We present a complete analytical derivation of the equations used for stationary and nonstationary wave systems regarding resonant sound transmission and reflection described by the phenomenological Coupled-Mode Theory. We calculate the propagating and coupling parameters used in Coupled-Mode Theory directly by utilizing the generalized eigenwave-eigenvalue problem from the Hamiltonian of the sound wave equations. This Hamiltonian formalization can be very useful since it has the ability to describe mathematically a broad range of acoustic wave phenomena. We demonstrate how to use this theory as a basis for perturbative analysis of more complex resonant scattering scenarios. In particular, we also form the effective Hamiltonian and coupled-mode parameters for the study of sound resonators with background moving media. Finally, we provide a comparison between Coupled-Mode theory and full-wave numerical examples, which validate the Hamiltonian approach as a relevant model to compute the scattering characteristics of waves by complex resonant systems.