Fourier-transformed method is used to investigate the problem of the rapid growth and decay of the disturbance energy of Rossby wave in barotropic atmosphere on the assumption that this kind of atmosphere is adiabatic, non-frictional, unforced, non-dissipative and quasigeostrophic on a beta plane. The analytical solution of linear barotropic potential vorticity equation is obtained for the first time and we go a step further to discuss the relations between the disturbance energy and the initial perturbation flow wave numbers, shear of basic flow and viscid coefficient by using the above analytical solution.