In this paper, 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 for the problem of a one-dimensional isolated on-channel resonance. This Hamiltonian formalization could be beneficial and could potentially model and parameterize a broad range of acoustic wave phenomena. We demonstrate how to use this theory as a basis for perturbation analysis of complex resonant scattering scenarios. In particular, we 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.