Directional properties of an array of sound receivers positioned in an impedance screen recess

Expressions for calculating the directional characteristics of an array of sound receivers positioned in a waveguide with impedance walls are obtained from the solution to the problem on the diffraction of a plane sound wave by the waveguide open end with impedance flanges. The waveguide can be of a finite length, and, in this case, it can be considered as an open cavity in an impedance screen. The solution of the integral equation for the sound pressure distribution over the opening area is reduced to the solution of an infinite system of algebraic equations for the coefficients of the field expansion in normal waveguide waves. Examples of calculated directional characteristics are presented for arrays with receivers positioned at different distances from the opening and for different values of the impedances of the waveguide walls and flanges.     

On acoustic radiation by a rigid object in a fluid flow

A problem of sound radiation by an absolutely rigid object, moving with respect to the surrounding fluid, is considered on the basis of the Lighthill's equation for aerodynamic sound. An integral representation of the radiated acoustic field is utilized, where the field is characterized as the sum of three fields, generated by a volume distribution of monopoles and by distributions of monopoles and dipoles on the surface of the rigid object. It is shown that, due to a discontinuity of Lighthill's stress tensor on the rigid boundary, a layer of surface divergence of hydrodynamic stresses on the boundary must be taken into account when evaluating the volume integral over Lighthill's quadrupole sources. When the contribution of the surface divergence is included in the solution of Lighthill's equation, amplitudes of the monopole and dipole sound radiated by the rigid object are shown to depend on the potential components of the normal velocity and the pressure on the rigid surface. The obtained solution is compared with Curle's solution for this problem, which establishes that the sound radiation by a rigid object is determined by the force exerted by the object upon the fluid. Both solutions are applied to two known problems of sound scattering and radiation by a rigid sphere in variable pressure and velocity fields. It is shown that predictions based on the obtained solution are equivalent to the results known from literature, whereas Curle's solution gives predictions contradicting the known results. It is also shown that the Ffowcs Williams and Hawkings equation, which coincides with Curle's equation for an immoveable rigid object, does not lead to the correct predictions as well.     

Sound generation and transmission in flow ducts with axial temperature gradients

This article describes a one-dimensional, linearized, analysis of fundamental mode sound generation and propagation in rigid-walled flow ducts with axial temperature variation. An acoustic wave equation, including damping effects and volume sources, is derived and its solution (in the absence of sources) by a numerical technique and an approximate analytical method is discussed. The “forced” wave equation is then solved (the existence of an oscillating solution to the “unforced” equation being assumed) for sound generation by a side-branch volume source in an infinite duct, and the results are applied to a duct of finite length. Reasonably good agreement is obtained between measurements and predictions of the sound pressure field in a flow duct, away from the source region.     

On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown [I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.     

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.     

Diffraction of sound by an elastic cylinder near the surface of an elastic halfspace

Diffraction of an acoustic wave by an elastic cylinder near the surface of an elastic halfspace is considered. The solution relies on a Helmholtz-type integral equation and uses the Green function of an elastic halfspace. The latter function is represented in the form of an integral over the Sommerfeld contour on the plane of a complex variable that has the meaning of the angle of the wave incidence on the halfspace boundary. An integral equation for the sound pressure distribution over the cylinder surface is derived. This equation is reduced to an infinite system of equations for the Fourier-series expansion coefficients of this distribution. The results obtained are valid for the diffraction of a cylindrical wave and a plane wave. They also describe the diffraction of a spherical wave when the transmitter and receiver are far from the cylinder and lie in one plane that is orthogonal to the cylinder axis.     

Microscopic theory of sound propagation in dielectric crystals

The time evolution of the atomic displacement field in a dielectric crystal subjected to an external force is studied in the domain of linear response by means of imaginary time Green's functions. For slowly varying disturbances two coupled equations have to be solved: a differential equation for the amplitude of an acoustic wave and a linearized Boltzmann equation. The latter results from the integral equation for the vertex part and includes an additional integral operator. The collision equation is solved for different relative magnitudes of the sound frequency and the frequencies for normal and Umklapp processes using the method developed by Weiss. Some of the expressions showing up in the velocity and damping of the sound wave are estimated numerically for rare gases with two-body forces in the form of the Morse potential.     

