185 research outputs found
High frequency homogenisation for elastic lattices
A complete methodology, based on a two-scale asymptotic approach, that
enables the homogenisation of elastic lattices at non-zero frequencies is
developed. Elastic lattices are distinguished from scalar lattices in that two
or more types of coupled waves exist, even at low frequencies. Such a theory
enables the determination of effective material properties at both low and high
frequencies. The theoretical framework is developed for the propagation of
waves through lattices of arbitrary geometry and dimension. The asymptotic
approach provides a method through which the dispersive properties of lattices
at frequencies near standing waves can be described; the theory accurately
describes both the dispersion curves and the response of the lattice near the
edges of the Brillouin zone. The leading order solution is expressed as a
product between the standing wave solution and long-scale envelope functions
that are eigensolutions of the homogenised partial differential equation. The
general theory is supplemented by a pair of illustrative examples for two
archetypal classes of two-dimensional elastic lattices. The efficiency of the
asymptotic approach in accurately describing several interesting phenomena is
demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure
Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes
The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is
a long-standing problem with no known closed-form solution except in specific
cases. Taking this as a starting-point, and extending to a general
mean-reverting process, we investigate the long- and short-time asymptotics
using a combination of Hopf-Cole and Laplace transform techniques. As a result
we are able to give a single formula that is correct in both limits, as well as
being exact in certain special cases. We demonstrate the results using a
variety of other models
The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms
An asymptotic theory is developed to generate equations that model the global
behaviour of electromagnetic waves in periodic photonic structures when the
wavelength is not necessarily long relative to the periodic cell dimensions;
potentially highly-oscillatory short-scale detail is encapsulated through
integrated quantities.
The theory we develop is then applied to two topical examples, the first
being the case of aligned dielectric cylinders, which has great importance in
the modelling of photonic crystal fibres. We then consider the propagation of
waves in a structured metafilm, here chosen to be a planar array of dielectric
spheres. At certain frequencies strongly directional dynamic anisotropy is
observed, and the asymptotic theory is shown to capture the effect, giving
highly accurate qualitative and quantitative results as well as providing
interpretation for the underlying change from elliptic to hyperbolic behaviour
Analytical approximation to the multidimensional Fokker--Planck equation with steady state
The Fokker--Planck equation is a key ingredient of many models in physics,
and related subjects, and arises in a diverse array of settings. Analytical
solutions are limited to special cases, and resorting to numerical simulation
is often the only route available; in high dimensions, or for parametric
studies, this can become unwieldy. Using asymptotic techniques, that draw upon
the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system
and obtain its representation as a product of terms, representing short-term,
long-term, and medium-term behaviour. A further reduction yields a simple
explicit formula, both intuitive in terms of its physical origin and fast to
evaluate. We illustrate a breadth of cases, some of which are `far' from the OU
model, such as double-well potentials, and even then, perhaps surprisingly, the
approximation still gives very good results when compared with numerical
simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe
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
Viscothermal Losses in Double-Negative Acoustic Metamaterials
[EN] The influence of losses in double-negative metamaterial slabs recently introduced by Graciá-Salgado et al. [Phys. Rev. B 88, 224305 (2013)] is comprehensively studied. Viscous and thermal losses are considered in the linearized Navier-Stokes equations with no flow. Despite the extremely low thicknesses of boundary layers associated with each type of losses, the double-negative behavior is totally suppressed for the rigid structures under analysis. In other words, almost 100% of the energy transmitted into the slab is dissipated by viscothermal effects, in agreement with experimental data. Simulations undertaken for larger structures, using scale factors of up to 20 times, show that double-negative behavior is never recovered. The huge dissipation obtained by these structures leads us to propose them as interesting alternatives to conventional absorbers for specific situations, e.g., when treating low frequencies or when the excitation is narrow banded.V. M. G.-C. and J. S.-D. acknowledge the support from the Spanish Ministerio de Economia y Competitividad (MINECO), and the European Union Fondo Europeo de Desarrollo Regional (FEDER) through Project No. TEC 2014-53088-C3-1-R.Cutanda-Henriquez, V.; Garcia Chocano, VM.; Sánchez-Dehesa Moreno-Cid, J. (2017). Viscothermal Losses in Double-Negative Acoustic Metamaterials. Physical Review Applied. 8(1):014029-1-014029-12. doi:10.1103/PhysRevApplied.8.014029S014029-1014029-128
- …