A. I. Korolkov

Diffraction by an impedance strip I. Reducing diffraction problem to Riemann–Hilbert problems

A 2D problem of acoustic wave scattering by a segment bearing impedance boundary conditions is considered. In the current paper (the first part of a series of two) some preliminary steps are made, namely, the diffraction problem is reduced to two matrix Riemann-Hilbert problems with exponential growth of unknown functions (for the symmetrical part and for the antisymmetrical part). For this, the Wiener--Hopf problems are formulated, they are reduced to auxiliary functional problems by applying the embedding formula, and finally the Riemann-Hilbert problems are formulated by applying the Hurds method. In the second part the Riemann-Hilbert problems will be solved by a novel method of OE-equation.

Diffraction by an impedance strip II. Solving Riemann–Hilbert problems by OE-equation method

The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven that the dependence of the solution on this parameter can be described by a simple ordinary differential equation (ODE1). The boundary conditions for this equation are known and the inverse problem of reconstruction of the coefficient of ODE1 from the boundary conditions is formulated. This problem is called the OE-equation. The OE-equation is solved by a simple numerical algorithm.

Diffraction by a thin cone in the parabolic approximation. method of the boundary integral equation

The problem of diffraction by a thin cone is considered. The parabolic equation of diffraction theory is used as the governing equation. Authors introduce the method of a boundary integral equation for the parabolic equation. The integral equation is derived by applying the Greens theorem. The kernel of the equation has a difference form, so the equation can be solved analytically using Fourier transform. As the result, some known formulae can be re-derived. Also, the integral equation can be solved numerically by iterations.

Boundary integral equation method for the parabolic equation on the curved surface

A 2D problem of diffraction by a curved surface with Neumann boundary conditions is considered. The incident wave is assumed to fall almost tangentially to the surface. The problem is studied in the parabolic approximation. A boundary integral equation of Hongs type is derived for the field. This equation belongs to Volterra class. A test problem of diffraction by a parabola is solved. An analytical solution of the integral equation can be found, and this solution is similar to well-known solution by V. A. Fock. A numerical solution of the equation is obtained by iterations.

Diffraction of high-frequency grazing wave on grating with screens of different height

A 2D problem of diffraction by a periodic diffraction grating composed of absorptive screens is studied. A period of the grating comprises two screens having different heights. The incident wave is assumed to have wavelength short comparatively to the period of the grating, and the incidence angle is assumed to be small. High diffraction orders are neglected, and the parabolic approximation is used to describe the wave process. The embedding formula, which expresses reflection coefficients in terms of the directivity of the edge Greens functions is proven. A spectral equation, which is an ordinary differential equation for the directivities of the edge Greens function is derived. The coefficient of the spectral equation is found by solving an Ordered Exponential (OE) equation numerically.