The existence of the periodic solution to the nonlinear ordinary differential equation (ODE) deduced from the nonlinear barotropic vorticity equation can be directly proved without expanding it by Taylor’s series at the point of equilibrium. Conditions for the existence of the periodic solution thus proved by using Liapunov’s second method greatly correspond to those for the barotropic stability. The rationality of the travelling wave solution and the expansion by Taylor’s series has been confirmed. Comparison is made between the numerical solution to the nonlinear ODE and the linear numerical solutio.