312 results found
Batchvarov A, Kahouadji L, Constante-Amores CR, et al., 2021, Three-dimensional dynamics of falling films in the presence of insoluble surfactants, Journal of Fluid Mechanics, Vol: 906, Pages: A16-1-A16-13, ISSN: 0022-1120
We study the effect of insoluble surfactants on the wave dynamics of vertically falling liquid films. We use three-dimensional numerical simulations and employ a hybrid interface-tracking/level-set method, taking into account Marangoni stresses induced by gradients of interfacial surfactant concentration. Our numerical predictions for the evolution of the surfactant-free, three-dimensional wave topology are validated against the experimental work of Park & Nosoko (AIChE J., vol. 49, 2003, pp. 2715–2727). The addition of surfactants is found to influence significantly the development of horseshoe-shaped waves. At low Marangoni numbers, we show that the wave fronts exhibit spanwise oscillations before eventually acquiring a quasi-two-dimensional shape. In addition, the presence of Marangoni stresses is found to suppress the peaks of the travelling waves and preceding capillary wave structures. At high Marangoni numbers, a near-complete rigidification of the interface is observed.
Tang K, Makwana MP, Craster RV, et al., 2020, Observations of symmetry induced topological mode steering in a reconfigurable elastic plate, arXiv preprint, arXiv:1910.08172
Wiltshaw R, Craster R, Makwana M, 2020, Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers, Wave Motion, Vol: 99, ISSN: 0165-2125
We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach also lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on some topical examples from topological wave physics.
Chaplain GJ, Ponti JMD, Aguzzi G, et al., 2020, Topological rainbow trapping for elastic energy harvesting in graded Su-Schrieffer-Heeger systems, Applied Physics Letters, Vol: 14, Pages: 054035 – 1-054035 – 15, ISSN: 0003-6951
We amalgamate two fundamental designs from distinct areas of wave control in physics, and place them in the setting of elasticity. Graded elastic metasurfaces, so-called metawedges, are combined with the now classical Su-Schrieffer-Heeger (SSH) model from the field of topological insulators. The resulting structures form one-dimensional graded-SSH-metawedges that support multiple, simultaneous, topologically protected edge states. These robust, enhanced localised modes are leveraged for applications in elastic energy harvesting using the piezoelectric effect. The designs we develop are first motivated by applying the SSH model to mass-loaded Kirchhoff-Love thin elastic plates. We then extend these ideas to using graded resonant rods, and create SSH models, coupled to elastic beams and full elastic half-spaces.
De Ponti JM, Colombi A, Riva E, et al., 2020, Experimental investigation of amplification, via a mechanical delay-line, in a rainbow-based metamaterial for energy harvesting, APPLIED PHYSICS LETTERS, Vol: 117, ISSN: 0003-6951
Makwana M, Wiltshaw R, Guenneau S, et al., 2020, Hybrid topological guiding mechanisms for photonic crystal fibers, Optics Express, Vol: 28, Pages: 30871-30888, ISSN: 1094-4087
We create hybrid topological-photonic localisation of light by introducing concepts from the field of topological matter to that of photonic crystal fiber arrays. S-polarized obliquely propagating electromagnetic waves are guided by hexagonal, and square, lattice topological systems along an array of infinitely conducting fibers. The theory utilises perfectly periodic arrays that, in frequency space, have gapped Dirac cones producing band gaps demarcated by pronounced valleys locally imbued with a nonzero local topological quantity. These broken symmetry-induced stop-bands allow for localised guidance of electromagnetic edge-waves along the crystal fiber axis. Finite element simulations, complemented by asymptotic techniques, demonstrate the effectiveness of the proposed designs for localising energy in finite arrays in a robust manner.
