Дифракция сферической звуковой волны на упругом цилиндре с неоднородным покрытием

В статье рассматривается задача о рассеянии плоской монохроматической звуковой волны, падающей произвольным образом на упругий круговой цилиндр с радиальнонеоднородным покрытием в присутствии плоской поверхности (абсолютно жесткой и акустически мягкой). Методом мнимых источников с применением теорем сложения для цилиндрических волновых функций получено аналитическое решение задачи. Волновые поля в содержащей среде и однородном упругом цилиндре находятся в виде разложений по волновым цилиндрическим функциям, а для нахождения полей смещений в неоднородных покрытиях построена краевая задача для системы обыкновенных дифференциальных уравнений второго порядка.Проведены численные расчеты частотных и угловых характеристик рассеянного поля для упругих однородных цилиндров с покрытием и без него, находящихся вблизи подстилающей плоскости. Выявлено существенное влияние непрерывно-неоднородных упругих покрытий на звукоотражающие свойства упругих цилиндрических тел.

On the basis of the known equations of motion of the system of coupled gyrostats by P.V. Kharlamov and the functions of state by S.L. Sobolev, the equations of rotation in a medium with resistance of a free system of two elastically connected rigid bodies with cavities completely filled with an ideal incompressible fluid were derived. Rigid bodies are connected by an elastic restoring spherical joint. Assuming that the center of mass of the rigid bodies is located on the third main axis of inertia and the fluid is ideal, the equation of disturbed motion of the considered mechanical system is obtained in the form of a countable system of ordinary differential equations. In the case of two Lagrangian gyroscopes with arbitrary axisymmetric cavities filled with an ideal fluid, a transcendental characteristic equation has been derived. Taking into account the fundamental tone of liquid oscillations, a characteristic equation of the sixth order was obtained, and on the basis of the Lenard-Schipar criterion, it was written in the innor form, and the conditions for the asymptotic stability of uniform rotation of Lagrange gyroscopes with a liquid were written out in the form of a system of five inequalities. These inequalities are presented in the form of the first, third, sixth, and eighth powers with respect to the coefficient of the spherical joint elasticity. It was proved that if the first tones of liquid oscillations in two cavities are greater than one and do not coincide, then this is sufficient for the higher inequality coefficients to be positive. It was shown that if the first oscillation tones coincide, only the degree of the last inequality decreases, while the higher inequality coefficients remain positive; therefore, internal resonance is impossible. Thus, when the first tones of fluid oscillations are greater than one, the asymptotic stability will always be possible with the increase in the elasticity coefficient. For ellipsoidal cavities, this means that they must be oblate along the axis of rotation. It was shown that in the absence of the spherical joint elasticity, the characteristic equation has a zero root, and the conditions of stability are already presented in the form of a system of four inequalities, which are only necessary.

Nonlinear nonstationary heat conduction problem for infinite circular cylinder under a complex heat transfer taking into account the temperature dependence of thermophysical characteristics of materials is solved numerically by the method of lines. Directing it to the Cauchy’s problem for systems of ordinary differential equations studied feature which takes place on the cylinder axis. Taken into account the dependence on the temperature coefficient of heat transfer that the different interpretation of its physical content makes it possible to consider both convective and convective-ray or heat ray. Using the perturbation method, the corresponding thermoelasticity problem taking into account the temperature dependence of mechanical properties of the material is construed. The influence of the temperature dependence of the material on the distribution of temperature field and thermoelastic state of infinite circular cylinder made of titanium alloy Ti-6Al-4V by radiant heat transfer through the outer surface has been analyzed.

