The interaction of flexural-gravity waves with periodic geometries

A periodic structure of finite extent is embedded within an otherwise uniform two-dimensional system consisting of finite-depth fluid covered by a thin elastic plate. An incident harmonic flexural-gravity wave is scattered by the structure. By using an approximation to the corresponding linearised boundary value problem that is based on a slowly varying structure in conjunction with a transfer matrix formulation, a method is developed that generates the whole solution from that for just one cycle of the structure, providing both computational savings and insight into the scattering process. Numerical results show that variations in the plate produce strong resonances about the ‘Bragg frequencies’ for relatively few periods. We find that certain geometrical variations in the plate generate these resonances above the Bragg value, whereas other geometries produce the resonance below the Bragg value. The familiar resonances due to periodic bed undulations tend to be damped by the plate.

2D hexagonal mesoporous platinum films exhibiting biaxial, in-plane pore alignment

The synthesis of 2D hexagonal mesoporous platinum films with biaxial, in-plane pore alignment is demonstrated by electrodeposition through an aligned lyotropic liquid crystal templating phase. Shear force is used to align a hexagonal lyotropic liquid crystalline templating phase of an inexpensive and a commercially available surfactant, C16EO10, at the surface of an electrode. Electrodeposition and subsequent characterisation of the films produced shows that the orientation and alignment of the phase is transferred to the deposited material. Transmission electron microscopy confirms the expected nanostructure of the films, whilst transmission and grazing incidence small angle X-ray scattering analysis confirms biaxial, in plane alignment of the pore structure. In addition further electrochemical studies in dilute sulfuric acid and methanol show that the pores are accessible to electrolyte solution as indicated by a large current flow; the modified electrode therefore has a high surface area, that catalyses methanol oxidation, and the pores have a very large aspect ratio (of theoretical maximum 2 × 105). Films with such aligned mesoporosity will advance the field of nanotechnology where the control of pore structure is paramount. The method reported is sufficiently generic to be used to control the structure and order of many materials, thus increasing the potential for the development of a wide range of novel electronic and optical devices.

Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions

New representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane, and for the Green's function for sound propagation in the canyon formed by two infinitely high, parallel rigid or sound soft walls and an impedance ground surface. The infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a Laplace-type integral. A pole subtraction technique is used to remove poles in the integrand which lie near the path of integration, obtaining a smooth integrand, more suitable for numerical integration, and a specific numerical integration method is proposed. Numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types. It is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts, and for problems of scattering by periodic surfaces.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Complex dynamics in simple systems with periodic parameter oscillations

We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of ϵ-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.

On the quasi-periodic nature of magnetopause flux transfer events

The recurrence rate of flux transfer events (FTEs) observed near the dayside magnetopause is discussed. A survey of magnetopause observations by the ISEE satellites shows that the distribution of the intervals between FTE signatures has a mode value of 3 min, but is highly skewed, having upper and lower decile values of 1.5 min and 18.5 min, respectively. The mean value is found to be 8 min, consistent with previous surveys of magnetopause data. The recurrence of quasi-periodic events in the dayside auroral ionosphere is frequently used as evidence for an association with magnetopause FTEs, and the distribution of their repetition intervals should be matched to that presented here if such an association is to be confirmed. A survey of 1 year's 15-s data on the interplanetary magnetic field (IMF) suggests that the derived distribution could arise from fluctuations in the IMF Bz component, rather than from a natural oscillation frequency of the magnetosphere-ionosphere system.

Periodic auroral events at the high-latitude convection reversal in the 16 MLT region

Combined optical and radar observations of two breakup-like auroral events near the polar cap boundary, within 74–76° MLAT and 1210 – 1240 UT (roughly 1540 – 1610 MLT) on 9 Jan. 1989 are reported. A two-component structure of the auroral phenomenon is indicated, with a local intensification of the pre-existing arc as well as a separate, tailward moving discrete auroral event on the poleward side of the background aurora, close to the reversal between well-defined zones of sunward and tailward ion flows. The all-sky TV observations do not indicate a connection between the two components, which also show different optical spectral composition. The 16 MLT background arc is located on sunward convecting field lines, as opposed to the 12–14 MLT auroral emission observed on this day. Although the magnetospheric plasma source (s) of the 16 MLT events are not easily identified from these ground-based data alone, it is suggested that the lower and higher latitude components, may map to the plasma sheet boundary layer and along open field lines to the magnetopause boundary, respectively. The events occur at the time of enhancements of westward ionospheric ion flow and corresponding eastward electrojet current south of 74° MLAT. Thus, they seem to be very significant events, involving periodic (10 min period), tailward moving filaments of field-aligned current/discrete auroral emission at the 16 MLT polar cap boundary.

On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

We study the topology of a set naturally arising from the study of β-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of β-expansions are equal and explicitly calculable.

Aligned platinum nanowire networks from surface-oriented lipid cubic phase templates

Mesoporous metal structures featuring a bicontinuous cubic morphology have a wide range of potential applications and novel opto-electronic properties, often orientation-dependent. We describe the production of nanostructured metal films 1–2 microns thick featuring 3D-periodic ‘single diamond’ morphology that show high out-of-plane alignment, with the (111) plane oriented parallel to the substrate. These are produced by electrodeposition of platinum through a lipid cubic phase (QII) template. Further investigation into the mechanism for the orientation revealed the surprising result that the QII template, which is tens of microns thick, is polydomain with no overall orientation. When thicker platinum films are grown, they also show increased orientational disorder. These results suggest that polydomain QII samples display a region of uniaxial orientation at the lipid/substrate interface up to approximately 2.8 ± 0.3 μm away from the solid surface. Our approach gives previously unavailable information on the arrangement of cubic phases at solid interfaces, which is important for many applications of QII phases. Most significantly, we have produced a previously unreported class of oriented nanomaterial, with potential applications including metamaterials and lithographic masks.