The distribution of net electric charge in graphene is investigated, using both a constitutive atomic charge-dipole interaction model and an approximate analytical solution to Laplace's equation. We demonstrate a strong size dependence of the charge distributions in finite-size, infinitelylong and multi-layered rectangular graphene sheets, respectively. We found that the charge density can be naturally enhanced up to 13 times at graphene's geometry edges. This edge charge enhancement effect becomes more significant when the length or the width of graphene increases.The charge enhancement ratio is found to follow a linear relationship with the number of layers.These results can be used to understand the newly experimentally observed electron emission, charge impurity and chemical doping phenomena in 2-dimension nanostructure. * Electronic address: wzzhao@yahoo.fr 1 arXiv:0909.3432v1 [cond-mat.mes-hall]

Resonant modes of an elliptic membrane are computed for a wide range of frequencies using a Galerkin formulation. Results are confirmed using Mathieu functions and finite-element methods. Algorithms and their implementations are described to handle Dirichlet or Neumann boundary conditions and draw animations or contour plots of the modal surfaces. The methods agree to four or more digit accuracy for the first one hundred modes. The effects of high function order and high frequency parameter upon the convergence of the modified Mathieu function series are discussed and quantified. The Galerkin method is conceptually simple and requires only an eigenvalue solver without the need of special functions.

The field scattered by each element in a finite array is decomposed into the contribution of the infinite or periodic array, a component due to the presence of the left edge of the array, and a component due to the right edge of the array. The scattered field due to either array edge is derived from a separate analysis of the appropriate semiinfinite array and is interpreted in terms of a decaying wave that is launched from the array end. © 1995 John Wiley & Sons, Inc.

The Biot-Tolstoy exact time-domain solution for the three-dimensional impulse response of an impenetrable wedge is extended to accommodate the isovelocity or density-contrast wedge. Fourier transformation of the time and axial variables, along with a Kantorovich-Lebedev transform applied to the cylindrical radial coordinate, leads to a solution in terms of residue series. When the wedge angle is a rational fraction of , the residue series can be reduced to a finite sum which is evaluated for some special cases. The total pressure field consists of geometrical acoustics contributions, as predicted by Snell's laws, plus a modified version of the Biot-Tolstoy diffraction field.

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