In paper the problem of the diffraction of a spherical monochromatic sound wave on a homogeneous isotropic elastic cylinder with a radially non-uniform anisotropic elastic coating. It is assumed that the body is located in a free space filled with an ideal liquid. The analytical solution of the problem is obtained. In the case of steady-state oscillations, the propagation of small perturbations in an ideal fluid is described by the Helmholtz scalar equation. The radiation field of a spherical source is written as a series expansion in cylindrical wave functions. The propagation of elastic waves in an isotropic elastic cylinder is described by the scalar and vector Helmholtz equations. The oscillations of an non-uniform anisotropic elastic cylindrical layer are described by the general equations of motion of a continuous medium. The asymptotic formula for the far field zone is obtained by the steepest descent method. Numerical calculations of the frequency characteristics of the scattered field are carried out for elastic cylinders with homogeneous and inhomogeneous transversally isotropic coatings, as well as for the case of a homogeneous isotropic coating. A significant influence and mutual influence of the inhomogeneity and anisotropy of the coating material on the acoustic properties of the scattering cylindrical body is revealed.

An exact analytic solution of the problem of diffraction of a plane acoustic wave on a cone of arbitrary aperture angle was obtained and studied in [1]. For the case of spherical wave diffraction on a cone a formula is known [2] which relates the solutions of the spherical and plane wave diffraction problems. This study will employ the results of [1, 2] to derive and investigate an exact analytical solution of the problem of diffraction of a spherical acoustic wave on a cone of arbitrary aperture angle. Results of numerical calculations will be presented and compared with analogous results for a plane wave.

Finite element study of a penny-shaped crack along the interface in a bi-material cylinder

We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. It is assumed throughout that the source is a perfect fluid with linear equation of state. We restrict the similarity group to possess both an Abelian G 2, and a single hypersurface orthogonal Killing vector field, and we restrict the fluid flow to be orthogonal to the orbits of the Abelian G 2. The temporal evolution of the models is forced to be power law, due to the similarity group, and the Einstein field equations reduce to a three-dimensional autonomous system of ordinary differential equations which is qualitatively analysed in order to determine the spatial structure of the models. The existence of two classes of well-behaved models is demonstrated. The first of these is asymptotically spatially homogeneous and matter dominated, and the second is vacuum dominated and either asymptotically spatially homogeneous or acceleration dominated, at large spatial distances.

A piecewise-homogeneous medium, consisting of anisotropic layers, is excited by a primary spherical sound wave due to a point source lying in the exterior of the medium or in one of its layers. The direct scattering problem is formulated by means of a modified scalar Helmholtz equation incorporating the anisotropic characteristics of the problem. The T-matrix of the problem is determined analytically by means of a suitable coordinate transformation. Then, the acoustic fields in all layers are obtained. Finally, specific reductions to special cases are presented.

An acoustic or electromagnetic wave propagating in a material medium experiences fluctuations in amplitude and phase that can be attributed in part to scattering from inhomogeneities due to variations in the refractive index of the transmission medium. Since such “patches of inhomogeneity” are often regarded as lenticular in shape, it is the purpose of this paper to extend, for both plane and spherical sound waves, contemporary wave-fluctuation theory (based on a Gaussian correlation function for refractive index) to include the circumstance of spheroidal inhomogeneities. Mean-square amplitude and phase fluctuations are considered for acoustic waves propagating in a statistically isotropic medium in which random deviations of the refractive index are small. Large-scale inhomogeneities of arbitrary spheroidal shape are treated, and the influence of transmitter-to-receiver range, acoustic wavelength, and “patch” dimensions on variations in wave amplitude and phase is derived for this general case. Particular attention is given the high-frequency ray limit (Bergmann region) and the low-frequency (Fraunhofer diffraction) region. The significant effects on wave fluctuations of both spherical and nonspherical inhomogeneities are compared for plane and spherical acoustic waves.

