The electromagnetic field of a dipole radiation source in a slowly moving conducting medium

We solve the problem of determining the electromagnetic field created by a dipole radiation source in a conducting, homogeneous, isotropic, slowly moving medium. The investigations are carried out depending on the orientation of the dipole source and the velocity vector of the medium.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 31–35.     

The far field excited by a moving oscillating source in the resonance case

Wave fields excited by moving oscillating sources are considered. In the resonance case, the asymptotics of the far field for large values of t is found. Explicit expressions for the growing components of the field are provided.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 61–70.     

A method of calculating the field of a point source in a waveguide

In the paper a representation is obtained of the field of a point source in a waveguide with quadratic dependence of the index of refraction in the form of a sum of normal, geometrical-optics waves and a remainder. Sufficient conditions on the number of separate normal and geometrical-optics waves are found. The remainder is expressed by a simple formula.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 146–156, 1980.     

The combined method for calculating the field generated by a point source in a surface waveguide

The two-dimensional problem of propagation of waves generated by a point source in a plane boundary-layer waveguide is investigated. The Dirichlet condition is given on the surface of the waveguide. With the help of cumbersome transformations, the solution is represented as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the general amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 110–120.     

Influence of a wall impedance insertion on the behavior of the acoustic field in a waveguide

A rigid waveguide with an impedance covering occupying a bounded portion of its surface is considered. The influence exerted by the impedance insertion on the behavior of the waveguide field downstream of the insertion is studied depending on the impedance value, type, and the relative size of the impedance insertion. Special waveguide field propagation modes are found, and the areas of local maxima in the field suppression level are examined in the case of a layer, circular, and rectangular waveguide.     

Sound field of a source moving in an inhomogeneous layer on a homogeneous halfspace

Equations are suggested in the present paper for the amplitudes and instantaneous frequencies of sound waves generated by a moving source. The equations are derived by means of the Space-Time Ray (STR) method and by the two-scale expansion method (the method of horizontal rays and vertical modes).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 35–48, 1986.     

Discharge simulation in a high-frequency quasistatic field

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 275–281, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Hyperbolic conservation laws with a moving source

The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single transonic shock wave for a general system u_t + f(u)_x = g(x, u). Suppose that the i^th characteristic speed is close to zero. We propose the following stability criteria: Here l_i and r_i are the i^th normalized left and right eigenvectors of , respectively. Through the local analysis on the evolution of the speed and strength of the transonic shock wave, the above criterion can be justified. It turns out that the speed of the transonic shock wave is monotone increasing (decreasing) most of the time in the unstable (stable) case. This is shown by introducing a global functional on nonlinear wave interactions, based on the Glimm scheme. In particular, together with the local analysis, we can study the shock speed globally. Such a global approach is absent in the previous works. Using this strategy, we prove the existence of solutions and verify the asymptotic stability (or instability). © 1999 John Wiley & Sons, Inc.     

Detection of multilayered media in the acoustic waveguide

This work is concerned with the inverse problem for ocean acoustics modeled by a multilayered waveguide with a finite depth. We provide explicit formulae to locate the layers, including the seabed, and reconstruct the speed of sound and the densities in each layer from measurements collected on the surface of the waveguide. We proceed in two steps. First, we use Gaussian type excitations on the upper surface of the waveguide and then from the corresponding scattered fields, collected on the same surface, we recover the boundary spectral data of the related 1D spectral problem. Second, from these spectral data, we reconstruct the values of the normal derivatives of the singular solutions, of the original waveguide problem, on that upper surface. Finally, we derive formulae to reconstruct the layers from these values based on the asymptotic expansion of these singular solutions in terms of the source points.     

Adiabatic asymptotics of reflection coefficients of a quantum electron moving in a two-dimensional waveguide

We study the adiabatic asymptotics of reflection coefficients of a quantum electron moving in a two-dimensional waveguide. The direction of the waveguide axis can slowly change, whereas the cut-section is a periodic function slowly varying along the waveguide axis. The motion of an electron is described by the free Helmoltz equation. We study turning points in a neighborhood of which considerable reflection of an electron is observed. We describe the asymptotic behavior of reflection coefficients and uniform asymptotic formulas for the wave function of electron; moreover, these formulas remain valid even if the turning points approach each other. An example of a waveguide with four turning points (with the so-called resonance tunneling) is considered. The quantization condition characterizing the asymptotic behavior of resonances is described. Bibliography: 26 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 93–119.     

The phase velocity of acoustic perturbations near a vibration excitation source

The plane problem of the forced vibrations of an elastic half-space due to the action of a time-periodic force, acting along the tangent to the surface of a medium is considered. It is shown that in the near zone the phase velocity of surface acoustic perturbations varies considerably with distance from the vibration source and considerably exceeds the Rayleigh wave velocity.     

The point source field near a concave boundary

A new representation for the field emitted by a short-wave source placed near a concave boundary is obtained by the hybrid method. A Dirichlet problem is considered. The representation obtained provides an efficient algorithm for numerical experiments. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 212–219.     

An estimate of the gradient characteristics and source intensity of an acoustic field

An algorithm is proposed for estimating characteristic wave processes in underwater acoustics. We obtain minimax mean-square estimates for functionals which determine the solutions and right sides of the Helmholtz equation. Relations are presented which are satisfied by the minimax estimates; results of numerical computations are also given.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 128–134, 1989.     

A high-frequency point source of oscillations near a concave mirror

A high-frequency expansion of the problem in question is obtained by the boundary-layer method. This expansion describes the complex phenomenon of a large number of ray reflections from the surface of the mirror.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 5–19, 1975.     

The acoustic field of an elastic band in contact with a liquid

We propose a method of computation and analyze the acoustic field of an elastic plate under forced vibrations over the shapes of a cylindrical bend on an infinite rigid screen at the interface of liquid media. The problem is reduced to solving a system of algebraic equations with respect to the real and imaginary components of the acoustic pressure at a series of fixed points of the plate.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 102–105.     

Variational formulations for scattering in a three‐dimensional acoustic waveguide

Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Combined representation of the field of a point sound source in an underwater waveguide and the asymptotic summation of normal waves

In this work an asymptotic method is given of computing high-frequency asymptotics of sonic fields in layered-inhomogeneous oceanic waveguides. According to this method, the field is represented in the form of a sum of groups of normal waves and ray terms distinguished in a special way.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 66–83, 1981.