The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

MATHEMATICA notebook for computing tetrahedral finite element shape functions and matrices for the Helmholtz equation

A MATHEMATICA notebook to compute the elements of the matrices which arise in the solution of the Helmholtz equation by the finite element method (nodal approximation) for tetrahedral elements of any approximation order is presented. The results of the notebook enable a fast computational implementation of finite element codes for high order simplex 3D elements reducing the overheads due to implementation and test of the complex mathematical expressions obtained from the analytical integrations. These matrices can be used in a large number of applications related to physical phenomena described by the Poisson, Laplace and Schrodinger equations with anisotropic physical properties.

Kelvin-Helmholtz instabilities at the magnetic cavity boundary of comet 67P/Churyumov-Gerasimenko

We investigate the plasma environment of comet 67P/Churyumov-Gerasimenko, the target of the European Space Agency's Rosetta mission. Rosetta will rendezvous with the comet in 2014 at almost 3.5 AU and follow it all the way to and past perihelion at 1.3 AU. During its journey towards the inner solar system the comet's environment will significantly change. The interaction of the solar wind with a well developed neutral coma leads to the formation of an upstream bow shock and, closer to the comet, the inner shock separating the solar wind, with cometary pick-up ions mass-loaded, from the inner cometary ions which are dragged outward through abundant collisions and charge exchange with the expanding neutral gas. As a consequence the interplanetary magnetic field is prevented from penetrating the innermost region of the comet, the so-called magnetic cavity. We use our magnetohydrodynamics model BATSRUS (Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme) to simulate the solar wind - comet interaction. The model includes photoionization, ion-electron recombination, and charge exchange. Under certain conditions our model predicts an unstable plasma flow at the inner shock. We show that the plasma shear flow around the magnetic cavity can lead to Kelvin-Helmholtz instabilities. We investigate the onset of this phenomenon with change of heliocentric distance and furthermore show that a previously stable magnetic cavity boundary can become unstable when the neutral gas is predominately released from the dayside of the comet.

Sound and music: an elementary treatise on the physical constitution of musical sounds and harmony, including the chief acoustical discoveries of Professor Helmholtz.

Mode of access: Internet.

Handbuch der physiologischen Optik / von H. von Helmholtz.

"Issued in 17 parts, 1886-1896." - L.C.

The problem of human life : containing the fundamental principles of the substantial philosophy; with a review of the six great modern scientists: Darwin, Tyndall, Huxley, Haeckel, Helmholtz, and Mayer ... /

Printed in two columns.

Hermann von Helmholtz als mensch und gelehrter /

Mode of access: Internet.

Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome /

Bibliographical footnotes.

Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

On Fractional Helmholtz Equations

MSC 2010: 26A33, 33E12, 33C60, 35R11