A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Wavelets and the generalization of the variogram

The experimental variogram computed in the usual way by the method of moments and the Haar wavelet transform are similar in that they filter data and yield informative summaries that may be interpreted. The variogram filters out constant values; wavelets can filter variation at several spatial scales and thereby provide a richer repertoire for analysis and demand no assumptions other than that of finite variance. This paper compares the two functions, identifying that part of the Haar wavelet transform that gives it its advantages. It goes on to show that the generalized variogram of order k=1, 2, and 3 filters linear, quadratic, and cubic polynomials from the data, respectively, which correspond with more complex wavelets in Daubechies's family. The additional filter coefficients of the latter can reveal features of the data that are not evident in its usual form. Three examples in which data recorded at regular intervals on transects are analyzed illustrate the extended form of the variogram. The apparent periodicity of gilgais in Australia seems to be accentuated as filter coefficients are added, but otherwise the analysis provides no new insight. Analysis of hyerpsectral data with a strong linear trend showed that the wavelet-based variograms filtered it out. Adding filter coefficients in the analysis of the topsoil across the Jurassic scarplands of England changed the upper bound of the variogram; it then resembled the within-class variogram computed by the method of moments. To elucidate these results, we simulated several series of data to represent a random process with values fluctuating about a mean, data with long-range linear trend, data with local trend, and data with stepped transitions. The results suggest that the wavelet variogram can filter out the effects of long-range trend, but not local trend, and of transitions from one class to another, as across boundaries.

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.

From the sun to the Earth: The 13 May 2005 coronal mass ejection

We report the results of a multi-instrument, multi-technique, coordinated study of the solar eruptive event of 13 May 2005. We discuss the resultant Earth-directed (halo) coronal mass ejection (CME), and the effects on the terrestrial space environment and upper Earth atmosphere. The interplanetary CME (ICME) impacted the Earth’s magnetosphere and caused the most-intense geomagnetic storm of 2005 with a Disturbed Storm Time (Dst) index reaching −263 nT at its peak. The terrestrial environment responded to the storm on a global scale. We have combined observations and measurements from coronal and interplanetary remote-sensing instruments, interplanetary and near-Earth in-situ measurements, remote-sensing observations and in-situ measurements of the terrestrial magnetosphere and ionosphere, along with coronal and heliospheric modelling. These analyses are used to trace the origin, development, propagation, terrestrial impact, and subsequent consequences of this event to obtain the most comprehensive view of a geo-effective solar eruption to date. This particular event is also part of a NASA-sponsored Living With a Star (LWS) study and an on-going US NSF-sponsored Solar, Heliospheric, and INterplanetary Environment (SHINE) community investigation.

Variational calculations of rovibrational states: a precise high-energy potential surface for HCN [and discussion]

We report the results of variational calculations of the rovibrational energy levels of HCN for J = 0, 1 and 2, where we reproduce all the ca. 100 observed vibrational states for all observed isotopic species, with energies up to 18000 cm$^{-1}$, to about $\pm $1 cm$^{-1}$, and the corresponding rotational constants to about $\pm $0.001 cm$^{-1}$. We use a hamiltonian expressed in internal coordinates r$_{1}$, r$_{2}$ and $\theta $, using the exact expression for the kinetic energy operator T obtained by direct transformation from the cartesian representation. The potential energy V is expressed as a polynomial expansion in the Morse coordinates y$_{i}$ for the bond stretches and the interbond angle $\theta $. The basis functions are built as products of appropriately scaled Morse functions in the bond-stretches and Legendre or associated Legendre polynomials of cos $\theta $ in the angle bend, and we evaluate matrix elements by Gauss quadrature. The hamiltonian matripx is factorized using the full rovibrational symmetry, and the basis is contracted to an optimized form; the dimensions of the final hamiltonian matrix vary from 240 $\times $ 240 to 1000 $\times $ 1000.We believe that our calculation is converged to better than 1 cm$^{-1}$ at 18 000 cm$^{-1}$. Our potential surface is expressed in terms of 31 parameters, about half of which have been refined by least squares to optimize the fit to the experimental data. The advantages and disadvantages and the future potential of calculations of this type are discussed.

