A unified linear wave theory of the Shallow Water Equations on a rotating plane Nathan Paldor 1 and Andrey Sigalov 2 1 Institute of Earth Science, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem, 91904 Israel nathan.paldor@huji.ac.il 2 Institute of Earth Science, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem, 91904 Israel sigalov@huji.ac.il Summary. The linearized Shallow Water Equations (LSWE) on a tangent (x, y) plane to the rotating spherical Earth with Coriolis parameter f (y) that depends arbitrarily on the northward coordinate y is considered as a spectral problem of a self-adjoint operator. This operator is associated with a linear second order equa- tion in x - y that yields all the known exact and approximate solutions of the LSWE including those that arise from different boundary conditions, vanishing of some small terms (e.g. the β-term and frequency) and certain forms of the Corio- lis parameter f (y) on the equator or in mid-latitudes. The operator formulation is used to show that all solutions of of the LSWE are stable. In some limiting cases these solutions reduce to the well-known plane waves of geophysical fluid dynamics: Inertia-gravity (Poincare) waves, Planetary (Rossby) waves and Kelvin waves. In addition, the unified theory yields the non-harmonic analogs of these waves as well as the more general propagating solutions and solutions in closed basins. 1 Introduction The Shallow Water Equations (SWE) are the most fundamental system of equations in geophysical fluid dynamics. They describes the dynamics of an incompressible inviscid rotating layer of fluid. We consider in this paper the linearized form of this system (LSWE) on the tangent plane to the sphere (commonly referred to as the f -plane or the β-plane, depending on the approximation employed) that yields the various types of waves observed in the atmosphere and ocean. Assuming that the bottom of the layer of fluid is flat the LSWE may be written in local x and y coordinates on the rotating earth as: ∂u ∂t = -f × u - g∇h, ∂h ∂t = -H∇· u, (1) where u =(u, v) is the horizontal velocity vector, whose u and v com- ponents are directed in the zonal and meridional directions, respectively, h