Mathematical analysis of guided water waves

Projet IDENTIn this article, we present a detailed mathematical analysis of the phenomenon of water waves guided by the coast. We work with the mathematical model issued from the linearized theory of gravity waves. Our results mainly concern the existence, the number and the high and low frequency behaviours of the guided modes. These results point out a large variety of phenomena which are deeply influenced by the geometry of the coast. We obtain them with the help of the spectral theory of unbounded selfadjoint operators. Some numerical results are presented at the end of the paper to illustrate the theory

Stochastic band structure for waves propagating in periodic media or along waveguides

We introduce the stochastic band structure, a method giving the dispersion relation for waves propagating in periodic media or along waveguides, and subject to material loss or radiation damping. Instead of considering an explicit or implicit functional relation between frequency $\omega$ and wavenumber $k$, as is usually done, we consider a mapping of the resolvent set in the dispersion space $(\omega, k)$. Bands appear as as the trace of Lorentzian responses containing local information on propagation loss both in time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The stochastic band structure can be obtained for any system described by a time-harmonic wave equation

Fluid-loaded metasurfaces

We consider wave propagation along fluid-loaded structures which take the form of an elastic plate augmented by an array of resonators forming a metasurface, that is, a surface structured with sub-wavelength resonators. Such surfaces have had considerable recent success for the control of wave propagation in electromagnetism and acoustics, by combining the vision of sub-wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. We explore one aspect of recent interest in this field: graded metasurfaces, but within the context of fluid-loaded structures. Graded metasurfaces allow for selective spatial frequency separation and are often referred to as exhibiting rainbow trapping. Experiments, and theory, have been developed for acoustic, electromagnetic, and even elastic, rainbow devices but this has not been approached for fluid-loaded structures that support surface waves coupled with the acoustic field in a bulk fluid. This surface wave, coupled with the fluid, can be used to create an additional effect by designing a metasurface to mode convert from surface to bulk waves. We demonstrate that sub-wavelength control is possible and that one can create both rainbow trapping and mode conversion phenomena for a fluid-loaded elastic plate model.Comment: 13 pages, 10 figure

Explicit asymptotic modelling of transient Love waves propagated along a thin coating

The official published version can be obtained from the link below.An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples.This work is sponsored by the grant from Higher Education of Pakistan and by the Brunel University’s “BRIEF” research award

Trapped modes in finite quantum waveguides

The Laplace operator in infinite quantum waveguides (e.g., a bent strip or a twisted tube) often has a point-like eigenvalue below the essential spectrum that corresponds to a trapped eigenmode of finite L2 norm. We revisit this statement for resonators with long but finite branches that we call "finite waveguides". Although now there is no essential spectrum and all eigenfunctions have finite L2 norm, the trapping can be understood as an exponential decay of the eigenfunction inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped states. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two stripes, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for being applied to microscopic quantum devices

Polarization-resolved sensing with tilted fiber Bragg gratings: theory and limits of detection

Polarization based sensing with tilted fiber Bragg grating (TFBG) sensors is analysed theoretically by two alternative approaches. The first method is based on tracking the grating transmission for two orthogonal states of linear polarized light that are extracted from the measured Jones matrix or Stokes vectors of the TFBG transmission spectra. The second method is based on the measurements along the system principle axes and polarization dependent loss (PDL) parameter, also calculated from measured data. It is shown that the frequent crossing of the Jones matrix eigenvalues as a function of wavelength leads to a non-physical interchange of the calculated principal axes; a method to remove this unwanted mathematical artefact and to restore the order of the system eigenvalues and the corresponding principal axes is provided. A comparison of the two approaches reveals that the PDL method provides a smaller standard deviation and therefore lower limit of detection in refractometric sensing. Furthermore, the polarization analysis of the measured spectra allows for the identification of the principal states of polarization of the sensor system and consequentially for the calculation of the transmission spectrum for any incident polarization state. The stability of the orientation of the system principal axes is also investigated as a function of wavelength

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008