Proctor M, Xiao X, Craster RV, et al., 2020, Near- and far-field excitation of topological plasmonic metasurfaces, Photonics, Vol: 7, ISSN: 2304-6732
The breathing honeycomb lattice hosts a topologically non-trivial bulk phase due to the crystalline-symmetry of the system. Pseudospin-dependent edge states, which emerge at the interface between trivial and non-trivial regions, can be used for the directional propagation of energy. Using the plasmonic metasurface as an example system, we probe these states in the near- and far-field using a semi-analytical model. We provide the conditions under which directionality was observed and show that it is source position dependent. By probing with circularly-polarised magnetic dipoles out of the plane, we first characterise modes along the interface in terms of the enhancement of source emissions due to the metasurface. We then excite from the far-field with non-zero orbital angular momentum beams. The position-dependent directionality holds true for all classical wave systems with a breathing honeycomb lattice. Our results show that a metasurfac,e in combination with a chiral two-dimensional material, could be used to guide light effectively on the nanoscale.
Batchvarov A, Kahouadji L, Magnini M, et al., 2020, Effect of surfactant on elongated bubbles in capillary tubes at high Reynolds number, Physical Review Fluids, Vol: 5, Pages: 093605 – 1-093605 – 21, ISSN: 2469-990X
The effect of surfactants on the tail and film dynamics of elongated gas bubbles propagating through circular capillary tubes is investigated by means of an extensive three-dimensional numerical study using a hybrid front-tracking/level-set method. The focus is on the visco-inertial regime, which occurs when the Reynolds number of the flow is much larger than unity. Under these conditions, “clean” bubbles exhibit interface undulations in the proximity of the tail, with an amplitude that increases with the Reynolds number. We perform a systematic analysis of the impact of a wide range of surfactant properties, including elasticity, bulk surfactant concentration, solubility, and diffusivity, on the bubble and flow dynamics in the presence of inertial effects. The results show that the introduction of surfactants is effective in suppressing the tail undulations as they tend to accumulate near the bubble tail. Here large Marangoni stresses are generated, which lead to a local “rigidification” of the bubble. This effect becomes more pronounced for larger surfactant elasticities and adsorption depths. At reduced surfactant solubility, a thicker rigid film region forms at the bubble rear, where a Couette film flow is established, while undulations still appear at the trailing edge of the downstream “clean” film region. In such conditions, the bubble length becomes an influential parameter, with short bubbles becoming completely rigid.
Chaplain GJ, Ponti JMD, Colombi A, et al., 2020, Tailored elastic surface to body wave Umklapp conversion, Nature Communications, Vol: 11, ISSN: 2041-1723
Elastic waves guided along surfaces dominate applications in geophysics, ultrasonic inspection, mechanical vibration, and surface acoustic wave devices; precise manipulation of surface Rayleigh waves and their coupling with polarised body waves presents a challenge that offers to unlock the flexibility in wave transport required for efficient energy harvesting and vibration mitigation devices. We design elastic metasurfaces, consisting of a graded array of rod resonators attached to an elastic substrate that, together with critical insight from Umklapp scattering in phonon-electron systems, allow us to leverage the transfer of crystal momentum; we mode-convert Rayleigh surface waves into bulk waves that form tunable beams. Experiments, theory and simulation verify that these tailored Umklapp mechanisms play a key role in coupling surface Rayleigh waves to reversed bulk shear and compressional waves independently, thereby creating passive self-phased arrays allowing for tunable redirection and wave focusing within the bulk medium.
Archer AJ, Wolgamot HA, Orszaghova J, et al., 2020, Experimental realization of broadband control of water-wave-energy amplification in chirped arrays, Physical Review Fluids, Vol: 5, Pages: 62801(R) – 1-62801(R) – 8, ISSN: 2469-990X
Water waves in natural environments are typically broadband, nonlinear anddynamic phenomena. Taking concepts developed for slow light in optics, weaddress the challenge of designing arrays to control the spatial distributionof wave energy, and amplify target frequencies at specified locations.Experiments on incident waves interacting with a chirped array of eightvertical cylinders demonstrate significant amplifications as predictednumerically, and provide motivation for application to energy harvesting.
