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

   https://doi.org/10.1098/rspa.2025.0647  

   Guzina, Bojan B; Meng, Shixu; Salasiya, Prasanna; Nguyen, Long (January 2026, Proceedings of the Royal Society A Mathematical Physical and Engineering Science)

   Hillston, Jane (Ed.)

   Abstract 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. 

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2. Locally Resonant Effective Phononic Crystals for Subwavelength Vibration Control of Torsional Cylindrical Waves

   https://doi.org/10.1115/1.4052748  

   Arretche, Ignacio; Matlack, Kathryn H. (June 2022, Journal of Vibration and Acoustics)

   Abstract Locally resonant materials allow for wave propagation control in the subwavelength regime. Even though these materials do not need periodicity, they are usually designed as periodic systems since this allows for the application of the Bloch theorem and analysis of the entire system based on a single unit cell. However, geometries that are invariant to translation result in equations of motion with periodic coefficients only if we assume plane wave propagation. When wave fronts are cylindrical or spherical, a system realized through tessellation of a unit cell does not result in periodic coefficients and the Bloch theorem cannot be applied. Therefore, most studies of periodic locally resonant systems are limited to plane wave propagation. In this article, we address this limitation by introducing a locally resonant effective phononic crystal composed of a radially varying matrix with attached torsional resonators. This material is not geometrically periodic but exhibits effective periodicity, i.e., its equations of motion are invariant to radial translations, allowing the Bloch theorem to be applied to radially propagating torsional waves. We show that this material can be analyzed under the already developed framework for metamaterials. To show the importance of using an effectively periodic system, we compare its behavior to a system that is not effectively periodic but has geometric periodicity. We show considerable differences in transmission as well as in the negative effective properties of these two systems. Locally resonant effective phononic crystals open possibilities for subwavelength elastic wave control in the near field of sources. 

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3. Two-dimensional stability analysis of finite-amplitude interfacial gravity waves in a two-layer fluid

   https://doi.org/10.1017/jfm.2022.145  

   Murashige, Sunao; Choi, Wooyoung (May 2022, Journal of Fluid Mechanics)

   We explore basic mechanisms for the instability of finite-amplitude interfacial gravity waves through a two-dimensional linear stability analysis of the periodic and irrotational plane motion of the interface between two unbounded homogeneous fluids of different density in the absence of background currents. The flow domains are conformally mapped into two half-planes, where the time-varying interface is always mapped onto the real axis. This unsteady conformal mapping technique with a suitable representation of the interface reduces the linear stability problem to a generalized eigenvalue problem, and allows us to accurately compute the growth rates of unstable disturbances superimposed on steady waves for a wide range of parameters. Numerical results show that the wave-induced Kelvin–Helmholtz (KH) instability due to the tangential velocity jump across the interface produced by the steady waves is the major instability mechanism. Any disturbances whose dominant wavenumbers are greater than a critical value grow exponentially. This critical wavenumber that depends on the steady wave steepness and the density ratio can be approximated by a local KH stability analysis, where the spatial variation of the wave-induced currents is neglected. It is shown, however, that the growth rates need to be found numerically with care and the successive collisions of eigenvalues result in the wave-induced KH instability. In addition, the present study extends the previous results for the small-wavenumber instability, such as modulational instability, of relatively small-amplitude steady waves to finite-amplitude ones. 

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4. Inverse design of nonlocal lattices with arbitrary multiband dispersion relations

   https://doi.org/10.1016/j.ymssp.2026.114252  

   Khan, Muhammad Bilal; Sugino, Christopher (May 2026, Mechanical Systems and Signal Processing)

   We present a numerical algorithm to implement arbitrary multiband dispersion relations in one-dimensional periodic lattices with N-degree-of-freedom unit cells using nonlocal interactions. First, we present a general model to calculate the dispersion relation for periodic lattices with N-DOF unit cells with linear, mass-spring-damper-type interactions. We show that the dispersion relation is obtained from a quadratic eigenvalue problem, in which nonlocal interactions yield wavenumber-dependent mass, stiffness, and damping matrices via a Fourier series. In principle, because these properties can be specified at every wavenumber, nonlocal interactions can yield any set of dispersion curves. We then develop a generalized numerical framework that enables the inversion of dispersion relations for metamaterials with arbitrary band structures, including active and damped components, variable mass distributions, and inertial interactions. By formulating the inverse problem as an optimization task, we incorporate constraints ensuring physical realizability while employing L1 regularization to promote sparsity in interaction matrices. Our approach successfully reconstructs desired dispersion relations for various test cases, demonstrating the ability to design lattices with customized multiband wave characteristics. The results highlight the robustness and versatility of our methodology, providing a powerful toolkit for advanced wave manipulation and nonreciprocal wave control. 

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5. Principles of leaky-mode photonic lattices: Band flips and Bloch mode dynamics

   https://doi.org/10.1117/12.2508984  

   Magnusson, Robert; Lee, Sun G.; Lee, Kyu J.; Hemmati, Hafez; Carney, Daniel J.; Bootpakdeetam, Pawarat; Ko, Yeong H. (March 2019, Proceedings Volume 10921, Integrated Optics: Devices, Materials, and Technologies XXIII; 109211E (2019))

   García-Blanco, Sonia M.; Cheben, Pavel (Ed.)

   We present principles of leaky-mode photonic lattices explaining key properties enabling potential device applications. The one-dimensional grating-type canonical model is rich in properties and conceptually transparent encompassing all essential attributes applicable to two-dimensional metasurfaces and periodic photonic slabs. We address the operative physical mechanisms grounded in lateral leaky Bloch mode resonance emphasizing the significant influence imparted by the periodicity and the waveguide characteristics of the lattice. The effects discussed are not explainable in terms of local Fabry-Perot or Mie resonances. In particular, herein, we summarize the band dynamics of the leaky stopband revealing principal Bragg diffraction processes responsible for band-gap size and band closure conditions. We review Bloch wave vector control of spectral characteristics in terms of distinct evanescent diffraction channels driving designated Bloch modes in the lattice. 

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