Attractive and repulsive Casimir vacuum energy with general boundary conditions

The infrared behaviour of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth–Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials.     

Confinement of atoms under general boundary conditions

The energy spectrum of a nonrelativistic quantum particle and hydrogen-like atom is considered under the most general conditions of confinement in a spatial box (vacuum cavity). It is shown that the rearrangement of the lowest energy levels occurring in this case turns out to be considerably more significant when compared with the case of confinement achieved by the impenetrable potential barrier. The role in the rearrangement of this level, played by the von Neumann-Wigner level repulsion effect, is emphasized. For an atom confined in a spherical cavity of radius R, it is also shown that, when the role of the cavity boundary is played by the surface layer of nonzero depth d, the atomic ground state possesses a deep and pronounced minimum for the physically reasonable width and depth of that surface layer, in which the binding energy turns out to be an order of magnitude larger than that of the lowest 1s-level of a free atom E _1s . Also, it becomes possible to achieve a mode when the binding energy of an atom is noticeably higher than E _1s at R on the order of 10–100 nm.     

Optimization of a clamped plate silencer

A previous theoretical study [L. Huang, J. Acoust. Soc. Am. 119, 2628-2638 (2006)] shows that, in a duct, a simply supported plate covering a side-branch rigid cavity can function effectively as a wave reflector over a broad range of low to medium frequencies. In this study, analytical formulation is extended to the boundary condition of clamped plate, which is easier to implement in practice. The theoretical model is tested experimentally using balsawood, which has a very high stiffness to mass ratio. The spectral peaks and shapes of the measured TL are in agreement with those calculated theoretically, attempts are also made to account for the considerable sound absorption in the rig. Further numerical studies based on the validated model show that, for a uniform plate, the optimal stopband is narrower and the lower band limit is worse than that of the simply supported configuration. However, a wave reflector using nonuniform, clamped plates with thinner ends out-performs the simply supported configuration in every aspect. Analyses show that the improvement is attributed to the increased acoustic radiation efficiency over the bulk length of the nonuniform plate, which behaves more like a rigid plate.     

Inverse scattering on the half-line for generalized ZS-AKNS system with general boundary conditions

The inverse scattering problem for a first order system of three equations on the half-line with nonsingular diagonal matrix multiplying the derivative and general boundary conditions is considered. It is focused the case of two repeated diagonal elements of diagonal matrix. The scattering matrix on the half line is defined and a unique restoration of the potential from the scattering matrix is proved. The possible application to integration of integro-differential four-wave interaction problem is also focused.     

Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

We study the bifurcation of radially symmetric solutions of +f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.Research supported in part by the NSF under Grant No. MCS-800 2337     

Acoustic analysis of a rectangular cavity with general impedance boundary conditions

A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh-Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh-Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.     

Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions

By using the intertwiner and face-vertex correspondence relation, we obtain the Bethe ansatz equation of the eight-vertex model with open boundary conditions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for the eight-vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries.     

A general method for deriving boundary conditions associated with reduced distribution functions

A systematic procedure based on projection operator methods for deriving boundary conditions on the reduced distribution functions arising from those imposed on the full distribution function is given. The method is illustrated by applying it to Brownian motion.     

Stochastic radiative transfer in a finite plane-parallel medium with general boundary conditions

The time-independent radiative transfer problem in a scattering and absorbing planar random medium with general boundary conditions and internal energy source is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The problem is solved in terms of the solution of the corresponding free-source problem with simple boundary conditions which is solved using Pomraning-Eddington approximation in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average partial heat fluxes are calculated in terms of the albedoes of the source-free problem. Results are obtained for isotropic and anisotropic scattering for specular and diffused reflecting boundaries.     

Development of composite plate for compact silencer design

A compact flow-through plate silencer is constructed for low frequency noise control using new reinforced composite plates. The concept comes from the previous theoretical study [L. Huang, Journal of the Acoustical Society of America 119 (2006) 2628–2638] which concerns a clamped supported plate enclosed by rigid cavities. When the grazing incident sound wave comes and induces the plate into the vibration, it will radiate sound and reflect sound. Such sound reflection causes a desirable noise reduction from low to medium frequency with wide broadband. The structural property of the very light plate with high bending stiffness is very crucial element in such plate silencer. In this study, an approach to fabricate new reinforced composite panel with light weight and high flexibility to increase the bending stiffness is developed in order to realize the function of this plate silencer practically. The plate silencer can be constructed in more compact size compared with the previous two-plate silencer with two rectangular cavities and the performance with the stopband of the range from 229 to 618 Hz, in which the transmission loss is higher than 10 dB over the whole frequency band without flow or with flow at the speed of 15 m/s, can be achieved. The experimental data also proves that the non-uniform clamped plates with thinner ends perform very well. To implement the use of such silencer practically in controlling noise at different dominant frequency ranges, a design chart has been established for searching the optimal bending stiffness and corresponding stopband at different geometries.     