Chaplain GJ, Pajer D, De Ponti JM, et al., 2020, Delineating rainbow reflection and trapping with applications for energy harvesting, New Journal of Physics, Vol: 22, Pages: 1-12, ISSN: 1367-2630
Important distinctions are made between two related wave control mechanisms that act to spatially separate frequency components; these so-called rainbow mechanisms either slow or reverse guided waves propagating along a graded line array. We demonstrate an important nuance distinguishing rainbow reflection from genuine rainbow trapping and show the implications of this distinction for energy harvesting designs, through inspection of the interaction time between slowed zero group velocity waves and the array. The difference between these related mechanisms is highlighted using a design methodology, applied to flexural waves on mass loaded thin Kirchhoff-Love elastic plates, and emphasised through simulations for energy harvesting in the setting of elasticity, by elastic metasurfaces of graded line arrays of resonant rods atop a beam. The delineation of these two effects, reflection and trapping, allows us to characterise the behaviour of forced line array systems and predict their capabilities for trapping, conversion and focusing of energy.
Chaplain GJ, Craster R, 2020, Ultrathin entirely flat Umklapp lenses, Physical Review B: Condensed Matter and Materials Physics, Vol: 101, Pages: 155430 – 1-155430 – 9, ISSN: 1098-0121
We design ultra-thin, entirely flat, dielectric lenses using crystal momentum transfer, so-called Umklapp processes, achieving the required wave control for a new mechanism of flat lensing; physically, these lenses take advantage of abrupt changes in the periodicity of a structured line array so there is an overlap between the first Brillouin zone of one medium with the second Brillouin zone of the other. At the interface between regions of different periodicity, surface, array guided waves hybridize into reversed propagating beams directed into the material exterior to the array. This control, and redirection, of waves then enables the device to emulate a Pendry-Veselago lens that is one unit cell in width, with no need for an explicit negative refractive index. Simulations using an array embedded in an idealized slab of silicon nitride (Si3N4) in air, operating at visible wavelengths between 420–500THz demonstrate the effect.
Makwana M, Laforge N, Craster R, et al., 2020, Experimental observations of topologically guided water waves within non-hexagonal structures, Applied Physics Letters, Vol: 116, Pages: 131603-1-131603-5, ISSN: 0003-6951
We investigate symmetry-protected topological water waves within a strategically engineered square lattice system. Thus far, symmetry-protected topological modes in hexagonal systems have primarily been studied in electromagnetism and acoustics, i.e. dispersionless media. Herein, we show experimentally how crucial geometrical properties of square structures allow for topological transport that is ordinarily forbidden within conventional hexagonal structures. We perform numerical simulations that take into account the inherent dispersion within water waves and devise a topological insulator that supports symmetry-protected transport along the domain walls. Our measurements, viewed with a high-speed camera under stroboscopic illumination, unambiguously demonstrate the valley-locked transport of water waves within a non-hexagonal structure. Due to the tunability of the energy's directionality by geometry, our results could be used for developing highly-efficient energy harvesters, filters and beam-splitters within dispersive media.
Haslinger SG, Lowe MJS, Huthwaite P, et al., 2020, Elastic shear wave scattering by randomly rough surfaces, Journal of the Mechanics and Physics of Solids, Vol: 137, Pages: 1-20, ISSN: 0022-5096
Characterizing cracks within elastic media forms an important aspect of ultrasonic non-destructive evaluation (NDE) where techniques such as time-of-flight diffraction and pulse-echo are often used with the presumption of scattering from smooth, straight cracks. However, cracks are rarely straight, or smooth, and recent attention has focussed upon rough surface scattering primarily by longitudinal wave excitations.We provide a comprehensive study of scattering by incident shear waves, thus far neglected in models of rough surface scattering despite their practical importance in the detection of surface-breaking defects, using modelling, simulation and supporting experiments. The scattering of incident shear waves introduces challenges, largely absent in the longitudinal case, related to surface wave mode-conversion, the reduced range of validity of the Kirchhoff approximation (KA) as compared with longitudinal incidence, and an increased importance of correlation length.The expected reflection from a rough defect is predicted using a statistical model from which, given the angle of incidence and two statistical parameters, the expected reflection amplitude is obtained instantaneously for any scattering angle and length of defect. If the ratio of correlation length to defect length exceeds a critical value, which we determine, there is an explicit dependence of the scattering results on correlation length, and we modify the modelling to find this dependence. The modelling is cross-correlated against Monte Carlo simulations of many different surface profiles, sharing the same statistical parameter values, using numerical simulation via ray models (KA) and finite element (FE) methods accelerated with a GPU implementation. Additionally we provide experimental validations that demonstrate the accuracy of our predictions.
