We present a new analytical model of atmospheric waves in the equatorial area that filters out inertia-gravity waves, to gain insight on the Madden-Julian Oscillation (MJO). Filtered models are usually derived by partitioning the flow into nondivergent and irrotational parts, which are expressed in terms of the streamfunction and velocity potential. Then certain approximations are introduced into the divergence and potential vorticity equations, with the result that inertia-gravity waves are filtered. Such procedures have led to the disadvantage that, in the process of filtering the inertia-gravity waves, the Kelvin waves are distorted (Moura 1976). In the present paper we take a different approach to the filtering problem. We partition the flow into Kelvin and non-Kelvin parts (Ripa 1994), expressing the non-Kelvin part in terms of a single potential function, which satisfies a master equation (Schubert et. al. 2008). The methods also yield an improvement of the longwave approximation (Gill 1980), in that they provide a more precise approximation of Rossby waves of all wavelengths. This is very promising for learning more about the MJO because an accurate representation of the MJO phenomenon requires well represented Rossby waves propagating west of the heat source, and a Kelvin wave propagating east of the heat source. We also derive an analytical solution for convectively coupled equatorial waves using the same filtering method as mentioned above.