (f < 0).
The associated 
is given approximately by
where 
is the arc distance along the basin
perimeter and 
an alongshore wave number.
The amplitude decays inwards with e-folding scale 
(hereafter the radius of deformation for mode n), and
is negligible in the basin interior provided that
. The latter condition is most easily
satisfied for baroclinic modes for which 
km
(for typical oceanic conditions at mid-latitude).
It is marginally satisfied for the barotropic mode for
oceanic basins ( 
km).
Gravity modes having the above boundary trapped character and
rotational speed in the same sense as f are called
Kelvin wave modes. Their trace speed alongshore
(ds/dt) is simply 
and hence they are non-dispersive.
The above analysis based on constant f breaks down at the
equator. However, it can be shown that the equator also acts
as a wave guide for ``two-sided'' Kelvin-like waves
(see Moore and Philander, Ch. 8, Vol. 6, The Sea, 1977).
Other modes, having form (3.5) with real and hence
oscillatory character in r as well as 
, are possible
in addition to the Kelvin modes. The modes with oscillating
character in r can propagate in either azimuthal direction.
However, an interesting feature is that the angular phase
speed ( ) is smaller for positive waves
( 
) then for negative
waves ( 
). Lamb (1945, Art. 210) shows
some numerical examples for 
.