Ungureanu B, Makwana M, Craster R, et al., 2020, Localising symmetry protected edge waves via the topological rainbow effect, Publisher: arXiv
We combine two different fields, topological physics and metamaterials to design a topological metasurface tocontrol and redirect elastic waves. We strategically design a two-dimensional crystalline perforated elastic platethat hosts symmetry-induced topological edge states. By concurrently allowing the elastic substrate to spatiallyvary in depth, we are able to convert the incident slow wave into a series of robust modes, with differing envelopemodulations. This adiabatic transition localises the incoming energy into a concentrated region where it can thenbe damped or extracted. For larger transitions, different behaviour is observed; the incoming energy propagatesalong the interface before being partitioned into two disparate chiral beams. This “topological rainbow” effectleverages two main concepts, namely the quantum valley-Hall effect and the rainbow effect usually associatedwith electromagnetic metamaterials. The topological rainbow effect transcends specific physical systems, hence,the phenomena we describe can be transposed to other wave physics. Due to the directional tunability of theelastic energy by geometry our results have far-reaching implications for applications such as switches, filtersand energy-harvesters.
Proctor M, Huidobro PA, Maier SA, et al., 2020, Manipulating topological valley modes in plasmonic metasurfaces, Nanophotonics, Vol: 9, Pages: 657-665, ISSN: 2192-8606
The coupled light-matter modes supported by plasmonic metasurfaces can be combined with topological principles to yield subwavelength topological valley states of light. We give a systematic presentation of the topological valley states available for lattices of metallic nanoparticles: All possible lattices with hexagonal symmetry are considered, as well as valley states emerging on a square lattice. Several unique effects which have yet to be explored in plasmonics are identified, such as robust guiding, filtering and splitting of modes, as well as dual-band effects. We demonstrate these by means of scattering computations based on the coupled dipole method that encompass the full electromagnetic interactions between nanoparticles.
Craster R, Maria de Ponti J, Colombi A, et al., 2020, Graded elastic metasurface for enhanced energy harvesting, New Journal of Physics, Vol: 22, Pages: 1-11, ISSN: 1367-2630
In elastic wave systems, combining the powerful concepts of resonance andspatial grading within structured surface arrays enable resonant metasurfaces to exhibitbroadband wave trapping, mode conversion from surface (Rayleigh) waves to bulk(shear) waves, and spatial frequency selection. Devices built around these conceptsallow for precise control of surface waves, often with structures that are subwavelength,and utilise Rainbow trapping that separates the signal spatially by frequency. Rainbowtrapping yields large amplifications of displacement at the resonator positions whereeach frequency component accumulates. We investigate whether this amplification, andthe associated control, can be used to create energy harvesting devices; the potentialadvantages and disadvantages of using graded resonant devices as energy harvesters isconsidered.We concentrate upon elastic plate models for which the A0 mode dominates, and takeadvantage of the large displacement amplitudes in graded resonant arrays of rods,to design innovative metasurfaces that trap waves for enhanced piezoelectric energyharvesting. Numerical simulation allows us to identify the advantages of such gradedmetasurface devices and quantify its efficiency, we also develop accurate models ofthe phenomena and extend our analysis to that of an elastic half-space and Rayleighsurface waves.
