## Abstract

The propagation of planetary waves on a sphere and on a $\beta $ plane has been extensively studied by Longuet-Higgins (1964a, 1965), in connexion with problems concerning the long period oscillations of the atmosphere and oceans. However, the concepts of energy and its propagation in planetary waves is not well understood, and in this paper we make a thorough study of the energy of these waves in the $\beta $ plane approximation with constant depth. In the case of non-divergent waves on a $\beta $ plane, two energy conservation equations are derived, both of which are consistent with energy flux in the direction of the group velocity $c_{\text{g}}$. The total energy density E of a plane wave is shown to be the sum of the kinetic energy density T and a 'Rossby energy' density V, where $V=-\frac{1}{2}\rho \beta \psi \eta $, $\psi $ is the stream function, and $\eta $ the northward particle displacement. The principle of virtual work is used to derive a similar result for a basin enclosed by rigid walls. The flux of E is given by $Ec_{\text{g}}$, and the equipartition principle $\overline{T}=\overline{V}=\frac{1}{2}\overline{E}$ is observed. The results are extended to the case of divergent planetary waves, the total energy density being E = T + U + V, where $U=(f^{2}/gh)\psi ^{2}$ and $\overline{T}+\overline{U}=\overline{V}=\frac{1}{2}\overline{E}$. A special case of reflexion of a plane wave at a straight wall is discussed at the end of the paper. It is shown that $\psi ^{\ast}=a\exp (\text{i}\omega t)$ is a non-trivial solution of the non-divergent planetary wave equation. In certain cases, $\psi ^{\ast}$ is needed to represent waves which are incident to or reflected from the wall.