The transition from laminar to turbulent flow is initiated by the amplification of Tollmien-Schlichting (TS) waves. Consequently, there is significant research interest in boundary layer stability control aimed at mitigating these waves to delay transition. This work presents a novel variational formulation to investigate the stability of channel flows interacting with a phononic subsurface. This wall-mounted metamaterial leverage local resonances to provide tunable dispersive properties, demonstrating potential for passive flow stabilization over broad frequency ranges [1,2]. Traditional formulations for stability problems involving compliant surfaces typically rely on the Orr-Sommerfeld equation coupled with wall effects characterized by a complex admittance. However, this methodology results in a decoupled solution, restricting the analysis to spatial stability (imposed frequency) and neglecting the intricate fluid-solid interplay. By employing a variational approach, we enforce the continuity of velocity and traction at the interface, obviating the need for an admittance function. The resulting discretized system yields a fully coupled generalized eigenvalue problem, enabling the determination of complex phase velocities and growth rates in both spatial and temporal (imposed wavenumber) frameworks. Furthermore, it fully accommodates the geometric complexity required to model metamaterial design, which can be both frequency and wavenumber dependent. The present numerical analysis focuses on the energy exchange between fluid instabilities and elastic waves within metamaterial structure. Specifically, we examine how phononic structure influences the synchronization between TS waves and structural modes. Preliminary results indicate that the fully coupled temporal analysis yields different results compared to uncoupled spatial analysis, even for identical metamaterial designs. This discrepancy suggests that full coupling is essential for accurate stability prediction, offering new insights for the development of passive flow and transition control strategies.