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.     

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.     

On the asymptotics and stability of the point spectrum of a waveguide with thin shielding obstacle

Some explicit conditions are found for the existence and absence of an eigenvalue in the interval (0, π ²) of the continuous spectrum of the Neumann problem for the Laplace operator in the unit strip with a thin (of width O(ε)) symmetric screen, which, as ε → +0, shrinks into a line segment perpendicular to the sides of the strip. An asymptotics of this eigenvalue is constructed, as well as the asymptotics of the reflection coefficient, which describes Wood’s anomalies, namely, quick changes of the diffraction pattern near a frequency threshold in the continuous spectrum. Bibliography: 32 titles.     

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.     

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.     

Simple waves and shock waves in a thin prestressed elastic rod

Zusammenfassung Ein halb-unendlicher dünner elastischer Stab wird einer monoton ansteigenden Störung am freien Ende unterworfen. Die unstetige Wellenbewegung für endliche Verzerrungen wird untersucht, und die Bedingungen zur Bildung von Stössen werden aufgestellt. Eine implizite Lösung für die Wellenbewegung im Gebiete einfacher Wellen wird mittels der Methode der Charakteristiken erhalten. Eine andere und explizite Lösung in diesem Gebiete wird gefunden durch Anwendung der Theorie sich fortpflanzender Diskontinuitätsflächen, als eine Taylor-Reihe um die Ankunftszeit der Wellenfront. Es wird gezeigt dass der Koeffizient des ersten Gliedes in der Entwicklung eine Lösung für eine homogene nichtlineare gewöhnliche Differentialgleichung ecster Ordnung darstellt. Die übrigen Koeffizienten befriedigen lineare, jedoch inhomogene gewöhnliche Differentialgleichungen mit veränderlichen Koeffizienten. Für den Fall einer rampenartigen Belastung und für Stossbildung an der Wellenfront wird für schwache Stösse durch Anwendung einer Methode von Friedrichs die Vereinigung von einfachen Wellen untersucht. Als Beispiel wird die Entlastung eines Stabes betrachtet, der einer anfänglichen Dehnung von 50% unterworfen ist.     

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.     

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.     

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.     

Thermoelastic waves in a constrained isotropic plate: incompressibility at uniform temperature

Thermoelastic waves are investigated which propagate throughan isotropic plate which is constrained to be incompressibleat uniform temperature. This is an example of a deformation–temperatureconstraint, also known as a strain–temperature constraint.The boundaries of the plate are taken to be traction free andeither isothermal or insulated. Dispersion relations are derivedand expanded asymptotically in the long-wave low-frequency limit.The short-wave limit is also discussed. The higher modes areinvestigated numerically. Graphical comparison with the unconstrainedmaterial is also presented.     

Structure of normal shock waves in a gas-particle mixture

Summary The structure of normal shock wave propagating in a gas particle mixture has been obtained by including the particle-volume effect.

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Zusammenfassung Unter der Berücksichtigung des Teilchenvolumeneffektes wird die Struktur einer senkrechten Stoßwelle untersucht, die sich in einem Gasteilchengemisch bewegt.

     

Bending normal waves in a crystal layer surrounded by a liquid

We obtain the dispersion relations that describe the spectrum of the “effluent” harmonic antisymmetric normal waves for an arbitrary direction in the plane of an orthotropic layer surrounded by a viscous or ideally compressible fluid. We present the results of computation of the lower branches of the dispersion spectrum for elastically equivalent directions of propagation in a layer of monocrystal Seignette salt in water. Four figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 123–131.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg–de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Wave-field asymptotics in the problem of diffraction by a planar junction of thin layers and boundary-contact conditions

The problem of diffraction by a planar junction of thin layers covering a perfectly conducting substratum is considered, and its asymptotic solution is constructed. The wave field in the vicinity of the junction of the layers is described by a function of the boundary layer. Based on the asymptotics obtained, the generalized impedance boundary condition, which simulates thin layers, and the contact conditions are derived. The uniqueness of the solution of a model problem is discussed. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 157–171. Original Translated by M. A. Lyalinov.