Identification technique for nonlinear boundary conditions of a circular plate

As a basic study for developing an identification technique for boundary conditions of machines and structures, a new technique for a circular plate is proposed. This technique has features that do not require data measured on the boundary and is applicable to nonlinear boundary conditions. In the proposed technique, the boundary is modelled by springs and dampers as well as effective mass and moment of inertia. Then their characteristics are determined by using the analytical solution together with the experimental data. Since the technique is based on the analytical solution, it is applicable to any structure, provided that its analytical solution can be derived. Numerical simulation is conducted to show that the procedure determines the boundary conditions accurately.     

基于边界元法的近平板圆孔气泡动力学行为研究

研究了带有圆孔的平板附近气泡动力学特性. 基于不可压缩势流理论, 建立了平板圆形破口附近气泡运动数值模型, 并针对气泡初始位置距离破口很近而导致计算结果发散的数值缺陷, 采用气泡壁和壁面融合的方法, 将流场分离为两个半无限域问题进行求解, 实现了在不同无量纲参数范围内的数值模拟, 数值结果与实验结果符合良好. 通过对圆孔附近气泡运动特性的研究发现, 圆孔对气泡的影响基本与壁面相反, 在膨胀阶段对气泡产生腔吸作用, 收缩阶段产生排斥, 在特定的工况下会产生对射流现象. 最后分析了气泡壁与壁面融合, 流场分离后的气泡动态特性以及各工况参数对其影响规律. 关键词： 气泡 边界元 射流 圆孔     

Vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions

Results are presented for the fundamental frequency of a rectangular plate having a thickness which varies in a bilinear fashion in the x-direction. Translational and rotational flexibilities are taken into account at all edges. A simple algorithm, which allows one to evaluate the fundamental frequency of vibration, is derived by making use of the Ritz method and expressing the fundamental displacement function in terms of a polynomial co-ordinate function which satisfies approximately the natural boundary conditions.     

Upwind schemes and boundary conditions with applications to Euler equations in general geometries

A unified treatment of several upwind shock capturing algorithms is presented. Each algorithm has a Riemann initial value problem as its basis. The treatment of boundaries involves solving the associated Riemann initial-boundary value problem. The first author's algorithm, applied to multidimensional Euler equations in general geometries, is then presented. Its worth is verified by various calculations, which include Mach 8 supersonic flow past a circular cylinder.     

On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces

This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded bodyD with general boundary conditions. First, mildL ¹-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with anH-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weakL ¹-solutions are proved by compactness, using theH-theorem from the cutoff case. Finally, anH-theorem is given also for the infinite-range case.     

Influence of boundary restraint on sound attenuation performance of a duct-membrane silencer

Low frequency noise in duct is a challenge for the traditional passive noise control techniques. Recently, a so-called duct-membrane silencer has attracted much research attention due to its simple configuration and potential application, however, the current studies are merely limited to the cases in which just the classical boundary conditions are considered. Actually, as an important factor affecting the modal characteristics of the membrane, and the existing studies are not enough to fully understand the vibro-acoustic characteristics of such silencer with complicated boundary conditions. Motivated by this, in this paper, the structural–acoustic coupling model of duct-membrane system is established by a modified Fourier series method in combination with Rayleigh–Ritz procedure, in which the transverse elastic boundary restraints are taken into account. Energy principle is formulated for the vibro-acoustic coupling of such duct-membrane silencer to obtain the system matrix equation. Numerical results are then presented to validate the proposed model, and the influence of boundary restraining stiffness on sound attenuation performance is also studied. To the best of authors’ knowledge, this work represents the first time that the elastic boundary restraints have been considered for such duct-membrane silencing system.     

Experimental validation of the sound transmission of rectangular baffled plates with general elastic boundary conditions

Several prediction methods have recently been developed for systematically studying the effects of general boundary conditions on the sound transmission loss (STL) of plate-like structures. But corresponding experimental validation studies remain scarce owing to the difficulty of obtaining accurate boundary conditions for practical structures. This paper presents a convincing experiment conducted on a baffled plate system to validate the STL prediction model in a previous paper by Yu et al. [Noise Control Eng. J. 58(2), 187-200, 2010]. A method is proposed to determine the boundary conditions of this system, and experimental STL compares well with the predictions based on the identified boundary condition.     

Spectral determinant on graphs with generalized boundary conditions

The spectral determinant of the Schr?dinger operator ( - Δ + V(x)) on a graph is computed for general boundary conditions. (Δ is the Laplacian and V(x) is some potential defined on the graph). Applications to restricted random walks on graphs are discussed. Received 9 July 2001