For f = constant, (3.2) can be satisfied for arbitrary
if 
= 0, in which case (3.1) reduces to an
exact geostrophic balance 
where
In fact, for the real world, f depends upon latitude
and hence
which vanishes only if is a function of latitude
(implying zonal geostrophic flow). Thus if the flow
(even though nearly geostrophic) has a component across
the contours of f (meridional component), then the field
of must change with time (see Eq. 2.12).
The possibility of wave modes with very small frequency as solutions of the LTEs was first recognized by Laplace, who termed these ``waves of the second class.'' They differ from the gravity wave counterparts in that they have large vorticity relative to horizontal divergence. For this reason a logical generic name is vortex modes; although from the standpoint of restoring ``force'' they are sometimes called inertial modes, since gravity plays only a secondary role.
Although one can derive both the gravity wave modes and the vortex modes simultaneously form the LTEs with allowance for variable f, a much simpler analysis of the vortex mode properties follows from the quasigeostrophic vorticity equation, as first recognized by Carl-G. Rossby (1936, J. Mar. Res.). We give a generalization of this methodology here. It is for reasons of the recognition of this methodology that the vortex modes are so commonly called Rossby waves. He was the first to recognize their importance in the dynamics of the ocean and the atmosphere.