Intercropping with pulses to concentrate nitrogen and sulphur in wheat

The effects of intercropping wheat with faba bean (Denmark, Germany, Italy and UK) and wheat with pea (France), in additive and replacement designs on grain nitrogen and sulphur concentrations were studied in field experiments in the 2002/03, 2003/04 and 2004/05 growing seasons. Intercropping wheat with grain legumes regularly increased the nitrogen concentration of the cereal grain, irrespective of design or location. Sulphur concentration of the cereal was also increased by intercropping, but less regularly and to a lesser extent compared with effects on nitrogen concentration. Nitrogen concentration (g/kg) in wheat additively intercropped with faba bean was increased by 8% across all sites (weighted for inverse of variance), but sulphur concentration was only increased by 4%, so N:S ratio was also increased by 4%. Intercropping wheat with grain legumes increased sodium dodecyl sulphate (SDS)-sedimentation volume. The effect of intercropping on wheat nitrogen concentration was greatest when intercropping had the most deleterious effect on wheat yield and the least deleterious effect on pulse yield. Over all sites and seasons, and irrespective of whether the design was additive or replacement, increases in crude protein concentration in the wheat of 10 g/kg by intercropping with faba bean were associated with 25-30% yield reduction of the wheat, compared with sole-cropped wheat. It was concluded that the increase in protein concentration of wheat grain in intercrops could be of economic benefit when selling wheat for breadmaking, but only if the bean crop was also marketed effectively.

Pea-barley intercrops use nitrogen sources 20-30% more efficiently than the sole crops

Field pea (Pisum sativum L.) and spring barley (Hordeum vulgare L) were intercropped and sole cropped to compare the effects of crop diversity on the use of nitrogen sources in European organic crop-ping systems. Across a wide range of growing condi-tions pea-barley intercropping showed that nitrogen sources were used from 17 to 31% more efficiently by the intercrop than by the sole crops. Intercropping technologies offers the opportunity for organic cropping systems to utilize N complementarity between component crops, without compromising total crop N yield levels

Pea-barley intercropping for efficient symbiotic N-2-fixation, soil N acquisition and use of other nutrients in European organic cropping systems

Complementarity in acquisition of nitrogen (N) from soil and N-2-fixation within pea and barley intercrops was studied in organic field experiments across Western Europe (Denmark, United Kingdom, France, Germany and Italy). Spring pea and barley were sown either as sole crops, at the recommended plant density (P100 and B100, respectively) or in replacement (P50B50) or additive (P100B50) intercropping designs, in each of three cropping seasons (2003-2005). Irrespective of site and intercrop design, Land Equivalent Ratios (LER) between 1.4 at flowering and 1.3 at maturity showed that total N recovery was greater in the pea-barley intercrops than in the sole Crops Suggesting a high degree of complementarity over a wide range of growing conditions. Complementarity was partly attributed to greater soil mineral N acquisition by barley, forcing pea to rely more on N-2-fixation. At all sites the proportion of total aboveground pea N that was derived from N-2-fixation was greater when intercropped with barley than when grown as a sole crop. No consistent differences were found between the two intercropping designs. Simultaneously, the accumulation Of Phosphorous (P), potassium (K) and sulphur (S) in Danish and German experiments was 20% higher in the intercrop (P50B50) than in the respective sole crops, possibly influencing general crop yields and thereby competitive ability for other resources. Comparing all sites and seasons, the benefits of organic pea-barley intercropping for N acquisition were highly resilient. It is concluded that pea-barley intercropping is a relevant cropping strategy to adopt when trying to optimize N-2-fixation inputs to the cropping system. (C) 2009 Elsevier B.V. All rights reserved.

Pea-barley intercropping and short-term subsequent crop effects across European organic cropping conditions

Grain legumes are known to increase the soil mineral nitrogen (N) content, reduce the infection pressure of soil borne pathogens, and hence enhance subsequent cereals yields. Replicated field experiments were performed throughout W. Europe (Denmark, United Kingdom, France, Germany and Italy) to asses the effect of intercropping pea and barley on the N supply to subsequent wheat in organic cropping systems. Pea and barley were grown either as sole crops at the recommended plant density (P100 and B100, respectively) or in replacement (P50B50) or additive (P100B50) intercropping designs. In the replacement design the total relative plant density is kept constant, while the additive design uses the optimal sole crop density for pea supplementing with 'extra' barley plants. The pea and barley crops were followed by winter wheat with and without N application. Additional experiments in Denmark and the United Kingdom included subsequent spring wheat with grass-clover as catch crops. The experiment was repeated over the three cropping seasons of 2003, 2004 and 2005. Irrespective of sites and intercrop design pea-barley intercropping improved the plant resource utilization (water, light, nutrients) to grain N yield with 25-30% using the Land Equivalent ratio. In terms of absolute quantities, sole cropped pea accumulated more N in the grains as compared to the additive design followed by the replacement design and then sole cropped barley. The post harvest soil mineral N content was unaffected by the preceding crops. Under the following winter wheat, the lowest mineral N content was generally found in early spring. Variation in soil mineral N content under the winter wheat between sites and seasons indicated a greater influence of regional climatic conditions and long-term cropping history than annual preceding crop and residue quality. Just as with the soil mineral N, the subsequent crop response to preceding crop was negligible. Soil N balances showed general negative values in the 2-year period, indicating depletion of N independent of preceding crop and cropping strategy. It is recommended to develop more rotational approaches to determine subsequent crop effects in organic cropping systems, since preceding crop effects, especially when including legumes, can occur over several years of cropping.