S. G. Haslinger

Transmission, trapping and filtering of waves in periodically constrained elastic plates

The paper discusses properties of flexural waves in elastic plates constrained periodically by rigid pins. A structured interface consists of rigid pin platonic gratings parallel to each other. Although the gratings have the same periodicity, relative shifts in horizontal and vertical directions are allowed. We develop a recurrence algorithm for constructing reflection and transmission matrices required to characterize the filtering of plane waves by the structured interface with shifted gratings. The representations of scattered fields contain both propagating and evanescent terms. Special attention is given to the analysis of trapped modes which may exist within the system of rigid pin gratings. Analytical findings are accompanied by numerical examples for systems of two and three gratings. We show geometries containing three gratings in which transmission resonances have very high quality factors (around 35 000). We also show that controlled lateral shifts of three gratings can give rise to a transmission peak with a sharp central suppression region, akin to the phenomenon of electromagnetic-induced transparency.

Symmetry and resonant modes in platonic grating stacks

We study the flexural wave modes existing in finite stacks of gratings containing rigid, zero-radius pins. We group the modes into even and odd classes, and derive dispersion equations for each. We study the recently discovered elasto-dynamically inhibited transmission (EDIT) phenomenon, and relate it to the occurrence of trapped waves of even and odd symmetries being simultaneously resonant. We show how the EDIT interaction may be steered over a wide range of frequencies and angles, using a strategy in which the single-grating reflectance is kept high, so enabling the quality factors of the even and odd resonances to be kept large.

Dynamic interfacial trapping of flexural waves in structured plates

The paper presents new results on the localization and transmission of flexural waves in a structured plate containing a semi-infinite two-dimensional array of rigid pins. In particular, localized waves are identified and studied at the interface boundary between the homogeneous part of the flexural plate and the part occupied by rigid pins. A formal connection has been made with the dispersion properties of flexural Bloch waves in an infinite doubly periodic array of rigid pins. Special attention is given to regimes corresponding to standing waves of different types as well as Dirac-like points that may occur on the dispersion surfaces. A single half-grating problem, hitherto unreported in the literature, is also shown to bring interesting solutions.

Localisation near defects and filtering of flexural waves in structured plates

The paper deals with localisation of flexural waves within gratings composed of either pinned points or rigid inclusions of finite radius in a structured plate. We study the filtering and resonant action of such systems. The effect of the finite size of inclusions on the dynamic localisation is analysed for the range of frequencies where only zeroth grating orders propagate. The structure of the resonant modes within gratings of inclusions is of special interest. In particular, we consider the circumstances under which such gratings can deliver for flexural waves a phenomenon similar to Electromagnetically Induced Transparency, where a resonant maximum of transmission is cut in two by a resonant minimum. We identify system designs which yield very high concentration of flexural fields within the interface that may lead to a further structural failure.

Localization in semi-infinite herringbone waveguides

The paper includes novel results for the scattering and localization of a time-harmonic flexural wave by a semi-infinite herringbone waveguide of rigid pins embedded within an elastic Kirchhoff plate. The analytical model takes into account the orientation and spacing of the constituent parts of the herringbone system, and incorporates dipole approximations for the case of closely spaced pins. Illustrative examples are provided, together with the predictive theoretical analysis of the localized waveforms.

Active Cloaking for Finite Clusters of Pins in Kirchhoff Plates

This paper considers active cloaking of a square array of evenly spaced pins in a Kirchhoff plate in the presence of flexural waves. Active sources, modeled as ideal point sources, are represented by the nonsingular Greens function for the two-dimensional biharmonic operator and have an arbitrary complex amplitude. These sources are distributed exterior to the cluster, and their complex amplitudes are found by solving an algebraic system of equations. This procedure ensures that selected multipole orders of the scattered field are successfully annulled. For frequencies in the zero-frequency stop band, we find that a small number of active sources located on a grid is sufficient for cloaking. For higher frequencies, we achieve efficient cloaking with the active sources positioned on a circle surrounding the cluster. We demonstrate the cloaking efficiency with several numerical illustrations, considering key frequencies from band diagrams and dispersion surfaces for a Kirchhoff plate pinned in a doubly per...

Structured interfaces for flexural waves – trapped modes and transmission resonances

The article combines the analytical models of scattering and Bloch waves for a stack of periodic gratings in an infinite elastic plate. The waves represent flexural deflections of the plate governed by a fourth-order partial differential equation. The emphasis is on the analysis of trapped modes and transmission resonances for different configurations of the grating stack and physical parameters of the flexural waves. Special attention is given to the phenomenon of Elasto-Dynamically Inhibited Transmission (EDIT). The analytical model is supplemented with comprehensive numerical examples.

Wave Localisation in Structured Elastic Plates

The paper presents the results of recent work on the modelling of flexural waves in elastic plates constrained periodically by rigid pins. It includes an outline of the analysis of the transmission problem for a stack of rigid pin gratings incorporating lateral shifts. We use a recurrence algorithm to determine reflection and transmission matrices which characterise the filtering of plane waves by the structured interface. The representations of scattered fields use the quasi-periodic Green’s function for a single grating. Both propagating and evanescent fields are taken into account. A special attention is given to the analysis of trapped modes which may exist within the system of rigid pin gratings. Analytical findings are accompanied by numerical examples.