From d’Alembert to Bloch and back: A semi-analytical solution of 1D boundary value problems governed by the wave equation in periodic media

Abstract

We propose a simple semi-analytical model for solving one-dimensional (1D) boundary value problems governed by the wave equation in periodic media at arbitrary frequency. When the boundary value problem is well-posed in that the excitation frequency does not match any of its resonant frequencies, any solution that satisfies the field equation and boundary conditions is the (unique) solution of the problem. This motivates us to seek the solution in the form of Bloch waves that by design solve the wave equation with periodic coefficients. For a given excitation frequency this assumption leads to a quadratic eigenvalue problem in terms of the wavenumber k which, in turn, produces a d’Alembert-type solution in terms of the “right”- and “left”- propagating (or evanescent) Bloch waves. In this vein, the solution of a boundary value problem is obtained by computing the amplitudes of the two Bloch waves from the prescribed boundary conditions. For completeness, we consider situations when the driving frequency is: (i) within a passband, (ii) inside a band gap, (iii) at the edge of a band gap (the so-called exceptional points), and (iv) at band crossing (repeated eigenfrequency). From simulations, we find that the semi-analytical solution – expressed in terms of either propagating, evanescent, or standing Bloch waves – is computationally effective and reproduces the numerical results with high fidelity. Consistent with related studies, we also find that the solution of the boundary value problem undergoes sharp transition in a frequency neighborhood of exceptional points. One of the most intriguing results of this study is our finding that near exceptional points, the global resonance of a periodic medium could be induced by “microscopically” adjusting the support locations so they coincide with a shared node of the germane Bloch eigenfunctions. This opens a door toward the design of adaptive wave control systems whose performance can be tuned by adjusting the boundary conditions – as opposed to altering the properties of a periodic medium.