Spectral problem in a waveguide with homogeneous bi-isotropic filling

A vector spectral problem in a rectangular waveguide with homogeneous bi-isotropic filling is considered. A generalized statement of this problem is proposed, which is used to develop a finite-element algorithm for calculating propagation constants with spurious solutions eliminated.     

Nonstationary normal waves in a thin,curved waveguide of variable cross section

The propagation of nonstationary normal waves in a thin waveguide with variable properties along its course is considered. Along space-time rays the eikonal and transport equations are integrated, and asymptotic formulas for these waves are obtained. It is observed that for some types of initial data the effect of overturning of a wave packet of a normal wave is possible.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 210–218, 1979.     

Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.     

Nonstationary normal waves in a thin,curved waveguide in a multiray zone (uniform asymptotics)

In a thin waveguide with properties that vary along its course, the propagation of nonstationary normal waves in the presence of caustics for space-time rays is considered. The connection of the critical section of the waveguide with such caustics is determined. Uniform asymptotic formulas are obtained for the wave field in a multiray zone, and the passage into geometric rays outside a neighborhood of the caustics is traced.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 127–137, 1980.In conclusion, we note that the matrix machinery used in the present work is not connected with the special nature of the present problem and can always be applied when the initial Ansatz contains a collection of special functions which go over into one another under differentiation.     

Local degeneration of waves in a thin waveguide

A thin, curved waveguide of variable width with an impedance condition on one of the walls is considered. Local degeneration of normal waves is studied. Coefficients for the transformation of waves as they pass through points of local degeneration are found.     

Propagation of shear waves in a semi-infinite elastic waveguide with a local constriction

A procedure is proposed for computing longitudinal-shear stresses and displacements in a semi-infinite waveguide with two lateral cutouts. Numerical results are reported for the stresses on the cutout contours and for the transmission coefficient of the elastic wave.Sumy Branch, Khar'kov Polytechnical Institute. Translated from Dinamicheskie Sistemy, No. 10, pp. 14–20, 1992.     

An approach to the analysis of the interaction of surface waves with depth-varying current fields

An approach is developed for the analysis of the interactions of surface waves with depth-varying current fields. The approach is based on an approximate dispersion relation for wave-current interactions derived from the governing equations of the problem (the inviscid Orr-Sommerfeld equations) coupled with the wave kinematic-wave action formulation of surface wave propagation. The approach is particularly useful in that it focuses directly on the wave parameters of interest (the amplitude, frequency, direction, and wavelength of the wave) and eliminates the requirement to solve the inviscid Orr-Sommerfeld equations to derive these parameters. The validity of the approach is demonstrated by comparisons with exact Orr-Sommerfeld solutions.     

An asymptotic approach to the problem of the distant propagation of waves

An asymptotic approach is used in this paper to examine solutions of the problem of distant propagation of Love waves in a medium consisting of an inhomogeneous layer lying on a homogeneous half-space. The dispersion properties of these waves and their attenuation are estimated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 125–135, 1986.     

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.     

Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide

The problem of the propagation of coupled surface electromagnetic waves in a two-layer cylindrical circular waveguide filled with an inhomogeneous nonlinear medium is considered. A nonlinear coupled TE-TM wave is characterized by two (independent) frequencies ω_e and ω_m and two propagation constants \({\widehat \gamma _e}\) and \({\widehat \gamma _m}\). The physical problem reduces to a nonlinear two-parameter eigenvalue problem for a system of nonlinear ordinary differential equations. The existence of eigenvalues (\({\widehat \gamma _e}\), \({\widehat \gamma _m}\)) in proven and intervals of their localization are determined.     

A stationary approach to the existence and completeness of long-range wave operators

The modified wave operators intertwining the Schrödinger operators – and – + V(x), where V(x) is a real long-range potential, are shown to exist and to be complete. The method employed is entirely stationary (time-independent) : One constructs complete stationary wave operators, utilizing a spectral representation (eigenfunction expansion) theory, and then shows that the time-dependent modified wave operators exist and are equal to the stationary ones already constructed.     

An invariant manifold approach to nonlinear normal modes of oscillation

Summary A method for determining the amplitude-dependent mode shapes and the corresponding modal dynamics of weakly nonlinear vibratory systems is described. The method is a combination of a Galerkin projection and invariant manifold techniques and is applied to a class of distributed parameter vibratory systems. In this paper the general theory for a class of conservative systems is outlined and applied to determine the nonlinear mode shapes and modal dynamics of a linear Euler-Bernoulli team attached to a nonlinear elastic foundation.     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.