AbstractFor the shallow‐water equations on an equatorial beta plane, the properties of low‐frequency Rossby waves and (mixed) Rossby–gravity waves are investigated. It is shown that in the low‐frequency limit the horizontal divergence of these solutions is zero and their geopotential satisfies φ = fψ, where f = βy is the Coriolis parameter and ψ is the stream function of the non‐divergent velocity field. This type of balance is rather different from the geostrophic balance satisfied by Kelvin waves. It can be used to formulate a balanced potential vorticity equation in the single variable ψ that, while filtering out Kelvin waves and inertia–gravity waves, exactly reproduces Rossby waves and Rossby–gravity waves in the low‐frequency limit. Copyright © 2010 Royal Meteorological Society