The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel , where and are the eigenvalues of the negative Laplacian in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane ? in R 2 surrounded by simply connected bounded domains ? j with smooth boundaries ?? j (j = 1, ..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Γ i (i = 1+k j?1, ..., k j ) of the boundaries ?? j are considered, such that and k 0 = 0. The basic problem is to extract information on the geometry of ? using the wave equation approach. Some geometric quantities of ? (e. g. the area of ?, the total lengths of its boundary, the curvature of its boundary, the number of the holes of ?, etc.) are determined from the asymptotic expansion of the trace of the wave kernel for small |t|.

The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel μ（t） =∑ω=1^∞ exp（-itEω^1/2）, where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the （x^1, x^2, x^3）-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ （t） on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j （j = 1,……, n）, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^＊ （i = 1 ＋ kj-1,……, kj） of the bounding surfaces S j are considered, such that S j = Ui-1＋kj-1^kj Si^＊, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω （e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature） are determined from the asymptotic expansion ofexpansion of μ（t） for small │t│.

Examination of Whey Degreasing by Modified Membrane Filtration

The largest amount of dairy by-products, especially the whey, comes from the manufacture of cheese. The whey proteins are used in several different industry technologies. The forage production is used for animal feeding in the forms of various flours mixed in feeds, and the food industry uses whey proteins as human nutrition, such as different dry soups, infant formulas and supplements. The fat components of whey may inhibit the efficient processing and might impair the use of whey in these technologies. Thus, the aim of the experiment was to investigate a cheap and economical separation of the lipid fraction of whey. This separation method was made by microfiltration, which is an inexpensive, effective and energy efficient method for this task. During the measurements, 0.2 μm and 0.45 μm microfiltration membranes were used in a laboratory tubular membrane filtration module, and the membrane separation method was combined and modified by using astatic mixer and/or air insufflation. The same pore size membranes were used in a vibrating membrane filtration equipment （VSEP）, too. The two different membrane filtration devices allowed the comparison of the effect of vibration and the effect of the static mixer and/or air insufflation. The flux values above 0.2 MPa transmembrane pressures strongly decreased on using the tubular membrane. Therefore, it can be determined that the use of the lower transmembrane pressures gave better flux combined with air insufflation and the use of static mixer. The flux values increased three times higher with using vibration during the microfiltration process than that without vibration. Comparing these methods, it can be concluded that the separation made on tubular membrane （0.2 μm） combined with statics mixer gave sufficient result according to the degreasing, retentions and flux values of the other components.

A study on wrinkling characteristics and dynamic mechanical behavior of membrane

An eigenvalue method considering the membrane vibration of wrinkling out-of-plane deformation is introduced, and the stress distributing rule in membrane wrinkled area is analyzed. A dynamic analytical model of rectangular shear wrinkled membrane and its numerical analysis approach are also developed. Results indicate that the stress in wrinkled area is not uniform, i.e. it is larger in wrinkling wave peaks along wrinkles and two ends of wrinkle in vertical direction. Vibration modes of wrinkled membrane are strongly correlated with the wrinkling configurations. The rigidity is larger due to the heavier stress in the part of wrinkling wave peaks. Therefore, wave peaks are always located at the node lines of vibration mode. The vibration frequency obviously increases with the vibration of wave peaks.

VIBRATION OF ARBITRARILY SHAPED MEMBRANES WITH ELASTICAL SUPPORTS AT POINTS

This paper presents a new method for solving the vibration of arbitrarily shaped membranes with ela.stical supports at points. The reaction forces of elastical supports at points are regarded as unknown external forces acting on the membranes. The exact solution of the equation of motion is given which includes terms representing the unknown reaction forces. The frequency equation is derived by the use of the linear relationship of the displacements with the reaction forces of elastical supports at points. Finally the calculating formulae of the frequency equation of circular membranes are analytically performed as examples and the inherent frequencies of circular membranes with symmetric elastical supports at two points are numerically calculated.

Backward Propagation of Otoacoustic Emissions

Normal mammalian ears not only detect but also generate sounds. The ear-generated sounds, i.e., otoacoustic emissions (OAEs), can be measured in the external ear canal using a tiny sensitive microphone. In spite of wide applications of OAEs in diagnosis of hearing disorders and in studies of cochlear functions, the question of how the cochlea emits sounds remains unclear. The current dominating theory is that the OAE reaches the cochlear base through a backward traveling wave. However, recently published works, including experimental data on the spatial pattern of basilar membrane vibrations at the emission frequency, demonstrated only forward traveling waves and no signs of backward traveling waves. These new findings indicate that the cochlea emits sounds through cochlear fluids as compression waves rather than through the basilar membrane as backward traveling waves. This article reviews different mechanisms of the backward propagation of OAEs and summarizes recent experimental results.

An Elastic Absorber Theory for a Thin Fabric Sheet

The current sound absorption theory which is based on Rayleigh model believes that fibrous material absorb sound by the fluid frictional energy dissipation between the air and the solid fibers. However, Rayleigh model is only useful for a quanlitative understanding of effects In a porous material but not for calculation of the acoustical properties of real absorbent. In this paper, a new vibration sound absorption theory which is totally different from classical theory was put forward. The specific acoustic impedance of fiber layers have been derived from the membrane vibration equation and the sound absorption coefficient calculated agree with test results. The new theory can explaIn the phenomenon that thIn fiber layers exhibit less sound absorption coefficient when it was as the cover fabric of sound absorber, but it is more efficient to sound absorption when it was hang as the curtains or have back cavity behind it.

Some studies on mechanics of continuous mediums viewed as differential manifolds

The continuous mediums are divided into two kinds according to their geometrical configurations,the first one is related to Euclidian manifolds and the other one to Riemannian manifolds/surfaces in the point of view of the modern geometry.Two kinds of finite deformation theories with respect to Euclidian and Riemannian manifolds have been developed in the present paper.Both kinds of theories include the definitions of initial and current physical and parametric configurations,deformation gradient tensors with properties,deformation descriptions,transport theories and governing equations of nature conservation laws.The essential property of the theory with respect to Euclidian manifolds is that the curvilinear coordinates corresponding to the current physical configurations include time explicitly through which the geometrically irregular and time varying physical configurations can be mapped in the diffeomorphism manner to the regular and fixed domains in the parametric space.It is quite essential to the study of the relationships between geometries and mechanics.The theory with respect to Riemannian manifolds provides the systemic ideas and methods to study the deformations of continuous mediums whose geometrical configurations can be considered as general surfaces.The essential property of the theory with respect to Riemannian manifolds is that the thickness variation of a patch of continuous medium is represented by the surface density and its governing equation is rigorously deduced.As some applications,wakes of cylinders with deformable boundaries on the plane,incompressible wakes of a circular cylinder on fixed surfaces and axisymmetric finite deformations of an elastic membrane are numerically studied.