Analysis of truncation resonances in 2D periodic discrete lattices

Abstract: Truncation resonances (TRs) are localized modes emerging from the truncation of infinitely periodic phononic crystals (PCs). In recent years, extensive studies on TRs in 1D PCs have identified beneficial characteristics such as strongly localized modes for various applications, e.g., energy harvesting and passive flow control. Even though 2D PCs offer additional control over wave propagation, such as engineered directivity, truncation resonances in 2D PCs remain completely unexplored. To address this research gap, we introduce the new Spatial-Spectral Decomposition (SSD) method to capture the complex dynamics of TRs in 2D PCs with arbitrary unit cell parameters, degrees of freedom, and boundary conditions, and use this method to study 2D TRs in two types of PCs based on a 2D square spring-mass lattice. We demonstrate that for various boundary conditions and unit cell parameters, 2D TRs can exist as edge modes and corner modes with various localization characteristics. Furthermore, localized modes can emerge that cannot be characterized as edge or corner modes, such as diagonally localized modes. We use the SSD method to analyze these modes and report on 2D TRs for the first time to the author’s knowledge. We relate the exponential decay rates associated with the 2D TRs to the 2D complex band structures, and analyze how elastic coupling in one direction influences localization characteristics (e.g., existence and decay rates of TRs) in the perpendicular direction. The SSD method and the analysis of 2D TRs presented here have potential applications in passive flow control and energy harvesting beyond 1D settings.

Citation: Y. Shi, V. Ramakrishnan, K. H. Matlack, "Analysis of Truncation Resonances of 2D Discrete Lattices", J. Sound Vib., 645, 120136 (2026)
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