For low Froude number regimes, and analogously for low Mach number regimes, the shallow water and barotropic Euler equations pose a well-known numerical challenge: the scale separation between fast acoustic waves and slow advective dynamics imposes severe stability restrictions and may induce non-physical oscillations or excessive diffusion when standard explicit methods are employed. In this work, we develop and analyze, in the sense of asymptotic consistency and uniform-in-ε stability, an implicit–explicit (IMEX) finite volume scheme with the Asymptotic Preserving (AP) property for the shallow water equations in the low-Froude limit, ensuring that the method remains stable and consistent without the need to explicitly resolve the acoustic scale. In this regime, the pressure-driven (acoustic) subsystem can be viewed, after linearization around near-equilibrium states, as a stiff wave system with characteristic speeds of order O(1/ε), while the transport dynamics remain O(1). This wave-system viewpoint guides the design of the IMEX splitting and provides a natural framework to discuss uniform-in-ε stability and the control of spurious high-frequency oscillations. As a complementary study, we consider a discontinuous Galerkin (DG) discretization of the associated wave subsystem to assess acoustic dispersion/diffusion and to motivate improved resolution of acoustic modes in the low-ε regime. The flux decomposition separates convective terms, treated explicitly, from the pressure contribution, which is treated implicitly, thereby preserving the correct dynamical balance in the asymptotic limit and avoiding the typical degeneration of non-AP schemes. A damped Jacobi-based multigrid smoother is employed, and an adaptive residual-based strategy is introduced to improve convergence with ε-robust and mesh-independent behavior. The effectiveness of the method is evaluated both numerically and theoretically. Additionally, we discuss the potential extension of this approach to an IMEX formulation for the barotropic Euler equations, as part of ongoing work.