"Days on Diffraction" have been held since 1968 and at that time were one day meetings that summed up the results of annual scientific researches in the famous Leningrad School on Diffraction Theory founded and led by V.A.Fock and V.I.Smirnov. 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 were concluded with a friendly picnic party on a beach near of St.Petersburg. This traditional picnic remained the keynote point of the social programme of every "Day on Diffraction". 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 then to three days. Simultaneously the scope of the meeting was widened and now various aspects of wave phenomena are included in the programme with the stress upon the use of asymptotic approaches. The Seminars "Day on Diffraction" are international since 1991. At present the Seminars usually accommodate about 30 participants from all over the world.

The number of participants that wished to participate in the 30-th anniversary Day on Diffraction was unusually large and the Organizing committee found it possible to enlarge the programme up to 50 presentations. However, the limitations on the size of the volume set a difficult task to select about 25 manuscripts that the book can contain. The final decision of the Organizing Committee was based on the judgement of the experts in the particular areas of mathematical physics and applied mathematics. We are grateful to Russian Foundation for Basic Research those support made possible this Proceedings to be published. We are also thankful to IEEE ED/MTT/AP St.Petersburg Chapter for continuous support of our Seminar.

V.S.Buldyrev,

V.M.Babich,

I.V.Andronov,

V.E.Grikurov,

A.P.Kiselev

Math. Institute of Russian Academy of Sciences, St.petersburg Branch, Russia

babich@pdmi.ras.ru

and

St.petersburg State University, Russia

c52-11@fs1.niimm.spb.su

*Uniform asymptotics of solutions of some second order differential
equation with a regular point of singularity is presented.*

__A.M.Il'in__

Univ. of Ufa, Russia

iam@eqmph.imm.intec.ru

*The problem d ^{2}u /dt^{2} + k^{2}u = hf
(u,du/dt), u (0) =a, du/ dt (0) = b, h > 0 is considered. Under some natural
conditions the asymptotic expansion of the solution up to any powers of
h^{ -1} on a time interval t=O (h^{ -N}), where N is arbitrary
natural number, is constructed and justified.*

__Andrzej Hanyga__

Institute for Solid Earth Physics, Univ. of Bergen, Norway

Andrzej.Hanyga@ifjf.uib.no

*It is demonstrated that Fermat's principle for an anisotropic elastic
medium can be expressed in terms of a Lagrangian. The Lagrangian corresponding
to a non-convex slowness surface is however singular. It is shown that
the second-order derivatives of the Lagrangian with respect to the group
velocity vector are unbounded at the cuspidal edges of the wavefront surface.
Concave parts of slowness surfaces give rise to extremal points which are
maxima.*

__J.M.H.Lawry__

Mathematical Institute, Univ. of Oxford, U.K.

lawry@maths.ox.ac.uk

*When a high-frequency time-harmonic line source is placed in a scalar
wavefield adjacent to a flat interface with a halfspace having a slower
wavespeed, a contribution to the transmitted wavefield is detected that
cannot be described asymptotically by real geometrical rays. This field,
known as S ^{*}, was detected numerically and found to be confined
to the exterior of a sector in the slower halfspace.*

*The field may be described by rays which emanate from the source
at complex angles into a complex coordinate space, and are refracted back
into the real domain from the analytic continuation of the boundary. The
complex ray method enables us to calculate both the asymptotic field and
the Stokes curves forming the boundary of the region in which it appears,
on the basis of the ray data alone and without reference to the exact solution.
This makes the method suitable for use in more general problems, such as
curved interfaces, for which analysis of the exact solution is not possible.
The structure of the complex rayfield reveals interesting features arising
from the fact that the data on the interface has branch points in complex
space.*

Steklov Mathematical Institute, St.Petersburg, Russia

nkirp@pdmi.ras.ru

*An algorithm is described for computation of the of higher-order
terms of ray theory for P waves in an inhomogeneous isotropic elastic medium.*

University of St. Petersburg, Russia

buld@mph.phys.spbu.ru, GNG@GG1623.spb.edu

*Let us consider a high-frequency stationary planar problem of diffraction
by obstacles with Generalized Impedance Boundary Conditions (GIBC) on their
surface. The head and the surface waves are well known to occur in various
diffraction problems on transpireouse obstacles. The purpose of our paper
is to derive the conditions for GIBC's coefficients which provide the existence
of head or surface waves.*

V.A.Steklov's Math. Institute., St.Petersburg, Russia

nkirp@pdmi.ras.ru

*A high-frequency diffraction of an incident creeping wave by a jump
of curvature is examined. *

Institute of Mechanical Engineering, V.O. St. Petersburg, Russia

kiselev@amath.usr.saai.ru

and

Visiting St.Petersburg on a One-Year Fellowship from the Royal Society, U.K.

