A. V. Shanin

Removing false singular points as a method of solving ordinary differential equations

A general formalism is described whereby some regular singular points are eectively removed and substantial simplications ensue for a class of Fuchsian ordinary dierential equations, and related confluent equations. These simplications follow provided the exponents at the singular points satisfy certain relations; explicit, illustrative examples are constructed to demonstrate the ideas.

Embedding formulae in diffraction theory

Embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables (angles of incidence and observation) into those dependent upon fewer angular variables. In terms of facilitating rapid computations across considerable parameter regimes, this is a considerable advantage. Our aim is to derive embedding formulae for scattering and diffraction problems in acoustics, electromagnetism and elasticity. Here we construct a general approach to formulating and using embedding formulae. We do this using complementary approaches: overly singular states and a physical interpretation in terms of sources. The crucial point we identify is the form of the auxiliary state used in the reciprocal theorem; this is unphysically singular at the edge and is reminiscent of weight–function methods used in fracture mechanics. Illustrative implementations of our approach are given using Wiener–Hopf techniques for semi–infinite model problems in both elasticity and acoustics. We also demonstrate our approach using a numerical example from acoustics and we make connections with high–frequency asymptotic methods.

Trapped modes in angular joints of 2D waveguides

The Helmholtz equation is considered in an angular joint of two semi-infinite planar waveguides with ideal Dirichlet boundary conditions at their walls and with as the junction angle. We examine trapped modes generated by the discrete spectrum of the Dirichlet Laplacian and, in particular, prove the existence of a critical angle such that, for , the total multiplicity of the discrete spectrum equals 1. We also provide an asymptotic lower bound for the multiplicity as is small and give numerical results.

Calculation of characteristics of trapped modes in T-shaped waveguides

The spectrum of the Dirichlet problem for the Laplace operator in a plane T-shaped waveguide is investigated. The critical width of the half-strip branch is determined such that, if the width is greater, the waveguide has no discrete spectrum. The existence of a critical width is proved theoretically.

Embedding formulae for diffraction by rational wedge and angular geometries

Embedding is the process of taking the far-field directivity from a diffraction problem (or problems) involving line sources or multipoles placed at a sharp edge, and then constructing the far field, for the same geometry, for more general incidence using only this canonical problem(s). Thus far, embedding has been limited to planar, parallel scattering surfaces, for instance, collections of parallel cracks or slits; it appeared that there was a fundamental limitation to embedding, disallowing its use for angular structures. In this article, we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape; a limitation is that the final formulae are for rational wedge angles.

High-frequency modes in a two-dimensional rectangular room with windows

We examine a two-dimensional model problem of architectural acoustics on sound propagation in a rectangular room with windows. It is supposed that the walls are ideally flat and hard; the windows absorb all energy that falls upon them. We search for the modes of such a room having minimal attenuation indices, which have the expressed structure of billiard trajectories. The main attenuation mechanism for such modes is diffraction at the edges of the windows. We construct estimates for the attenuation indices of the given modes based on the solution to the Weinstein problem. We formulate diffraction problems similar to the statement of the Weinstein problem that describe the attenuation of billiard modes in complex situations.

Weinstein's Diffraction Problem: Embedding Formula and Spectral Equation in Parabolic Approximation

A short-wave problem of reflection and radiation by an open end of a two-dimensional planar waveguide is studied. The incident mode is assumed to have frequency close to the cut-off. The problem is studied in the parabolic approximation. A recently developed approach based on the embedding formula and the “spectral” equation for the directivity of an edge Greens function is applied to the problem.

Nonlinear generation of airborne sound by waves of ultrasonic frequencies

The results of research into the design and optimization of laboratory sources of intense airborne ultrasound are reported. Two types of sources are studied: multielement arrays of small-size piezoelectric radiators and single membrane transducers of a capacitor type. The measured characteristics of the ultrasound fields and the audible sound fields generated in air due to the nonlinear interaction of high-frequency waves are presented. Applications of nonlinear acoustic problems in air are discussed.

A generalization of the separation of variables method for some 2D diffraction problems

A new method is proposed for solving diffraction problems having piecewise linear ideal boundaries. According to this method, the initial Helmholtz equation is replaced by a system of matrix equations of order 1. The coefficients of these equations are rational matrix functions of the coordinates. The properties of the coordinate matrix system are close to that of the ordinary differential equation, therefore the new method can be treated as a generalization of the separation of variables.

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.