Controlling low-frequency interface transmission via zero-energy evanescent modes

Mechanical metamaterials often exploit zero-energy modes (ZMs)—deformations that store no elastic energy—to achieve unusual static and dynamic properties. While bulk ZMs have been widely studied, the role of boundary-localized evanescent ZMs in low-frequency wave transport remains less understood. This work investigates how such evanescent ZMs control wave transmission across interfaces in polarized pantographic chains. By breaking spatial inversion symmetry in a simplified pantographic model, we transform a uniform extensional ZM into an evanescent mode localized at one boundary, simultaneously localizing a self-stress state (SSS) at the opposite edge. We demonstrate that the specific polarization combination at an interface dictates its low-frequency transmittance: interfaces where evanescent ZMs face each other result in total reflection, while interfaces where SSSs face each other allow nearly perfect transmission. A homogenized one-dimensional strain-gradient continuum model is derived, revealing that this stark contrast originates from the differing asymptotic orders of the higher-order boundary forces associated with the ZM and SSS evanescent modes. Furthermore, we show that the transmission is highly sensitive to local structural details: introducing a weak spring at a reflective ZM-hosting interface can switch its response to total transmission. These findings establish a fundamental mechanism for controlling low-frequency waves via polarized nonlocal elasticity and highlight the critical importance of interface-specific design in metamaterials.