*We consider the problem of diffraction by an isolated jump of curvature
in an otherwise smooth boundary. Effects of impedance along the critical
angle of reflection are investigated by Kirchhoff's method. For angles
of reflection close to that of resonance we present a new transitional
solution. Diffraction coefficients are presented.*

Dept. of Theoretical Mechanics, Univ. of Nottingham, U.K.

richard.tew@nottingham.ac.uk

*We consider the diffraction of a high-frequency plane wave by a finite,
convex obstacle. The obstacle is assumed to be slender in such a way that
the inner diffraction problem at the two tips is the full Helmholtz equation
with Neumann data specified on a non-trivial boundary curve (a parabola,
in fact). This is in contrast to the cases of sharp or blunt tip geometries,
where the classical Sommerfeld and Fock-Leontovic theories, respectively,
are appropriate. We obtain these theories as limiting cases of our solution
and we give a full account of creeping wave excitation, propagation and
tip diffraction on such a slender body*.

Institute on Physics, St.Petersburg Univ., Russia

boris@snoopy.phys.spbu.ru, simon@dal.usr.pu.ru

*Transient solutions of the inhomogeneous wave equation are obtained.
The sources are distributed on a specific expanding circle moving with
the constant velocity. The wavefunction is represented in terms of modes
in the cylindrical coordinate system. Application of the scalar solution
to description of the electromagnetic field is discussed.*

Laboratory for Math. Modeling of Wave Phenomena, Institute of Mechanical Engineering, RAS, St.Petersburg, Russia

kiselev@amath.usr.saai.ru

and

Dept. Math. Physics, Physics Faculty, St.Petersburg Univ., Russia

perel@mph.phys.spbu.ru

*We discuss asymptotic solutions of the wave equation exponentially
localized both in space and time. These solutions known as quasiphotons,
were described earlier by V. M. Babich and V. V. Ulin. For the case of
constant coefficients we present explicit solutions from the Bateman class
for which quasiphotons are their asymptotics. Some of these solutions are
truly localized `in large' while some others having the same local behavior
grow at infinity.*

Dept. of Electrical and Electronic Eng.,Faculty of Sci. & Eng., Chuo Univ., Japan

shirai@shirai.elect.chuo-u.ac.jp

and

Graduate School of Engineering, Tohoku Univ., Japan

*The transient field excited by an impulsive line source near a dielectric
half space is solved and analyzed via the spectral theory of transients
(STT). In this formulation, the field is described as a spectral integral
comprising as an angular superposition of time-dependent plane wave. Via
function-theoretic techniques, this STT integral is evaluated exactly,
giving rise to closed-form field solutions which agree with those derived
via the Cagniard-deHoop technique. We use the derived expressions to explore
the propagation characteristics of the reflected and transmitted fields.*

St-Petersburg Branch of Shirshov's Oceanological Institute of RAS, Russia

gotlib@io.spb.su

*Existence of solutions of Helmholtz equation exponentially decaying
away from a periodical boundary in the upper half-plane is proved. These
solutions can exist for some special form of the boundary under Dirichlet
or Neumann boundary conditions. In both cases the boundary has a form of
the resonator chain connected by narrow splits with the upper half-plane.*

Math. Dept. 2, ETU, St.Petersburg, Russia

Evgeni.Korotyaev@pobox.spbu.ru

*We consider the Hill operator T = -d ^{2}/dx^{2}+q(x),
acting on L^{2}(R), where q from L^{2} is a 1-periodic
real potential with zero average. The spectrum of T is absolutely continuous
and consists of intervals separated by the gaps G_{n}=(a^{ -}_{n},
a^{ +}_{n} ). Let m_{n}, n > 0, be the Dirichlet
eigenvalue of the equation -y''+qy=m y, y(0)=y(1)=0. Introduce the vector
g_{n}=(g_{cn}, g_{sn}) with components g_{cn}=(a^{
+}_{n}+a^{ --}_{n})/2-m_{n} and*

St.-Petersburg Institute of Fine Mechanics and Optics, Russia

tanya@loom2.softjoys.ru

*The Schrodinger operator in R ^{d} with an analytic potential,
having a non-degenerated minimum (well) at the origin, is considered. Under
a Diophantine condition on the frequencies, the full asymptotic series
(the Plank constant h tending to zero) for a set of eigenfunctions and
eigenvalues in some zone above the minimum is constructed; the Gaussian-like
asymptotics being valid in a neighborhood of the origin which is independent
of h. The existence of an exact solution with the mentioned asymptotics
is proven in a domain containing the origin and independent of h.*