Haslinger SG, Lowe MJS, Huthwaite P, et al., 2019, Appraising Kirchhoff approximation theory for the scattering of elastic shear waves by randomly rough defects, Journal of Sound and Vibration, Vol: 460, Pages: 1-16, ISSN: 0022-460X
Rapid and accurate methods, based on the Kirchhoff approximation (KA), are developed to evaluate the scattering of shear waves by rough defects and quantify the accuracy of this approximation. Defect roughness has a strong effect on the reflection of ultrasound, and every rough defect has a different surface, so standard methods of assessing the sensitivity of inspection based on smooth defects are necessarily limited. Accurately resolving rough cracks in non-destructive evaluation (NDE) inspections often requires shear waves since they have higher sensitivity to surface roughness than longitudinal waves. KA models are attractive, since they are rapid to deploy, however they are an approximation and it is important to determine the range of validity for the scattering of ultrasonic shear waves; this range is found here. Good agreement between KA and high fidelity finite element simulations is obtained for a range of incident/scattering angles, and the limits of validity for KA are found to be much stricter than for longitudinal wave incidence; as the correlation length of rough surfaces is reduced to the order of the incident shear wavelength, a combination of multiple scattering and surface wave mode conversion leads to KA predictions diverging from those of the true diffuse scattered fields.
Proctor M, Craster RV, Maier SA, et al., 2019, Exciting Pseudospin-Dependent Edge States in Plasmonic Metasurfaces, ACS PHOTONICS, Vol: 6, Pages: 2985-2995, ISSN: 2330-4022
Bennetts LG, Peter MA, Craster RV, 2019, Low-frequency wave-energy amplification in graded two-dimensional resonator arrays, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 377, ISSN: 1364-503X
Theodorakis PE, Smith ER, Craster RV, et al., 2019, Molecular dynamics simulation of the super spreading of surfactant-laden droplets. A review, Fluids, Vol: 4, Pages: 1-23, ISSN: 2311-5521
Superspreading is the rapid and complete spreading of surfactant-laden droplets on hydrophobic substrates. This phenomenon has been studied for many decades by experiment, theory, and simulation, but it has been only recently that molecular-level simulation has provided significant insights into the underlying mechanisms of superspreading thanks to the development of accurate force-fields and the increase of computational capabilities. Here, we review the main advances in this area that have surfaced from Molecular Dynamics simulation of all-atom and coarse-grained models highlighting and contrasting the main results and discussing various elements of the proposed mechanisms for superspreading. We anticipate that this review will stimulate further research on the interpretation of experimental results and the design of surfactants for applications requiring efficient spreading, such as coating technology.
Ungureanu B, Guenneau S, Achaoui Y, et al., 2019, The influence of building interactions on seismic and elastic body waves, EPJ Applied Metamaterials, Vol: 6, Pages: 1-12, ISSN: 2272-2394
We outline some recent research advances on the control of elastic waves in thin and thick plates, that have occurred since the large scale experiment [S. Brûlé, Phys. Rev. Lett. 112, 133901 (2014)] that demonstrated significant interaction of surface seismic waves with holes structuring sedimentary soils at the meter scale. We further investigate the seismic wave trajectories of compressional body waves in soils structured with buildings. A significant substitution of soils by inclusions, acting as foundations, raises the question of the effective dynamic properties of these structured soils. Buildings, in the case of perfect elastic conditions for both soil and buildings, are shown to interact and strongly influence elastic body waves; such site-city seismic interactions were pointed out in [Guéguen et al., Bull. Seismol. Soc. Am. 92, 794–811 (2002)], and we investigate a variety of scenarios to illustrate the variety of behaviours possible.
