Propagation of torsional waves in a circular elastic rod

Zusammenfassung Das Problem eines unendlichen kreiszylindrischen Halbstabes, dessen Endquerschnitt mit beliebigen Torsionsspannungen belastet ist, wird auf Grund der strengen dreidimensionalen Elastizitätstheorie gelöst. Die Lösungsmethode besteht aus einer Integraltransformationstechnik mit asymptotischer Inversion für eine frequenzmodulierte Stufenfunktion. Die Ergebnisse zeigen, dass die Gültigkeit eines «dynamischen Saint-Venant-Prinzips» von der betreffenden Frequenzkomponente abhängt.

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This work was initiated while the author was still employed by the IBM Corp. at Endicott, New York.     

Propagation of Scholte-Gogoladze surface waves along a curved fluid-solid interface

A high-frequency ray theory is presented for a type of small-amplitude waves (termed Scholte-Gogoladze waves), localized in a thin layer around an interface between elastic and fluid domains. The interface is assumed to be smooth, with typical radii of curvature much larger than the excitation wavelength. The technique employed in the paper is based on a boundary-layer version of the classical WKB-expansion (see V. M. Babich and N. Ya. Kirpichnikova, The Boundary-Layer Method in Diffraction Problems (Springer, Berlin 1979)). Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 229–246.     

Propagation of elastic waves along the fibers in oriented glass-reinforced plastics

The dispersion equation is derived for axisymmetric (longitudinal and torsional) normal vibrations in oriented glass-reinforced plastics. The low-frequency case is examined in detail. The frequency limits of applicability of the method of substituting a homogeneous medium for the composite (GRP) are estimated. The dependence of the vibration propagation velocity on glass content is calculated.Leningrad Zonal Scientific-Research Institute of Experimental Design of Public and Residential Buildings. Translated from Mekhanika Polimerov, No. 6, pp. 976–983, November–December, 1973.     

The propagation of elastic waves in a piezoelectric

Formulas of the ray method are developed in the case of the propagation of high-frequency waves in a piezoelectric. It follows from the equations of the paper that in first approximation the energy of the wave field propagates along rays.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 155–161, 1974.     

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.     

Solitary waves in a quartic nonlinear elastic rod

The bright and dark solitons described by the nonlinear Schrödinger equation (NLSE) are given for a quartic nonlinear elastic rod. It has also been found that the KdV soliton does not exist in this system.     

Propagation of internal waves in a three-layer liquid

The propagation o f internal harmonic waves in a three-layer liquid is examined in a linear approximation for arbitrary wave numbers. A travelling wave sohaion is obtained and a dispersion equation is derived. The roots of the dispersion equation and the dependence of the solution on the ratio of the densities of the liquids are examined in detail. The phase and group velocities are plotted as functions of wavelength and graphs of the various wave modes are presented. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 170–174, 1999.     

Characteristics of wave propagation in piezoelectric bent rods with arbitrary curvature

The wave propagation in the piezoelectric bend rods with arbitrary curvature is studied in this paper. Basic three-dimensional equations in an orthogonal curvilinear coordinate system (r, θ, s) are established. The Bessel functions in radial co-ordinate and triangle series in the angular co-ordinate are used to describe the displacements and electrical potential. Characteristics of dispersion, distributions of displacements and electrical potential over the cross section are calculated, respectively. In the numerical examples, the effects of the ratio of the two ellipse axes on the dispersion relations of the first three modes are observed. The characteristics of the distribution of displacements and electric potential in the cross section, along the radial and s direction are investigated.     

Propagation of longitudinal viscoelastic waves in a three-layer medium

The nonstationary problem of propagation of a longitudinal plane one-dimensional stress wave through a plane-parallel viscoelastic layer of finite thickness separating two linear elastic half-spaces with different properties is solved in the linear formulation. A plane wave traveling in one of the half-spaces is normally incident on the boundary of the layer (one-dimensional problem). The field in the other elastic half-space, excited as a result of the multiple reflection of the fronts from the boundaries of the layer, is investigated. Graphs of the small displacements at a given point of the elastic half-space are presented. The solution of the problem is based on the dynamic correspondence principle formulated by Bland [3].Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 151–156, January–February, 1971.     

Propagation of TM waves in a Kerr nonlinear layer

TM electromagnetic waves propagating through a nonlinear homogeneous isotropic unmagnetized dielectric layer located between two homogeneous isotropic half-spaces are studied. The nonlinearity in the layer obeys the Kerr law. The problem is reduced to a system of nonlinear ordinary differential equations. A dispersion relation for the propagation constants is derived. The results are compared with those in the case of a linear layer.     

Propagation of acoustic waves in a fibrous fluid-filled material

The homogenization method is used for the analysis of acoustic wave propagation in a unidirectional fibrous material for the acoustic control of the fiber arrangement in manufacturing composites. Acoustic equations for a rigid periodic structure filled by a nonviscous fluid are obtained by two-dimensional asymptotic expansions. A regular square and triangular arrangement of the fibers with a round cross-section are considered. The analysis reveals that the velocity of acoustic waves is significantly affected by both the volume content and the fiber arrangement.Moscow State Academy of Chemical Engineering, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 651–655 September–October, 1997.     

Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid‐solid acoustic interaction when a piezo‐ceramic elastic body ( Ω ⁺ ) is embedded in an unbounded fluid domain ( Ω ^? ). The corresponding physical process is described by boundary‐transmission problems for second order partial differential equations. In particular, in the bounded domain Ω ⁺ , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω ^? , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.