An acoustic wave propagating under water experiences fluctuations in phase and amplitude attributable, in the main, to scattering from inhomogeneities arising from variations in the temperature field. Since “patches of thermal inhomogeneity” in the sea are generally regarded as lenticular in shape, it is the purpose of this paper to extend, for both plane and spherical sound waves, contemporary wavefluctuation theory to include the circumstance of spheroidal inhomogeneities. Mean-square amplitude and phase fluctuations are considered for acoustic waves propagating in a statistically uniform medium in which random deviations of the refractive index are small. Large-scale inhomogeneities of arbitrary spheroidal shape are treated, and the influence of transmitter-to-receiver range, acoustic wavelength, and “patch” dimensions on variations in waveamplitude and phase is derived for this general case. Particular attention is given the high-frequency ray limit (Bergmann region) and the low-frequency (Fraunhofer diffraction) region. The significant effects on wavefluctuations of nonspherical inhomogeneities are compared for plane and spherical acoustic waves. [This work was supported by the LMSC Independent Research Program.]

В статье рассматривается задача рассеяния звуковых волн абсолютно жестким цилиндром с радиально-неоднородным изотропным упругим покрытием в плоском волноводе. Полагается, что волновод заполнен однородной идеальной жидкостью, одна его граница является абсолютно жесткой, а другая - акустически мягкой, законы неоднородности материала покрытия цилиндра описываются дифференцируемыми функциями, гармоническая звуковая волна возбуждается заданным распределением источников на сечении волновода. В случае установившихся колебаний распространение малых возмущений в идеальной жидкости описывается уравнением Гельмгольца. Колебания неоднородного изотропного упругого цилиндрического слоя описываются общими уравнениями движения сплошной среды. Для нахождения поля смещений в неоднородном покрытии построена краевая задача для системы обыкновенных дифференциальных уравнений второго порядка. Первичное поле возмущений представлено совокупностью собственных волн волновода. Давление рассеянного цилиндрическим телом поля ищется в виде потенциала простого слоя. Построена функция Грина для уравнения Гельмгольца, удовлетворяющая заданным граничным условиям на стенках волновода и условиям излучения на бесконечности. Функция плотности распределения источников ищется в виде разложения в ряд Фурье. Для нахождения коэффициентов этого разложения получена бесконечная линейная система уравнений. Проведено усечение бесконечной системы и ее решение найдено методом обратной матрицы. Получены аналитические выражения для рассеянного акустического поля в разных областях волновода.

The method of partial overlapping regions is demonstrated as a modification of the alternating Schwarz's technique. The presented approach uses overlapping region concept to reduce a boundary-value problem for the scalar or vector Helmholtz's equation to a Fredholm's second kind integral equation which can be easily algebraized by means of Galerkin's technique. Applications of the method to printed-circuit transmission line analysis and antenna array analysis are presented and discussed.

DYNAMIC RESPONSE OF AN INFINITE CYLINDER TO ASYMMETRIC PRESSURE ON ITS LATERAL SURFACE

Diffraction of water waves by porous breakwaters is studied based on the linear potential wave theory. The formulation of the problem includes a newly derived relation for the fluid motion through thin porous structures in addition to the conventional governing equation and boundary conditions for small-amplitude waves in ideal fluids. The porous boundary condition, indirectly verified by collected experimental data, is obtained by assuming that the flow within the porous medium is governed by a convection-neglected and porous-effect-modeled Euler equation. A vertically two-dimensional problem with long-crested waves propagating in the normal direction of an infinite porous wall is first solved and the solution is compared with available experimental data. The wave diffraction by a semiinfinite porous wall is then studied by the boundary-layer method, in which the outer approximation is formulated by virtue of the reduced two-dimensional solution. It is demonstrated that neglect of the inertial effect of the porous medium leads to an overestimate of the functional performance of a porous breakwater.

The problem of diffraction of a plane elastic wave by a gradient transversely isotropic layer is considered. Using the method of overdetermined boundary value problem in combination with the Fourier transform method, the system of ordinary differential equations of the second order with boundary conditions of the third type is obtained which is solved by the grid method. Results of calculations obtained using the above-mentioned technique for the case of piecewise linear profiles for the Young modulus of the layer are given.