Scattering of sound by an inhomogeneous thermoelastic spherical layer

The diffraction of sound by a radially layered isotropic thermoelastic spherical shell is considered. The system of equations for small perturbations of a hollow thermoelastic sphere is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by a spline-collocation method. Expressions are obtained which describe the wave fields outside the spherical layer. The results of calculations of the frequency and angular dependences of the amplitude of the scattered sound field in the far zone are presented.     

Hankel operators and problems of best approximation of unbounded functions

For each function f, f VMO, there exists a unique function f₀, analytic in the circle and such that f–f₀=_f{gVMO_A}. We define the operator of best approximation (nonlinear) A, Af=f₀, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H², then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.     

Interpolating Blaschke products and the ideals of the algebra H∂

To each vector-function f, f H(Z²) one can associate two ideals of the algebra. In the note one characterizes those functions f for which some interpolation Blaschke product is contained in I(f) (or J (f)). One also characterizes those functions u, u H such that if for some f, then u I(f) (in the corona theorem a similar statement is proved for the case u =1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 196–201, 1983.     

Diffraction of a plane acoustic wave by a non-uniform thermoelastic cylindrical layer bounded by inviscid heat-conducting fluids

The diffraction of sound by a radially laminated isotropic thermoelastic cylindrical shell is considered. The system of equations for small perturbations of a hollow thermoelastic cylinder is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the spline-collocation method. Expressions are obtained describing the wave field outside the cylindrical layer. Results of calculations of polar radiation patterns of the amplitude of the scattered acoustic wave in the far zone are presented.     

Cavitation interaction between fluid counterflows

Some features of self-oscillatory regimes for the cavitation interaction of coaxial water and annular gas jets with a water counterflow are studied. The dependencies of the self-oscillation frequency, the maximum displacement of a cavity toward the incident flow, and the displacement amplitude on the pipe diameter and the Froude number are found. A conclusion on the onset of self-oscillations is substantiated. The possibility of a significant increase in the counterflow range due to the gas injection in the vicinity of the counterflow is shown.     

On the problem of funnel formation during liquid outflow from vessels

The process of funnel formation during unsteady outflow of initially still water under gravity through one or two circular holes in the bottom of a prismatic vessel rectangular in plan is analyzed experimentally and numerically. The process is studied for various locations of the sinkholes, for vessel bottoms and walls with areas that differ in roughness, and for several values of the initial water depth h ₀. A sufficient, but not necessary, condition for the absence of vortex funnels is formulated. On the basis of a method proposed for determining the fluid rotation direction in vortex funnels, regions are found such that above sinkholes located in them vortex funnels with a given rotation direction are formed, as well as lines such that for sinkhole centers positioned on them stable vortex funnels are not formed. These results are confirmed by numerical calculations and experiments. At different times t the structure of the circumferential velocity field in the sinkhole cross-section is investigated. In the sinkhole crosssection, at several levels above it, and in the outflowing jet the dependence of the integral moment of the circumferential velocities on time and the initial depth is found. The time dependences of the flow-rate characteristics (flow-rate coefficient and volume flow-rate) are obtained for various initial depths and compared with results known in hydraulics. On the range of determining parameters investigated it is established that the fluid depth in the vessel at which a vortex funnel begins to be formed is independent of the initial water depth. The numerical calculations were carried out using STAR-CD software (license of the Institute of Mechanics of Moscow State University) tested in experiments on water outflow from prismatic vessels circular and rectangular in plan.     

Fission barriers and other characteristics of nuclei from the uranium region

Fission barriers in nuclei belonging to the uranium region and their other characteristics are calculated on the basis of the FaNDF⁰ energy density functional. In particular, the neutron-separation energies S _n and S _2n, the proton-separation energies S _p, and the beta-transition energiesQ _β are calculated for uranium, neptunium, and plutonium isotopes. In addition, the deformation energies and parameters of these nuclei are presented along with their radii. A comparison with the predictions of the Skyrme–Hartree–Fock method implemented with several versions of the Skyrme energy density functionals is performed. The role of the octupole deformation β ₃ is studied for the ²³⁸U nucleus. It is shown that this deformation does not have any significant effect on the first-barrier height B ⁽¹⁾ _f or ground-state properties. At the same time, the second-barrier height B(2) f decreases by a factor of about two upon taking into account β ₃. A phase transition at A ~ 260 is found for the three isotopic chains being considered: this point is a bifurcation point at which B ⁽¹⁾ _f (A) forks into two curves. Of these, the curve B ⁽²⁾ _f (A) splits from it, prolonging the former curve for B ⁽¹⁾ _f (A) almost continuously, whereas the curve for B ⁽¹⁾ _f (A) itself goes down sharply.     

Sound propagation through a discretely inhomogeneous thermoelastic plane layer adjacent to heat-conducting liquids

An analytical solution of the problem of the propagation of a plane sound wave through a discretely inhomogeneous thermoelastic layer adjacent to inviscid heat-conducting liquids is obtained. Results of calculations of the dependences of the transmission coefficient on the wave incidence angle and frequency for discretely inhomogeneous and continuously inhomogeneous thermoelastic layers are given. It is shown that a thermoelastic layer with continuously inhomogeneous thickness can be simulated using a system of homogeneous thermoelastic layers.     

Confinement of atoms with Robin’s condition: Spontaneous symmetry breaking

The properties of the ground state of the hydrogen atom in a spherical vacuum cavity with general boundary “not going out” conditions (i.e., when the probability current through the boundary vanishes) are studied. It is shown that in contrast to the confinement of an atom by a potential barrier, in this case depending on the parameters of the cavity, the atom could be in stable equilibrium at the center of the cavity or shift towards its periphery: spontaneous breaking of spherical symmetry occurs. The phase diagram of the shift and the dependence of the shift value and the binding energy of the ground state of the atom on the cavity parameters are presented. At the same time, the deformation properties of the electron wave function (WF) for an asymmetric distortion are so nontrivial that a non-zero shift occurs even when an electron is repulsed from the cavity boundary.