Huidobro PA, Galiffi E, Guenneau S, et al., 2019, Fresnel drag in space-time-modulated metamaterials, Publisher: arXiv
A moving medium drags light along with it as measured by Fizeau and explained by Einstein's theory of special relativity. Here we show that the same effect can be obtained in a situation where there is no physical motion of the medium. Modulations of both the permittivity and permeability, phased in space and time in the form of travelling waves, are the basis of our model. Space-time metamaterials are represented by effective bianisotropic parameters, which can in turn be mapped to a moving homogeneous medium. Hence these metamaterials mimic a relativistic effect without the need for any actual material motion. We discuss how both the permittivity and permeability need to be modulated in order to achieve these effects, and we present an equivalent transmission line model.
Chaplain GJ, Craster R, 2019, Flat lensing by graded line meta-arrays, Publisher: AMER PHYSICAL SOC
Makwana M, Craster R, Guenneau S, 2019, Topological beam-splitting in photonic crystals, Publisher: OPTICAL SOC AMER
Makwana M, Craster R, Guenneau S, 2019, Topological beam-splitting in photonic crystals, Optics Express, Vol: 27, Pages: 16088-16102, ISSN: 1094-4087
We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric and magnetic polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air, in particular we place several inclusions within a cell that is then extended periodically upon a square lattice. Hexagonal lattice structures are more commonly used in topological valleytronics but, as we discuss, three-way splitting is only possible using a square, or rectangular, lattice. To achieve splitting and transport around a sharp bend we use accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are either arranged around a triangle or square; we demonstrate the mechanism of splitting and why it does not occur for one of the cases. The theory is developed and full scattering simulations demonstrate the effectiveness of the proposed designs.
Palmer S, Xiao X, Pazos-Perez N, et al., 2019, Extraordinarily transparent compact metallic metamaterials, Nature Communications, Vol: 10, ISSN: 2041-1723
The design of achromatic optical components requires materials with high transparency and low dispersion. We show that although metals are highly opaque, densely packed arrays of metallic nanoparticles can be more transparent to infrared radiation than dielectrics such as germanium, even when the arrays are over 75% metal by volume. Such arrays form effective dielectrics that are virtually dispersion-free over ultra-broadband ranges of wavelengths from microns up to millimeters or more. Furthermore, the local refractive indices may be tuned by altering the size, shape, and spacing of the nanoparticles, allowing the design of gradient-index lenses that guide and focus light on the microscale. The electric field is also strongly concentrated in the gaps between the metallic nanoparticles, and the simultaneous focusing and squeezing of the electric field produces strong ‘doubly-enhanced’ hotspots which could boost measurements made using infrared spectroscopy and other non-linear processes over a broad range of frequencies.
Martin RJ, Kearney MJ, Craster R, 2019, Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes, Journal of Physics A: Mathematical and Theoretical, Vol: 52, ISSN: 1751-8113
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.
Acoustic flat lensing is achieved here by tuning a phononic array to have indefinite medium behavior in a narrow frequency spectral region along the acoustic branch in the irreducible Brillouin zone (IBZ). This is confirmed by the occurrence of a flat band along an unusual path in the IBZ and by interpreting the intersection point of isofrequency contours on the corresponding isofrequency surface; coherent directive collimated beams are formed whose reflection from the array surfaces create lensing. Theoretical predictions using a mass-spring lattice approximation of the phononic crystal (PC) are corroborated by time-domain experiments, airborne acoustic waves generated by a source with a frequency centered about 10.6 kHz, placed at three different distances from one side of a finite PC slab, constructed from polymeric spheres, yielding distinctive focal spots on the other side. These experiments evaluate the pressure field using optical feedback interferometry and demonstrate precise control of the three-dimensional wave trajectory through a sonic crystal.
Vanel AL, Craster RV, Schnitzer O, 2019, Asymptotic modelling of phononic box crystals, SIAM Journal on Applied Mathematics, Vol: 79, Pages: 506-524, ISSN: 0036-1399
We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a three-dimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbour interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs and crystals, that agree favourably with finite-element simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole- with dipole-type forcing; for the former the pressure field is asymptotically enhanced whilst for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann's Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array.
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