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

Much like their counterparts in homogeneous 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 antiplane shear waves in phononic crystals) as a test bed and develop a unit cell-of-periodicity-based, reduced-order model (ROM) of the SB waves that is capable of describing their dispersion, waveforms and 'skin depth'. The centrepiece 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 quantifying the evanescence away from 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 obtained by 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 SB waves via optimal design of the surface cut. Our analysis also includes an account for the power flow and skin depth of SB waves, both of which are critical for the energetic relevance of boundary layers.