On the surface Bloch waves in truncated periodic media: scalar-wave primer

Abstract

Much like their counterparts in homogenous elastic solids, waves in periodic media can be broadly classified into Floquet-Bloch body waves, and evanescent surface waves. Our goal is to elucidate the latter boundary layers, termed surface Bloch (SB) waves, affiliated with rational surface cuts and homogeneous Neumann data. To this end we adopt a two-dimensional (2D) scalar wave equation with periodic coefficients (describing anticline shear waves in phonoic crystals) as a test bed and develop a unit cell-of-periodicity-based, reduced order model of the SB waves that is capable of describing both their dispersion, waveforms, and ``skin depth''. The centerpiece of our analysis is a quadratic eigenvalue problem (QEP) for the effective unit cell of periodicity -- deriving from a geometric interplay between the mother Bravais lattice and orientation of the surface cut -- that seeks the complex wavenumber normal to the cut plane given (i) the excitation frequency and (ii) wavenumber in the direction of the cut plane. In this way the sought boundary layer is derived via superposition of the evanescent QEP eigenstates, whose relative amplitudes are obtained by imposing the homogeneous boundary condition. With the QEP eigenspectrum at hand, evaluation of an SB wave -- in terms of both dispersion characteristics and evanescent waveforms -- entails only a low-dimensional eigenvalue problem. This feature caters for rapid exploration of the effect of (periodic) surface undulations, and so enables manipulation of the SB waves via optimal design of the surface cut. Our analysis also includes an account for the power flow and ``skin depth'' of a surface Bloch wave, both of which are critical for the energetic relevance of boundary layers.