We consider the test suite for the Shallow Water (SW) equations on the sphere suggested in [27, 28]. This series of tests consists of zonally propagating wave solutions of the linearized Shallow Water (LSW) equations on the full sphere. Two series of solutions are considered. The first series [27] is referred to as "barotropic". It consists of an extension of the Rossby-Haurwitz test case in [33]. The second series [28] referred to as (Matsuno) "baroclinic", consists of a generalisation of the solution to LSW in an equatorial chanel introduced by Matsuno [17]. The Hermitian Compact Cubed Sphere (HCCS) model which is used in this paper is a Shallow Water solver on the sphere that was introduced in [4]. The spatial approximation is a center finite difference scheme based on high order differencing along great circles. The time stepping is performed by the explicit RK4 scheme or by an exponential scheme. For both test cases, barotropic and baroclinic, the results show a very good agreement of the numerical solution with the analytic one, even for long time simulations.