"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
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 d2u /dt2 + k2u = 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.
An algorithm is described for computation of the of higher-order terms of ray theory for P waves in an inhomogeneous isotropic elastic medium.
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.
A high-frequency diffraction of an incident creeping wave by a jump of curvature is examined.
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.
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.
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.
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.
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.
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.
We consider the Hill operator T = -d2/dx2+q(x),
acting on L2(R), where q from L2 is a 1-periodic
real potential with zero average. The spectrum of T is absolutely continuous
and consists of intervals separated by the gaps Gn=(a -n,
a +n ). Let mn, n > 0, be the Dirichlet
eigenvalue of the equation -y''+qy=m y, y(0)=y(1)=0. Introduce the vector
gn=(gcn, gsn) with components gcn=(a
+n+a --n)/2-mn and
gsn=||Gn|2/4-gcn2|sn,
where the sign sn=+ or sn=- for all n > 0. Using
nonlinear functional analysis in Hilbert space, we prove that the mapping
g: q -> g(q)={gn}, n=1,..., is the real analytic isomorphism.
In the second part for q from L2(0,1) the trace formula is proved.
The Schrodinger operator in Rd 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.
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.
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.
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.
The interaction of plane waves with a plane boundary between two weakly anisotropic elastic halfspaces is investigated.
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.
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.
The paper deals with the generalized one-dimensional nonlinear Schrodinger equation iut + uxx + |u|2pu - c|u|2qu = 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.
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.
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.
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.
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.
