We show that when linear azimuthal perturbations on the surfaces of a fluid shell are regrouped according to α^{m}, they can be divided into Bell model terms, coupling terms, and the newly identified thin-shell correction terms, where α is the ratio of R_{out} to R_{in}, and m is the mode number of a given unstable mode on the surfaces. It is also revealed that α^{m} is a convenient index variable of coupling effects, with which we show that the evolution of instability is composed of three stages, i.e., strongly coupled stage, transition stage, and uncoupled stage. Roughly, when α^{m}<6, the fluid shell is in the strongly coupled stage, where both coupling effects and the newly identified thin-shell corrections play important roles. Strong feed through is expected to be observed. The uncoupled stage is reached at α^{m}∼36, where Bell's model of independent surface holds. In between is the transition stage, where mode competitions on the two surfaces are expected to be observed. These results afford an intuitive picture which is easy to use in guiding the design of experiments. They may also help to quickly grasp major features of instability experiments of this kind.