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Table Of Integrals Series And Products

Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work. The fascinating wave phenomenon of ‘bound states in the continuum’ spans different material and wave systems, including electron, electromagnetic and mechanical waves. In this Review, we focus on the common physical mechanisms underlying these bound states, whilst also discussing recent experimental realizations, current applications and future opportunities for research.

Bound states in the continuum

United States. Army Research Office (Institute for Soldier Nanotechnologies under contract no. W911NF-07-D0004)

https://www.mit.edu/~soljacic/trapped-light_Nature.pdf

Observation of trapped light within the radiation continuum

http://ieeexplore.ieee.org/iel5/5/31047/01444179.pdf

Fundamentals of acoustics

Oscillating-water-column wave energy converters and air turbines: A review

Hydrodynamic characteristics of an oscillating water column device

Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

Complementary approximations to wave scattering by vertical barriers

An explicit solution is provided for the scattering of an obliquely incident flexural-gravity wave by a narrow straight-line crack separating two semi-infinite thin elastic plates floating on water of finite depth. By first separating the solution into the sum of symmetric and antisymmetric parts it is shown that a simple form for each part can be derived in terms of a rapidly convergent infinite series multiplied by a fundamental constant of the problem. This constant is simply determined by applying an appropriate edge condition. Curves of reflection and transmission coefficients are presented, showing how they vary with plate properties and angle of incidence. It is also shown that in the absence of incident waves and for certain relations between their wavelength and frequency, symmetric edge waves exist which travel along the crack and decay in a direction normal to the crack.

Wave scattering by narrow cracks in ice sheets floating on water of finite depth

https://msvlab.hre.ntou.edu.tw/grades/Integral-equation(2009)/Y.J.Lin/Near-trapping%20of%20waves%20by%20circular%20arrays%20of.pdf

Near-trapping of waves by circular arrays of vertical cylinders

Rayleigh–Bloch surface waves are acoustic or electromagnetic waves which propagate parallel to a two-dimensional diffraction grating and which are exponentially damped with distance from the grating. In the water-wave context they describe a localized wave having dominant wavenumber β travelling along an infinite periodic array of identical bottom-mounted cylinders having uniform cross-section throughout the water depth. A numerical method is described which enables the frequencies of the Rayleigh–Bloch waves to be determined as a function of β for an arbitrary cylinder cross-section. For particular symmetric cylinders, it is shown how a special choice of β produces results for the trapped mode frequencies and mode shapes in the vicinity of any (finite) number of cylinders spanning a rectangular waveguide or channel. It is also shown how one particular choice of β gives rise to a new type of trapped mode near an unsymmetric cylinder contained within a parallel-sided waveguide with locally-distorted walls. The implications for large forces due to incident waves on a large but finite number of such cylinders in the ocean is discussed.

Richard Porter

Papers

Hydrodynamic characteristics of an oscillating water column device

Complementary approximations to wave scattering by vertical barriers

Wave scattering by narrow cracks in ice sheets floating on water of finite depth

Near-trapping of waves by circular arrays of vertical cylinders

Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides