A generalization of the Wave Finite Element Method is proposed for the linear stability analysis of thermoacoustic systems, incorporating high-order modes associated with cross-sectional wave propagation and fluid–structure interaction. Formulated in the Laplace domain, this methodology enables the estimation of transfer matrices that integrate into network models for stability predictions. By extending beyond the classical plane wave assumption in network modeling, the framework accurately captures wave propagation in complex geometries where high-order acoustic modes play a crucial role. Additionally, it accounts for fluid–structure interaction, demonstrating how coupling between the fluid and structure modifies wave propagation and influences the system stability. The approach is validated through case studies of increasing complexity and the results confirm the importance of incorporating high-order modes and fluid–structure interaction effects into predictive models. By providing a computationally efficient alternative to full-domain numerical simulations, the proposed wave-based framework enhances the accuracy of reduced-order models, improving their predictive capabilities for thermoacoustic stability analysis and enabling design-oriented studies such as the optimization of flexible wall configurations for passive instability control.