A PARAMETRIC STUDY ON THE AXISYMMETRIC MODES OF VIBRATION OF MULTI-LAYERED SPHERICAL SHELLS WITH LIQUID CORES OF RELEVANCE TO HEAD IMPACT MODELLING

The free vibration of spheres composed of inviscid compressible liquid cores surrounded by spherical layers of linear elastic, homogeneous and isotropic materials are studied using three-dimensional elasticity equations. The exact three-dimensional equations are first derived for an N -layered sphere with a liquid core and an extensive parametric study is then presented for the first few natural frequencies of the spheroidal modes of vibration. Non-dimensional frequency parameters are compared with values obtained using lower order membrane and shell theories. It is shown that for a remarkably wide range of geometric and material parameters, which encompasses values typical for the human head, the first ovalling mode of a fluid-filled shell behaves like a membrane filled with incompressible fluid and a simple closed-form expression is derived which closely approximates the natural frequencies obtained using the exact three-dimensional equations.     

An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.     

One and two mode behaviors of surface phonon-polaritons of ternary mixed crystal films

One and two mode behaviors of surface phonon-polaritons of ternary mixed crystal (TMC) films are studied in the framework of the modified random-element-isodisplacement model and the Born-Huang approximation, based on the Maxwell's equations with the usual boundary conditions. The numerical results for the frequencies and splitting energies of the surface phonon-polaritons as functions of the composition in several II-VI and III-V compound semiconductor ternary mixed crystal films are obtained. The “two-mode” and “one-mode” behaviors for different types of systems are clearly shown in the curves of the splitting energies of surface phonon-polaritons. The theoretical conclusion obtained is agreement with the reported experiment results for bulk TMC systems.     

Parametric instability and Hamiltonian chaos in cavity semiclassical electrodynamics

We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high-Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ/2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high-Q microwave cavity are possible objects for observing parametric instability and dynamical chaos. Zh. éksp. Teor. Fiz. 115, 740–753 (February 1999)     

Skin-effect approximation of a full sphere for a large radius-frequency product (cosmic situations)

In a previous paper (Zollmann S., Schmutzer E., Kluge G., Czech. J. Phys. B29 (1979), 1224) the quasistationary exact theory of the skin effect of a conducting spherical shell was developed. The mathematical structures for the field functions in the interior, shell, and exterior are rather complicated. For a better understanding of the physical results, which above all are interesting for the polarity reversals of magnetic stars, the exact solutions are here approximated for a rather large radius-frequency product (radius and frequency enter in a characteristic combination). As a result radial electromagnetic waves with interesting characteristics occur in the sphere. Numerical estimates are performed.We thank Dr. G. Kluge very much for his discussions on this problem.     

Vibro-acoustic response of an elliptical plate-cavity coupled system to external shock loads

A general fully coupled three-dimensional vibro-acoustic model is developed to investigate the forced non-stationary acousto-structural response of a thin elastic plate of elliptical planform which is backed by a reverberant, rigid, and finite (closed) elliptic cylindrical acoustic enclosure, while under the action of general external transverse loads of arbitrary temporal and spatial variations. The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables in elliptic coordinates is used to obtain the transformed solutions as a linear combination of even and odd modes in terms of products of radial and angular Mathieu functions. A linear system of coupled algebraic equations is ultimately obtained, which is truncated and then solved numerically by implementing Durbin’s numerical Laplace transform inversion scheme. Detailed numerical simulations are conducted for the temporal histories of plate center-point displacement and on-axis cavity acoustic pressure for air-coupled elliptic aluminum plates of selected aspect ratios when subjected to external loadings of practical interest (i.e., an impulsive point load, a uniformly distributed pulse load, and a blast load). Also, acoustic radiation into the backing enclosure is examined by using appropriate 2D images of the internal sound field for selected cavity depths and plate eccentricities. The presented results confirm that the acousto-elastic characteristics of the coupled plate-cavity system are significantly influenced by the plate aspect ratio, cavity depth and the transverse loading configuration. Validity of the work is established through the computations made by using a commercial finite element package.     

