Volume 226, Nº 6 (2017)

A quaternion field is a pair p = {α, u} of a function α and a vector field u given on a 3d Riemannian manifold Ω with boundary. A field is said to be harmonic if ∇α = rot u in Ω. The linear space of harmonic fields is not an algebra with respect to quaternion multiplication. However, it may contain commutative algebras, which is the subject of the paper. Possible applications of these algebras to the impedance tomography problem are touched upon.

Results of a numerical experiment of solving an ill-posed Cauchy problem for the wave equation are discussed. An instrumental function for the regularizing algorithm applied here is given, and an analysis of stability is carried out.

The paper is a continuation of previous papers of the authors dealing with the exploration of shortwave diffraction by smooth and strictly convex bodies of revolution (the axisymmetric case). In these problems, the boundary layer method contains two large parameters: one is the Fock parameter M and the second is Λ that characterizes the oblongness of the scatterer. This naturally gives the possibility of using the two-scaled asymptotic expansion, where both M and Λ are regarded as independent. The approximate formulas for the wave field in this situation depend on the mutual strength between the large parameters and may vary. In the paper, we carry out numerical experiments with our formulas, in the case where the Fock analytical solution is in good coincidence with the exact solution of a model problem, in order to examine the influence of the parameter Λ on the wave field. It follows from our numerical experiments that the influence of the oblongness of the scatterer on the wave field is really insignificant if the method of Leontovich–Fock parabolic equation does not meet mathematical difficulties.

In this paper, extensions of the results obtained in our previous paper devoted to the diffraction of waves from a Hertzian dipole (point source) located over an impedance wedge are presented. The surface waves components and the edge wave, produced by a surface wave from a point source coming to the edge, are discussed. The geometric optics laws for the surface waves reflected by and transmitted across the edge are also addressed.

This article deals with short-wave diffraction by a strongly elongated body of revolution (an axially symmetric problem). In this case, the classical method of the Leontovich–Fock parabolic equation (equations of the Schrödinger type, to be more precise) appears to be nonapplicable, because the corresponding recurrent system of equations loses its asymptotic nature, and the equations themselves, including the main parabolic equation, gain singularity in their coefficients. This paper introduces a new boundary layer in a neighborhood of the light-shadow boundary, which is determined by other scales than the Fock boundary layer. In the arising main parabolic equation, the variables are not separated, so it turns out to be impossible to construct a solution in analytic form. In this case we state a nonstationary scattering problem, where the arc length along the geodesic lines (meridians) plays the part of time, and the problem itself is resolved by numerical methods. Bibliography: 11 titles.

The 2D problem of diffraction by a curved surface with ideal boundary conditions is considered in terms of the parabolic equation of diffraction theory. A boundary integral equation of Volterra type in Cartesian coordinates is introduced. Using the latter, the problem of diffraction by a parabola is analyzed. It is shown that the solution of this problem coincides with the asymptotic solution for the problem of diffraction by a cylinder obtained by V. A. Fock. The Efficiency of the numerical solution of the boundary integral equation is demonstrated for diffraction on a perturbation of a straight boundary.