© Day on Diffraction. Faculty of Physics, SPbU, 1998

Preface

This book contains the full texts of selected talks presented at the International Seminar "Day on Diffraction 98". The Seminars "Day on Diffraction" are annually held by the Faculty of Physics of St.Petersburg State University and St.Petersburg Branch of V.A.Steklov Mathematical Institute in the late dates of May or early June. The "Day on Diffraction 98" was opened on June 2, 1998 and marked the 31st anniversary of the Seminar.

"Day on Diffraction" have been held since 1968 and at that time was one day meetings that summed up the results of annual scientific researches in the famous Leningrad School on Diffraction Theory, founded and led by Prof.V.I.Smirnov and Prof. V.A.Fock, whose 100-th birthday is celebrated this year.

V.A.Fock (1898 - 1974)

The works of V.I.Smirnov and S.L.Sobolev on functionally-invariant solutions of wave equation in the beginning of 30-s, the method of partial separation of variables suggested by V.I.Smirnov in 1937, the widely known works of V.A.Fock on diffraction of radio-waves around the surface of the Earth of 40-s laid the corner-stone of Leningrad-St.Petersburg School. The main body of the diffraction school in 60-s - 70-s was constituted of the students of prof. G.I.Petrashen' who proceed with and developed the Diffraction Theory after the World War II.

The scientific talks and discussions of the first "Day on Diffraction" in 1968 were concluded with a friendly picnic party on a beach near of St.Petersburg. Since that time every "Day on Diffraction" ended by picnic party in woods near the University campus. In a time scientists from other cities of the former Soviet Union began to take part in the Seminar and the meeting was enlarged to two and more days. Simultaneously the scope of the Seminar was widened and now various aspects of wave phenomena are included in the programme with the stress upon the use of asymptotic approaches. "Day on Diffraction" are international since 1991. At the present, the Seminars usually accommodate about 50 participants from all over the world.

We are grateful to Russian Foundation for Basic Research those support made possible organization of the Seminar and this Proceedings to be published. We are also thankful to IEEE ED/MTT/AP St.Petersburg Chapter for continuous support of our Seminar and to URSI for mode A sponsoring.

The Organizing Committee

CONTENTS

• I David Abrahams, Jane B Lawrie

Ducts with High Order Boundary Conditions Application of a General Orthogonality Relation to Acoustic Scattering in Ducts with High Order Boundary Conditions

This article is concerned with a class of problems involving the propagation of waves in ducts whose walls are described by high-order boundary conditions and have abrupt changes in material properties or geometry. A new orthogonality relation is presented for the general class of problems. Via two specific examples, it is shown that the orthogonality relation allows mode matching to follow through in the same manner as for simpler (Dirichlet, Neumann or Robin) boundary conditions. That is, it yields coupled algebraic systems, for the coefficients of the eigenfunction expansions, which are easily solvable. By this means complicated problems, are tractable.

• Ivan Andronov

Edge Waves Along a Narrow Straight Crack in Fluid Loaded Elastic Plate

The edge waves in elastic plates were first noted in \cite{kon} and for the weak fluid loading were studied in \cite{k-l}. There the crack along which the edge waves were running was described by classical point model \cite{k}. However in \cite{wm} it was shown that such model is applicable only for exponentially (with respect to the thickness of the plate) narrow cracks. In this paper the generalized model suggested in \cite{gilb}, \cite{ak-j} is accepted. Numerical analysis of the dispersion equations for the symmetric and asymmetric waves allows some specific effects to be discovered

• L.A.Dmitrieva, Yu.A.Kuperin

Spectral Properties of the Hill Operator Perturbed by a Periodic Potential and Their Applications to the Induced Superlattices

A local perturbation theory for the spectral analysis of the periodic Schr\"odinger operator with two periodic potentials whose periods are commensurable has been constructed. It has been shown that the perturbation of the periodic 1D Hamiltonian by an additional small periodic potential leads to the following deformation: all gaps in the spectrum of the unperturbed periodic Hamiltonian bear shifts while any band splits by arising additional gaps into a set of smaller spectral bands. The spectral shift, the position of additional gaps and their widths have been calculated explicitly. The applications to the regime of a nanoelectronic device based on Mott-Peierls stimulated transition have also been discussed.

• Natalia G. Gelfreikh

High­Frequency Stationary Problem of Diffraction by a Thin Elastic Cylinder

A diffraction problem of acoustic wave by a thin elastic cylindrical shell is considered. The wave field has to satisfy the Helmholtz equation as well as the radiation conditions at infinity and the condition for equality of normal components of the speeds of the particles of the medium and the shell. The last one and the equations of the theory of the thin elastic shell allows us to derive the boundary condition for the wave field. It contains the tangential derivatives up to the sixth order with the coefficients which are the functions of the frequency, the physics and geometrical properties of the system, including the thickness of the shell. High-frequency asymptotic series for the wave field are constructed.

