Phase-Controlled Logic in a Y-Shaped Phononic Network with Dynamic Junction Inertia

We investigate wave scattering and programmable logic operations in a Y-shaped network of coupled harmonic chains connected through a tunable junction mass. Using stationary scattering theory, we derive analytical expressions for the transmission amplitude under coherent excitation of the two input branches with arbitrary relative phase. The interplay between wave interference and the spectral response of the junction enables the implementation of Boolean logic operations through a threshold criterion applied to the transmitted intensity. We further explore three models of dynamic effective mass, with frequency, amplitude, and velocity-dependent responses, which provide distinct mechanisms for tuning the transmission spectrum and the operating regions of the logic gates. The nonlinear models exhibit self-consistent transmission and bistable behavior, offering enhanced flexibility for programmable phononic logic. The proposed architecture provides a versatile platform for wave-based information processing in mechanical metamaterials devices.