A general initial value problem is solved for wave perturbations of small amplitude on a zonal atmospheric flow with constant vertical shear and vanishing temperature lapse rate, using frictionless, hydrostatic, adiabatic and geostrophic theory and beta-plane dynamics.
For every west-east wavelength, the solution is seen to be constituted of a finite number of discrete spectrum modes and a continuous spectrum contribution. (As in many other hydrodynamical problems, the relevant operator of the form L−λM=0 is singular and the differential operators L and M are of equal order.)
As was previously demonstrated, for almost all wavelengths one of the eigenvalue modes is unstable. The present analysis shows that, for the exceptional wavelengths for which no such unstable normal mode exists, the continuous spectrum contribution is unstable.