Abstract: Analysis is focused on the physical properties of coupled waves which result from interaction of the free Rossby waves for atmosphere and ocean respectively.It is shown that in the shallow water models exist two kinds of unstable coupled waves,which depend on the atmospheric and oceanic background state.One is the weak interaction coupled wave with the background state of baroclinic atmosphere and deep mixed layer.Other is the strong one with that of barotropic atmosphere and shallow mixed layer.The dispersion relation also shows that unstable coupled waves are responsive for various physical processes.Three mathematical methods are offered to test the truncation errors from truncated models.At the same time,a reasonable method is taken so as to make the free Rossby waves of atmosphere and ocean be correct.