Finally, eliminating    between eqs.(26) and (27) gives the following equation governing the vertical structure of the wave mode:

Eq.(28) has one solution which we will dispense with from the outset i.e.   . Referring back to eqs.(20) - (24) it can be seen that this solution corresponds to geostrophic and hydrostatic flow in the    direction with    and    both equal to zero. Another point worth noting at this stage is the appearance of the factor   . In the full QB set the term attached to this factor would not appear since both    and    would be equal to unity. If either of the terms to which and    are attached was to be neglected, then it would be appropriate to neglect both in order to avoid generating a spurious term in this equation. 

Firstly though, it is instructive to consider the exact linear system with   . The vertical structure equation (28) may then be simplified by using the transformation:

where    is the new dependent variable. Using this, together with eq.(9), leads to:

provided that   . Eq.(29) admits wave-like solutions and if    then the following dispersion relation is satisfied:

where    and 

   is an acoustic frequency and    represents a form of wave aspect ratio (i.e. ratio of the horizontal to vertical wavelength ).