Frequency change of partial spherical waves induced by time change of medium permittivity

The transformation of a plane harmonic wave caused by a time jump of the permittivity inside a dielectric sphere is investigated. The exact expressions for the transformed field are obtained as a solution of an initial and boundary value electromagnetic problem for the Maxwell’s equations based on the longitudinal and the transverse spherical vector functions.     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.     

Fast algorithms for spherical harmonic expansions,III

We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on “Fast algorithms for spherical harmonic expansions”. The requisite precomputations become manageable when organized as a “depth-first traversal” of the program’s control-flow graph, rather than as the perhaps more natural “breadth-first traversal” that processes one-by-one each level of the multilevel procedure. We illustrate the results via several numerical examples.     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.     

多模短波长激光在圆形球面腔中的传输

用数值积分方法求解了在圆柱坐标中的横向多模菲涅尔(Fresnel)积分方程.在初始复数光场振幅取均匀分布和随机分布的条件下对圆形球面共焦腔、稳定腔和平面腔的TEM_n0模进行了计算分析.计算结果表明,球面腔单一模的数值结果同解析解的结果符合较好,多种模同时存在时的结果不仅符合多模空间分布特性而且其基模是主要的.     

A spherical cavity in an Einstein Universe

Suitable metric forms for the regions inside and outside a spherical cavity in an Einstein universe are derived by means of perturbation. It is shown that for low proper pressure, the cavity behaves like negative Schwarzschild mass. Finally, the possibility of carrying over to the exact theory a proposed definition of the gravitational field in matter is examined.     

Analysis of the H-Plane T-Junction Millimeter Wave Waveguide Circulator with a Ferrite Sphere

This paper presents an approximate but efficient field treatment of the new easy-to-fabricate ferrite sphere based H-plane waveguide circulator for potentially low-cost millimeter wave communication systems. A new three-dimension modeling strategy using a self-inconsistent mixed coordinates based mode matching technique is developed, i.e. the solutions of the Helmholz wave equations in the ferrite sphere and in the surrounding areas are deduced in the form of infinite summation of spherical, cylindrical and general Cartesian modes respectively. The point matching method is then used on the interface to numerically obtain the coefficients of different orders basis functions of the field. Therefore, the field distributions as well as the characteristics of the circulator are numerically calculated and the good agreement is observed between the numerical results and the measured data.     

Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions

This paper presents a general account of the indentation responses of a one-dimensional hexagonal quasicrystal half-space pressed by an axisymmetric rigid punch. Based on Green's functions of the half-space subjected to point sources on the surface, the mixed boundary value problem is transformed to integral equations and solved exactly using the results of the potential theory method. Explicit expressions for the generalised pressures and indentation forces are derived for three common indenters (cylinder, cone and approximate sphere) in a systematic manner. For conical and spherical indenters, relations between the contact radius and indentation loads are determined. The coupling phonon–phason fields in the half-space under indentation are accurately expressed in terms of elementary functions. Numerical calculations are performed and discussions on related physical phenomena are given. The present exact solutions can serve as benchmarks for approximate or numerical analyses and can guide the experimental characterisation of material properties of quasicrystals.     

多重散射方法研究轴对称空腔覆盖层的声学特性

发展了一种多重散射方法研究声学覆盖层的半数值半解析模型,分析了影响轴对称空腔结构声学性能的主要能量耗散机制。在球坐标条件下推导出轴对称空腔结构的位移和应力场基函数,通过对空腔表面基函数的数值积分,得到散射波和入射波之间的传输矩阵方程,结合分层介质声传播理论计算了周期性空腔结构覆盖层的反射、透射和吸声性能。研究结果表明;空腔共振是低频能量耗散的主要形式,边界条件对材料空腔结构的谐振特性影响很大,利用双空腔耦合共振可以拓宽材料的低频吸声频带;背衬对材料的高频吸声影响较小,材料的高频能量损耗取决于空腔的散射和波型转换特性。      

