The critical frequencies of propagating waves in linearly isotropic cylindrical waveguides

We obtain the equations for determining the critical frequencies of normal waves with zero wave numbers and study them. We present the results of computation of the critical frequencies for a waveguide made of monocrystalline strontium sulfate, and also the results of a study of the variation of the critical frequencies with the generalized index of shear anisotropy for waves characterized at the critical frequencies by the longitudinal elastic displacements alone. Two figures. Bibliography: 8 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 87–95.     

Wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section

In this paper, the problem of electroelastic waves propagating in piezoelectric hollow cylinders of sector cross section is studied for the case when the boundary surfaces of sector cut are covered by non-extensible membranes. The three-dimensional linear equations of motion for the piezoelectric cylinder are analytically integrated and different boundary conditions on the cylindrical surfaces yield frequency equations, which relate the frequencies of elastic waves to their wavenumbers. Numerical results for waveguides with various boundary conditions are presented to illustrate the approach. Analysis of the dispersion spectra is carried out, and cutoff frequencies are obtained and characterized; mode asymptotic behavior and amplitude distributions of wave characteristics are analyzed. The main effects of their transformation by variation of the angular measure and the ratio of inner and outer radii are discussed. The results obtained are in good agreement with the results for the special case of a hollow semicircular cylinder.     

Non-Hopfian Relatively Free Groups

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.     

Computer simulations of the fission process of charged nanometre droplets

The process of fission of charged liquid droplets is an important stage of multiply charged ion formation in the electrospray ionization (ESI) process. ESI is currently the most powerful ionization method in the mass spectrometry of large molecules such as biopolymers. Fission plays a crucial role in the ion formation essentially in the range of nanosized droplets, since it determines the charge states of product ions. Usually the Rayleigh equation is used to determine critical conditions, at which the fission process takes place. This equation gives the value of the critical radius of the charged droplet as a function of its charge and specific surface energy. The Rayleigh equation does not give us the opportunity to determine charges and sizes of the fission fragments. In the present work we propose a mathematical model that describes the fission process of the charged liquid droplet in an external electrostatic field. The model is based on the quasi-equilibrium assumption that the most probable shape of the droplet corresponds to the minimum of the free energy of the system. The change in the internal kinetic energy and dissipative losses for the viscous liquid are also taken into account in the model.     

Special functions of generalized complex variables in problems of diffraction of elastic shear waves in anisotropic media

We consider the special sk functions of a certain pair of generalized complex variables introduced in the text. These functions form the base functions of particular solutions of the interior and exterior type for the equation describing elastic shear vibrations in an anisotropic medium with a plane of elastic vibrations symmetry. The functions are represented the form of absolutely convergent series and recursion relations, derivative formulas, addition theorems, and asymptotic forms are obtained. Solutions of the metaharmonic equation in cylindrical Bessel functions represent special cases of the solutions constructed here.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 79–83, 1987.     

Diffraction of pulse shear waves in a tunnel cavity of curvilinear section in an anisotropic massif

The problem of diffraction of longitudinal shear waves in the form of periodic triangular pulses by a cylindrical tunnel cavity in a rectilinearly orthotropic unbounded massif is reduced to a number of problems on the diffraction of harmonic shear waves of different relative length. Each of the problems on stationary diffraction is solved in series in the basis solutions of shear vibrations equations by using an affine coordinate transformation of the original domain. It is proposed to use a point method of least squares to seek the coefficients of the series. The influence of the degree of anisotropy of the elastic properties of the massif on the shear stress concentration at the boundary of a circular section is analyzed for a different position of the incident triangular pulse front relative to center of the cavity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 70–76, 1989.     

Reflection and Refraction of Normal Waves in a Composite Rectangular Orthotropic Waveguide

A numerical-analytic method employing series over basis systems of propagating and edge standing normal waves is used to study dynamic edge effects in the scattering of the lowest order travelling wave at the contact surface between two semi-infinite components of a prismatic waveguide with a rectangular cross section made of single crystals of Rochelle salt and sodium ammonium selenate dihydrate.     

Bis-para-semiquinoid type double cyclopalladation in the series of six-membered two-nitrogen bridged annulene-dihydroannulene ligands

The cyclometallation reaction known in the series of annulene heteroorganic ligands was extended to six-membered dihydroannulenes with a dinitrogen bridge. Double cyclopalladation of 4-methyl-4-trichloromethylcyclohexa-2,5-dien-1-one azine yielded the first representative of a new class of cross-conjugated diazadipalladatetracycles,viz., 5,10-bis(acetylacetonato)-2,7-dimethyl-2,7-bis(trichloromethyl)-2,7-dihydro-4b,9b-diaza-5,10-dipalladaindeno[2,1-a]indene, isolated as a diastereomer mixture of the achiralmeso-form (E-isomer) and a racemate (Z-isomer). This reaction offers a method for transition metal-mediated activation of non-reactive C−H bonds at position 2 of cyclohexa-2,5-dienylidene systems and a route toward the very rare chiral polyheteroelement system with rotational symmetry. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1236–1239, June, 1997.