Electromagnetic wave scattering by small bodies

A reduction of the Maxwell's system to a Fredholm second-kind integral equation with weakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic size of the bodies tends to zero. The technique developed is used for solving the many-body EM wave scattering problem by rigorously reducing it to solving linear algebraic systems, completely bypassing the usage of integral equations. An equation is derived for the effective field in the medium, in which many small particles are embedded. A method for creating a desired refraction coefficient is outlined.     

一种可稳定计算Pekeris波导中声场的波数积分方法

提出一种可稳定计算Pekeris波导中声场的波数积分方法,并在此基础上开发出一个数值模型,可用于提供Pekeris波导中声场的精确、稳定的数值解。在这个方法中,由于与深度有关的波动方程齐次解中所有的上行波与下行波均采用了合理的归一化表示,从而得到的系统方程是无条件稳定的。在简正波方法中,割线积分一般只对近场有显著影响。因此,传统的简正波模型一般都忽略割线积分对声场的贡献。但是,如果某号简正波离割线非常近,则割线积分对非常远距离的声场仍可能有显著影响。在这种情况下,传统的简正波模型由于忽略割线积分的贡献,从而得到的声场结果是不准确的。本文通过数值算例比较本文提出的波数积分模型与传统的简正波模型。数值结果表明,本文提出的模型可以提供精确、稳定的Pekeris波导中声场的数值解,而在某些情况下传统的简正波模型得到的声场结果是不准确的。因此,本文提出的模型可以作为Pekeris波导中声传播问题的标准模型使用。      

Exact solutions to one-dimensional acoustic fields with temperature gradient and mean flow

An exact solution for one-dimensional acoustic fields in ducts in the presence of an axial mean temperature gradient and mean flow is presented in this paper. The analysis is valid for mean Mach numbers such that the square of the mean Mach number is much less than one. The one-dimensional wave equation for ducts with axial mean temperature gradient and mean flow is derived. By appropriate transformations, the wave equation is reduced to an analytically solvable hypergeometric differential equation for the case of a linear mean temperature profile. The developed solution is applied to investigate the dependence of sound propagation in a duct on factors such as temperature gradient and mean flow. The results obtained using the analytical solution compare very well with the numerical results. The developed solution is also compared with an existing analytical solution.     

Transformation of a monochromatic plane wane by the pulse-periodic time modulation of an infinite medium

An exact solution to the problem of the transformation of a monochromatic plane wave by a finite train of equally spaced rectangular pulses of permittivity and conductivity of an infinite medium is considered. The permittivity pulse train is shifted relative to the conductivity pulse train by an arbitrary time. The problem is studied analytically in terms of the second-order Volterra integral equation describing the electromagnetic wave transformation in a medium with time-dependent parameters. The equation is solved using the resolvent technique. Expressions for the amplitude of the transformed electric field component for any time instant at any spatial point are derived and analyzed.     

Reconstruction of the dynamic contact stresses in an elastic layer from the displacements of its free surface

An inverse problem on the reconstruction of the wave field of contact stresses produced by an external load in an elastic layer from the displacements of its free surface is considered for the model of forced steady-state vibrations in the approximation of plane deformations. The solution is constructed using two approaches: (1) a reduction of the problem to the Fredholm integral equation of the first kind with the use of the Tikhonov regularization and (2) an expansion of the solution in a discrete set of waves. It is shown that both approaches are approximately equivalent in the model under consideration. Possibilities for an adequate reconstruction of the source field from far-zone measurements of a finite number of propagating wave modes are analyzed.     

Diffraction of sound by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace

An exact solution is obtained to the problem of sound diffraction by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace. The problem is solved using the Helmholtz integral equation in which the field of a point source in the halfspace with an elastic boundary is used as the Green function. The diffracted field is represented as a series expansion in spherical harmonics. The expansion coefficients are determined from a set of independent algebraic systems of equations. The matrix coefficients of these systems are determined as integrals of the products of the associated Legendre polynomials on the complex plane with respect to the real and complex angles of the sound incidence on the halfspace boundary. To decrease the number of such integrals, expansions using the Klebsh-Gordon coefficients are applied. As a result, algorithms for calculating the scattered field in the halfspace are obtained.     

