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.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

Acoustic scattering by an inhomogeneous layer on a rigid plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane

This paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance. The solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary, and the discussion of the paper is motivated by this application. The paper starts by deriving representations, as Laplace-type integrals, of the solution and its first partial derivatives. The evaluation of these integral representations by Gauss-Laguerre quadrature is discussed, and theoretical bounds on the truncation error are obtained. Specific approximations are proposed which are shown to be accurate except in the very near field, for all angles of incidence and a wide range of values of surface impedance. The paper finishes with derivations of partial results and analogous Laplace-type integral representations for the case of a point source.

Beyond urban legends : an emerging framework of urban ecology, as illustrated by the Baltimore Ecosystem Study

The emerging discipline of urban ecology is shifting focus from ecological processes embedded within cities to integrative studies of large urban areas as biophysical-social complexes. Yet this discipline lacks a theory. Results from the Baltimore Ecosystem Study, part of the Long Term Ecological Research Network, expose new assumptions and test existing assumptions about urban ecosystems. The findings suggest a broader range of structural and functional relationships than is often assumed for urban ecological systems. We address the relationships between social status and awareness of environmental problems, and between race and environmental hazard. We present patterns of species diversity, riparian function, and stream nitrate loading. In addition, we probe the suitability of land-use models, the diversity of soils, and the potential for urban carbon sequestration. Finally, we illustrate lags between social patterns and vegetation, the biogeochemistry of lawns, ecosystem nutrient retention, and social-biophysical feedbacks. These results suggest a framework for a theory of urban ecosystems.

Diversity for resilience : multi-scale application of agro-ecology

The integration of ecological principles into agricultural systems presents major opportunities for spreading risk at the crop and farm scale. This paper presents mechanisms by which diversity at several scales within the farming system can increase the stability of production. Diversity of above- and below-ground biota, but also genetic and phenotypic diversity within crops, has an essential role in safeguarding farm production. Novel mixtures of legume-grass leys have been shown to potentially provide significant benefits for pollinator and decomposer ecosystem services but to realise the greatest improvements carefully tailored farm management is needed such as mowing or grazing time, and the type and depth of cutivation. Complex farmland landscapes such as agroforestry systems have the potential to support pollinator abundance and diversity and spread risk across production enterprises. At the crop level, early results indicate that the vulnerability of pollen development, flowering and early grain set to abiotic stress can be ameliorated by managing flowering time through genotypic selection, and through the buffering effects of pollinators. Finally, the risk of sub-optimal quality in cereals can be mitigated through integration of near isogenic lines selected to escape specific abiotic stress events. We conclude that genotypic, phenotypic and community diversity can all be increased at multiple scales to enhance resilience in agricultural systems.

Curl free pressure gradients over orography in a solution of the fully compressible Euler equations with implicit treatment of acoustic and gravity waves

Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modelling flow over orography is introduced which guarantees curl free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain following grids. Curl-free gradients are achieved by using the co-variant components of velocity over orography rather than the usual horizontal and vertical components. In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves. Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of terrain-following layers. The model with implicit gravity waves is stable in strongly stratified conditions, with N∆t up to at least 10 (where N is the Brunt-V ̈ais ̈al ̈a frequency). A warm bubble rising over orography is simulated and the curl free pressure gradient formulation gives much more accurate results for this test case than a model without this mimetic property.

Managing artisanal and small-scale mining in forest areas: perspectives from a poststructural political ecology

Artisanal and small-scale mining (ASM) is an activity intimately associated with social deprivation and environmental degradation, including deforestation. This paper examines ASM and deforestation using a broadly poststructural political ecology framework. Hegemonic discourses are shown to consistently influence policy direction, particularly in emerging approaches such as Corporate Social Responsibility and the Forest Stewardship Council. A review of alternative discourses reveals that the poststructural method is useful for critiquing the international policy arena but does not inform new approaches. Synthesis of the analysis leads to conclusions that echo a growing body of literature advocating for policies to become increasingly sensitive to local contexts, synergistic between actors at difference scales, and to be integrated across sectors.