Institute on Physics, St.Petersburg University, Russia

Ivan.Andronov@pobox.spbu.ru

*The generalized point model of narrow crack in fluid loaded elastic
plate is interpreted from the physical point of view. Numerical results
computed for this model are presented and discussed.*

St.-Petersburg University, Russia

Peter.Tovstik@pobox.spbu.ru

*The free low-frequency vibrations of the thin elastic shell in the
form of an elliptical cylinder are studied. The eigenvalues form groups
which of them contains four very close to each other eigenvalues with the
same asymptotic expansion in powers of a small parameter. Moreover each
group consists of two subgroups within which the distances between eigenvalues
are much smaller. The numerical examples and the corresponding theoretical
explanations are given.*

Institute of Physics, St.Petersburg Univ., Russia

Yuri.Kiselev@pobox.spbu.ru

and

St.Petersburg Univ.,Russia

*Numerical simulation of the solution of (2-D, SH) inverse problem
on recovery of elastic parameters of local (~ wave length) inhomogeneity
by diffraction tomography method based upon the Born approximation is considered.
The direct problem is solved by the finite difference method. The satisfactory
accuracy for recovery of non-weak contrast inhomogeneities (~ 50%) located
in piecewise-homogeneous layered medium is obtained with the use of three
source-receiver pairs.*

Institute of Geophysics, Univ. of Hamburg, Germany

zillmer@dkrz.de

and

Institute for Physics, St. Petersburg Univ., Russia

*The interaction of plane waves with a plane boundary between two
weakly anisotropic elastic halfspaces is investigated.*

Institute for Problems in Mechanical Engineering, St.Petersburg, Russia

mov@snark.ipme.ru

*The liner problem of stationary flow about semisubmerged bodies is
studied. General theory permits existence of a sequence of such values
of velocity when the uniqueness of solution is violated. The purpose of
the paper is to construct examples of problems that have nonunique solution. *

Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia

sturova@hydro.nsc.ru

*Diffraction of internal waves by a submerged body in an uniform current
of a two-layer fluid is considered. The layers are infinitely deep, and
the flows are two-dimensional. The linearized potential theory is used
for the inviscid and incompressible fluid. The explicit solution for the
circular cylinder is given in the form of rapidly converging series. This
is achieved through the use of certain recursive relations.*

Dept. Math. Physics, St.Petersburg Univ., Russia

Valery.Grikurov@pobox.spbu.ru

*The paper deals with the generalized one-dimensional nonlinear Schrodinger
equation iu _{t} + u_{xx} + |u|^{2p}u - c|u|^{2q}u
= 0 c > 0, q > p, which is a model of laser propagation throught nonlinear
optic materials with saturation. We are focus on the effect of soliton's
"self-compression" (i.e., rapid upswitching of its amplitude due to small
perturbations), which is peculiar for this equation if p > 2 . The paper
summarize the results of different numerical approaches to the problem
in question.*

Institute of Applied Physics RAS, Nizhny Novgorod, Russia

gromov@hydro.appl.sci-nnov.ru

*Frequency modulated nonlocalized stationary wave solutions in a general
type equation of the third-order approximation of the nonlinear dispersion
wave theory (third-order nonlinear Schrodinger equation) are studied. The
case of a polynomial type dependence of additional frequency on wave amplitude
is considered. Sech-like and algebraic soliton solutions are obtained.*

Dept. of Theoretical Physics and Mechanics, St. Petersburg Institute of Fine Mechanics, Russia

*Scalar diffraction problem by a planar surface deformed on a finite
segment is considered. The method reducing the problem to an integral equation
on a segment is developped. This methods is applicable both for the diffraction
by a hill and hollow. The deformatin is assumed to be such that the surface
is representable in the form of equation y=f(x). Numerical computations
according to the suggested method are performed and the results are compared
to Kirchhoff approximation and computations by other approximate methods.*

University of Nizhny Novgorod, Russia

and

Radiophysical Research Institute, Nizhny Novgorod, Russia

zabr@nirfi.sci-nnov.ru

*A study of the current distribution and input impedance of a loop
antenna immersed in a resonant magnetoplasma was performed. The problem
of determining the current distribution is reduced to the set of integral
equations with the logarithmic kernel. On the basis of solution of these
equations, an approximate expressions for the antenna current and input
impedance are obtained. The results are given in the form suitable for
numerical computing*.

Moscow State University, Department of Physics, Russia

shanin@ort.ru

*The problem of excitation of wave field in an equilateral triangle
area with impedance boundary conditions is examined by means of functional
equations of Maljuzhinetz type.*