两个相邻目标对平面波、高斯波束的光散射

基于等效原理和互易性定理研究了两个靠近目标对平面波、高斯波束的光散射问题,给出了这一复合光散射模型的二阶散射结果。通常一阶散射结果容易求解,但由于耦合效应的复杂性,很难给出二阶散射结果的解析形式。为了解决这一问题,应用互易性定理给出了求解任意相邻介质目标二阶散射场的公式,同时借助等效原理将求解散射场公式中的体积分简化为面积分的形式,从而降低了求解难度。求解了两相邻球形粒子的复合散射场,并将求解结果与应用时域积分方程法求得的结果进行了比较。同时,还讨论了束腰半径、目标位置对散射截面及偏振度的影响。     

Steady-state sound field in enclosures

The classical normal-mode theory expresses the steady-state soundfield in an enclosure produced by a sound source as a series of normal modes ofvibration.Experimental facts are not often explained by this theory,and it wasconjectured that the normal-mode expression is not the complete solution ofthe wave equation in the enclosure,but only the reverberant part of it,and thereshould be an additional term representing the direct spherical radiation to makethe solution complete.The problem is examined by critically reviewing the de-rivation of the normal-mode expression,and by theoretical analysis of thesteady-state sound field in the room and experimental measurements therein.The conjecture is thus confirmed,and it is definitely shown that the sound fieldshould contain the direct wave as well as the standing waves(normal modes)formed by the confinement of the boundary surfaces.Relevant mathematicalexpressions are derived.     

Solution of the Vlasov equation for collective modes in nuclei

A semiclassical theory of giant resonances based on the Vlasov equation is developed. The linearized Vlasov equation is solved for the bound motion of particles in a central potential with an external time-dependent multipole field. The solution obeys an RPA-type integral equation. If the time-dependent part of the self-consistent field is neglected, the solution of the Vlasov equation has a simple analytical form. The strength function for each multipole can be expressed in terms of the natural frequencies of classical orbits and of radial integrals over the classical motion. The method is illustrated by studying the isoscalar monopole, quadrupole and octupole response in medium-heavy nuclei without residual interaction. There are remarkable similarities between the solutions of the semiclassical problem and the corresponding quantum problem. For a central potential with Saxon-Woods shape there is an interesting shift and concentration of strength in the quadrupole and octupole response functions.     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

Velocity redistribution by standing waves

This paper considers the modifications of the atomic velocity distribution imposed by a standing wave. The recoil due to induced and spontaneous processes provides an effective force on the particles. We formulate the general problem of a two-level atom in the field of two counter-propagating waves. We derive the rate equation limit for these equations and show how it can be generalized to treat a broad-band source of radiation. The connection with generalized relaxation theory is discussed. The ensuing integral equations are solved numerically for various cases of cooling and heating.     

Die Methode der Normallösungen für die Vlasov-BGK-Gleichung

The linearized Vlasov equation with collision damping is solved by the method of normal modes of Van Kampen and Case. The system considered is an infinitely extended nonrelativistic nondegenerate electron gas with neutralizing ion background and neutral particles. There is no magnetic field. Collision damping is taken into account by complete Bhatnagar-Gross-Krook collision integrals for electron-electron collisions and electronion collisions; in the case of a partially ionized gas elastic collisions between electrons and neutrals can be included likewise. The Vlasov-BGK operator is transformed into an integral operator yielding complex singular normal modes even if the equilibrium distribution is a Maxwellian. The adjoint integral equation belongs to a more complicated type than that of a collision-free system. Its solutions must be orthogonalized. The set of all normal modes is complete rendering the exact solution of the initial value problem possible. The completeness is shown by transformations and regularization of the singular integral equation of the initial value problem, the techniques of Case not being applicable because of the complicated type of this equation.     

Direct computation and aeroacoustic modelling of a subsonic axisymmetric jet

A numerical algorithm for acoustic noise predictions based on solving Lilley's third order wave equation in the time-space domain is developed for a subsonic axisymmetric jet. The sound field is simulated simultaneously with the source field calculation, which is based on a direct solution of the compressible Navier-Stokes equations. The computational domain includes both the nearfield and a portion of the acoustic farfield. In the simulation, the detailed sound source structure is provided by the nearfield direct numerical simulation (DNS), while the sound field is obtained from both the DNS and the numerical solution to the non-linear Lilley's equation. The source terms of Lilley's equation are used to identify the apparent sound source locations in the idealized axisymmetric low-Reynolds number jet. The sound field is mainly discussed in terms of instantaneous pressure fluctuations, frequency spectra, acoustic intensity and directivity. A good agreement is found between the predictions from the axisymmetric Lilley's equation and the DNS results for the sound field. Limitations and perspectives of the simulation are also discussed.