Let
then from (1.3)
and from (1.5)
Thus (1.4) converts to
where 
is a separation parameter. Hence
is a solution of
subject to the conditions (from the boundary conditions)
which follows fromm (1.6), (1.7), (1.8a) and (1.9).
Equations (2.5) to (2.7) constitute an eigenvalue problem for
and the associated , analogous to those
considered in previous sections, but with a more general upper
boundary condition.
It can be shown that all the eigenvalues must be
real. They are also positive, since from (2.5) to (2.7) is is
readily shown that
The dimensions of are 
or
has dimensions of speed; thus it is
suggestive to put
Using (2.1), Eqs. (1.1) and (1.2) become
and (1.11) yields
Thus the equations controlling 
, 
, 
all have
the same form, but with a differing celerity 
. These
are a special case of the Laplace tidal equations (LTE)
without the forcing terms, and are restricted to constant
depth.
The eigenvalue problem (2.5) to (2.7) is simpler than that
previously encountered due to the absence of 
. The
lowest order mode, which we denote n = 0, is characterized
by very small 
and 
is nearly constant.
With this approximation (2.8) yields
or
which we recognize as the wave speed of long surface gravity waves (in the absence of rotation). We identify the mode n = 0 as the barotropic mode.
All other vertical modes have n zeros of 
and much
larger (smaller ). These are the baroclinic
modes. The rigid lid condition yields a good approximation
for these modes. The baroclinic are roughly
which is based on the WKB approximation. More exact values can be obtained by numerical solution of the eigenvalue problem (2.5) to (2.7).