卷 243, 编号 5 (2019)

The Maxwell equations for smoothly inhomogeneous media are addressed. The first-order term of the Hadamard-type expansion for the wavefield of a pulsed point source is presented. The Fourier transform in t of this expression allows for a high-frequency asymptotic formula for the wavefield of a point source of electromagnetic oscillations.

The dynamical system

\( {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla \ln \rho \cdot \nabla u=0& in\kern1em {\mathbb{R}}_{+}^2\times \left(0,T\right),\\ {}u\left|{}_{t=0}\right.={u}_t\left|{}_{t=0}\right.=0& in\kern1em {\mathbb{R}}_{+}^2,\\ {}{u}_y\left|{}_{y=0}\right.=f& for\kern1em 0\le t\le T,\end{array}} \)

is under consideration, where \( {\mathbb{R}}_{+}^2:= \left\{\left(x,y\right)\in {\mathbb{R}}^2\left|y\right.>0\right\} \); ρ = ρ(x, y) is a smooth positive function; f = f(x, t) is a boundary control; u = u^f(x, y, t) is a solution. With the system one associates a response operator \( R:f\mapsto {u}^f\left|{}_{y=0}\right. \). The inverse problem is to recover the function ρ via the response operator. A short presentation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided.

If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way of making use of them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

Hörmander has presented a remarkable example of a solution of the homogeneous wave equation, which has a singularity at a running point. An analytic investigation of this solution is performed for the case of three spatial variables. The support of this solution is described, its behavior near the singular point is studied, and its local integrability is established. It is observed that the Hörmander solution is a specialization of a solution found by Bateman five decades in advance.

The Cauchy problem for the wave equation in Ω × ℝ with data given on some part of the boundary ∂Ω × ℝ is considered. A reconstruction algorithm for this problem based on analytic expressions is given. This result is applicable to the problem of determining a nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.

The parabolic equation describes the propagation of localized waves along the boundary with singularities. Some reformulation of the “separation of variables” scheme is presented, which enables us to obtain a rich set of solutions of the corresponding boundary value problems.

A formal approach for the construction of the Green’s function in a polygonal domain with the Dirichlet boundary conditions is proposed. The complex form of the Kontorovich–Lebedev transform and the reduction to a system of integral equations is employed. The far-field asymptotics of the wave field is discussed.

The paper is devoted to the calculation of the Morse index on geodesic lines upon smooth surfaces embedded into the 3D Euclidean space. The interest in this theme is created by the fact that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise. Two types of geodesic flows are considered upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. The position of the points where geodesic lines meet caustics, i.e., focal points, is found and it is proved that all focal points are simple (not multiple) irrespective of the geometric structure of the caustics arisen. The mathematical techniques in use are based on the complexification of the geometrical spreading problem for a geodesics/rays tube.

A two-dimensional problem that is a model for sound propagation in a narrow water wedge near the shore of a sea is studied. A solution to the Helmholtz equation, which is asymptotically a normal wave propagating along “water” wedge to the “shore,” is constructed explicitly. The solution satisfies the Helmholtz equation in the quadrant one side of which is “the surface of the water” and the second is perpendicular to it, starts at the top of the wedge and goes into the “bottom.” Boundary conditions on wedge boundaries and at infinity in the “bottom” are satisfied.