Eigenmodes of a Toroidal Cavity with a Conducting Separating Wall

The eigenmodes of a toroidal cavity with a separating wall are examined in the light of theory and experiment. Excellent agreement between the predicted resonant frequency of 1.148 GHz corresponding to a 4-period generating function and the experimentally observed frequency is found. The exact solution of Maxwell's equations in toroidal systems with a separating wall is presented and the eigenfrequencies are experimentally measured on a toroidal cavity with major and minor radii of respectively 12.5 cm and 10.0 cm.     

The vibration modes of concentrically supported free circular plates

Vibration of a circular plate internally concentrically supported is basic in structural mechanics. The complete first six frequencies and mode shapes are determined for clamped, simply-supported, sliding, and ring supports using exact characteristic frequency equations. Complex mode switches occur as the support radius is varied. Asymptotic formulas for large support radii are derived. Different types of singularities as the support radius shrinks to zero are distinguished and classified.     

Modelling of isotropic double negative media for microwave applications

A composite medium consisting of two sublattices of dielectric spherical particles of high permittivity and different radii embedded in a dielectric matrix of smaller permittivity are considered. It has been shown that such a composite medium reveals properties of an isotropic double negative media (DNG) in a limited frequency range, when resonance oscillations of H_III mode in one kind of particles and E_III mode in another kind of particles are excited simultaneously. The E_III resonance and the H_III resonance give rise to the magnetic dipole momentum and the electric dipole momentum correspondingly. Averaging the magnetic momentum and the electric momentum over the cells belonging to the appropriate spherical particles gives the negative permittivity and permeability. The model of diffraction of a plane electromagnetic wave on a dielectric sphere is presented and compared with the mixing rule based consideration. The results obtained are rather close. Distribution of the electromagnetic wave outside the sphere is calculated. Influence of the dispersion of the sphere size and the dielectric permittivity on the effective parameters of the DNG material is estimated.     

Introduction to light scattering by Gaussian random particles

Background, current status, and future prospects are offered for “Light scattering by Gaussian random particles: Ray-optics approximation” [1]. The stochastic geometry of the random particle is called the Gaussian random sphere. The radial distance of the Gaussian sphere is lognormally distributed. Two logarithmic radial distances at a given great-circle angle apart relate to one another according to the covariance function. Sample Gaussian particles can be conveniently generated using a Legendre polynomial expansion for the covariance function and a spherical harmonics expansion for the logarithmic radial distance. The ray-optics approximation consists of the geometric-optics and forward-diffraction parts fully accounting for polarization. It is valid for particles much larger than the wavelength of incident light and with central phase differences much larger than unity. The numerical ray-tracing algorithms are general and, in principle, applicable computationally to arbitrarily shaped non-spherical particles.     

Analysis of acoustic directivity of spherical cap transducers

In order to study the acoustic radiation from the spherical cap transducer,a theoretical model is used by solving the wave equation in spherical coordinates using the method of separation of variables,based on the spherical harmonic Fourier transform and boundary condition.The calculation formulas for the far field radiated pressure and directivity of spherical cap are derived.Some theoretical results are presented in the form of far-field directivity patterns of the spherical cap transducer for various polar angles of the spherical cap,radii of sphere baffle and operating frequencies.Both the diameter of the sphere baffle and the wavelength determine the directivity of acoustic radiation from a spherical cap.When the frequency is lower or the wavelength is longer than the diameter of the sphere baffle,the acoustic radiation from a spherical cap is omnidirectional.With the increase of the frequency or the diameter of the sphere baffle,the acoustic radiation from a spherical cap is more directional and the beamwidth more tends to spherical cap angle.Furthermore,the ripple in the beam is more obvious.The high frequency spherical cap transducer was fabricated and the directivity pattern was tested and the measurement data is shown to coincide with the theoretical results.This research can provide a guideline for designing the spherical cap transducers and arrays.