Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

Saving the planet but losing the landscape: the impact of renewable energy policies on rural Britain

The main instrument of the Government's renewable energy policy is to promote wind power through regulation and subsidy. This gives rise to anomalies in rural planning when turbines are erected in sensitve areas in which other forms of development are strictly controlled. The situation is reviewed in the context of economic viability and considered also against the alternative of growing fuel crops. The latter are currently hampered by lack of Government support but could fulfil a useful secondary role of sustaining the agricultural sector and with it the management of lowland landscapes.