• D.J. Gilbert

Asymptotic Methods in the Spectral Analysis of Singular Differential Operators

We review the main features of the theory of subordinacy for one-dimensional Schr\"{o}dinger operators by considering two standard cases, the half-line operator on $L_{2}[0,\infty)$ and the full-line operator on $L_{2}({\bf R})$. It is assumed that the potential function is locally integrable, that 0 is a regular endpoint in the half-line case, and that Weyl's limit point case holds at the infinite endpoints. We also consider some extensions of the theory to related differential and difference operators, and discuss its application, in conjunction with other asymptotic methods, to some typical problems in spectral analysis

• Aleksei P. Kiselev, Maria V. Perel

Asymptotic Theory of a Local Degeneration of Love and Rayleigh Modes

A propagation of Love and Rayleigh modes in a weakly anisotropic elastic structure with slow lateral variations of its parameters is considered in the case of a local degeneration. That is, when the phase curve of a certain mode and that of a certain mode of another type intersect at some point. Asymptotic formulas are found describing wave processes in a vicinity of this point, which are then matched with the adiabatic modal solutions. A transition matrix is presented

• R.C.K.Leung, C.C.K.Tang and N.W.M.Ko

Effect of Phase Difference on Perturbed Vortex Pairing

A study of the effect of phase difference of perturbation waves on two interacting vortex rings is reported. The role of phase difference in determining the stable and unstable mode of pairing of the perturbed vortex rings is identified. Based on the results obtained, some observations of flow structure dynamics of a circular jet are explained

• A.A. Abramov, A.L. Dyshko, N.B. Konyukhova , T.V. Levitina

On Numerical Solving of Several Mathematical Physics Problems by Separation of Variables

A brief survey of papers is presented. On the basis of the methods elaborated for solving both singular and multi-parameter spectral problems, several problems of acoustics and electrodynamics for spheroids and ellipsoids, as well as those of quantum physics for potentials separable in ellipsoidal and paraboloidal coordinates are considered

• Dmitri V. Lebedev, Vladimir L. Oleinik

Spectrum Band Asymptotics of Schr¨odinger Operator in Model Domain with Periodic Structure

Our report considers spectrum properties of the Schr\"odinger operator in model domains with mixed dimension (1D-2D-1D). The existence of spectrum band lying near the negative value $\mu_0$ is proven for the Schr\"odinger operator in periodic domain composed from 1D and 2D subdomains with sufficiently large period. Here $\mu_0$ is the eigenvalue of the Schr\"odinger operator in domain composed from one 2D subdomain and two semiaxes. The asymptotic formula for spectrum band is given in the case when period tends to infinity. In order to connect 1D and 2D subdomains we use standard approach of extension theory. We consider only real potentials and deal with self-ajoint operators

• Christine F. Hermik, Valeriya V. Vasilieva, Olga D. Shishkina

Study of Internal Waves Caused by Nonstationary Motion of Floating Body in Stratified Fluid

Some particular natural situations provide nonstationary internal waves generation in stratified fluids. One of the particular cases of such a situation is induced internal waves propagation. Estimations of variation of nonstationary wave parameters (phase angles) were fulfilled. Amplitude variations were studied only for free internal waves propagation in the shelf zone. Behavior of nonstationary induced internal waves (both phase and amplitude) propagating in the near-surface pycnocline was studied

• V.V. Borisov, I.I. Simonenko

Formation and Destruction of Waves with Singularities on the Front

• Alexander S.Starkov

The Inverse Scattering Problem in the Waveguide

The inverse scattering problem for two dimensional Schrodinger equation in periodical waveguide is considered. The first $N$ reflection coefficients of propagating normal waves are chosen as input data. The theorem that an error in the potential approximation is less than $c N^{-d}$ with some positive constants $c$ and $d$ is proved

• T. M. Zaboronkova, A. V. Kudrin and L. L. Popova

Source­excited Field Representations for Cylindrical Magnetic field­aligned Channels in a Magnetoplasma

General representations of source-excited electromagnetic field on a cy\-lindri\-cal magnetic-field-aligned plasma channel are obtained for the case when sources arbitrarily depend on the azimuthal angle. The fields are expressed in alternative forms by the application of modal methods and methods pertaining to a Green's function technique. Solutions derived in these two ways are discussed and interpreted as guided-wave representations along various coordinate directions.

• Matthias Zillmer, Dirk Gajewski, Boris M. Kashtan

The Quasi­isotropic Approximation of Ray Theory

The propagation of body waves in inhomogeneous, weakly anisotropic media is investigated with the ray method. The 'quasi-isotropic approximation' uses the ray series together with a special definition of the elasticity tensor. The transport equations for the shear waves are two coupled ordinary differential equations. Different forms of the transport equations are derived. The 'quasi-isotropic approximation' is shown to lead to the transport equations which are known from the 'coupling ray theory'