Analysis of truncation resonances in 2-D periodic discrete lattices

Truncation modes (TMs) are localized modes residing in phononic band gaps, and they have recently been explored for passive flow control and energy harvesting applications. However, current knowledge is limited to TMs in 1-D periodic lattices. Motivated by this, here we systematically investigate 2-D TMs of discrete phononic lattices with two types of rectangular unit cells. We present a decomposition method based on the transfer function framework to analyze 2-D TMs. Our study reveals that, even for simple unit cell configurations, the characterization of 2-D TMs can be complicated and unintuitive. Unlike 1-D TMs, a wavenumber–frequency pair may not fully describe the localization characteristics of 2-D TMs, which leads to various types of localization characteristics, such as diagonal localization. Consequently, we explore conditions of meaningful extension of 1-D TM concepts, e.g., decay rate, directionality, to 2-D TMs using our decomposition method, demonstrate how boundary conditions restrain 2-D TMs in certain partial bandgaps or full bandgaps, and show how boundary conditions at edges perpendicular to the TM affect its characteristics through orthogonal coupling. Finally, we show that the proposed decomposition method can be generalized to characterize 2-D TMs in more complex lattices.