Lamb Equations Research Articles

Numerical and experimental study of acoustic wave propagation in glass plate/water/128YX-LiNbO3 structure

There is a need for digital microfluidic systems that can integrate pretreatments and sensors on the same substrates in the medical and biotechnology fields. Surface acoustic waves can be used to manipulate and mix droplets. Sensors integrated on the droplet-manipulating surfaces can measure droplet properties. The adhesion of liquids to the substrate surfaces necessitates the development of disposable devices. We therefore propose a three-layer structure of glass plate/water/128YX-LiNbO3. A disposable device can be realized by replacing the inexpensive glass. However, the details of the propagation mode in these structures have not yet been studied. In this paper, numerical analysis and experiments were carried out. For the numerical analysis, the Rayleigh–Lamb equations for the glass plate immersed in liquid were used. Comparisons between the numerical analysis and experimental results indicate that the symmetric mode of the Lamb wave propagates in the glass plate.

Mathematical modeling of belt and rigid conveyer’s drum contact with non-affect Coulomb friction

Conveyor’s belt with the drum contact task is paid a great researcher attention. It had solved in stresses with the mix boundary condition and Coulomb friction low action applies. Obtained solutions, as usual, do not satisfy to the displacement on drum surface boundary condition. The attempt to solve the system of Lamb equations in cylindrical coordinates in condition h/R<<1 values where h the belt thin and R drums radius is done. The Prandtl‘s technique and substitution methods are used to analytical solution obtain. As a result, the displacement expressions for rest and slide drum arcs are obtained. In search of rest arc length, radial and peripheral displacement equalities are used. Applied rest arc efforts are equal in values and have an opposite direction. The equality means the fact that traction efforts do not transfer under rest drum arc. The obtained results are coincided with the ones obtained by Zhukovskiy N.E. for flexible thread model. The fact to simultaneously satisfy to boundary conditions on inner and outer belt sides have to do the conclusion for boundary belt seam existing. Expressions for rest drum arc length without applied friction law are obtained. The displacement and corresponding stresses graphics are demonstrated.

Permanence and boundedness of solutions of quasi-polynomial systems

In this paper we consider analytical properties of a class of general nonlinear systems known as Quasi-Polynomial systems. We analyze sufficient conditions for permanence and boundedness of solutions, and illustrate our approach with an application to Lamb equations describing the evolution of electromagnetic modes in an optical maser.

Modeling of three-dimensional Lamb wave propagation excited by laser pulses

As a type of broadband source of ultrasonic guided waves, laser pulses can be used to launch all modes of interests. In this paper, Lamb waves are excited by imposing heat flux mimicking the supply of the heat from laser pulses, and effects by defects on the received Lamb waves in a plate are investigated by means of the finite element method. In order to alleviate the heavy computational cost in solving the coupled finite element equations, a sub-regioning scheme is employed, and it reduces the computational cost significantly. A comparison of Lamb waves generated by unfocused and line-focused laser sources is conducted. To validate numerical simulations, the group velocity of A0 mode is calculated based on the received signal by using the wavelet transform. The result of A0 mode group velocity is compared with the solution of Rayleigh–Lamb equations, and close agreement is observed. Lamb waves in a plate with defects of different lengths are examined next. The out-of-plane displacement in the plate with a defect is compared with the displacement in the plate without defects, and the wavelet transform is used to determine the arrival times of Lamb waves traveling at the A0 mode group velocity. A strong correlation is observed between the extent of defects and the magnitude of wavelet coefficients.

An analysis of the simulated acoustic emission sources with different propagation distances, types and depths for rail defect detection

Acoustic Emission (AE) technique is an effective nondestructive detecting method, and has a promising application for rail defect detection. So far, little attention has been paid to propagation distances, types, and depths of AE sources, which are important for rail defect detection accurately. This paper presents an experimental study on the simulated AE sources with different propagation distances, types and depths for rail defect detection. Three simulated AE sources with different frequencies are seeded on the cross section of rail, and the depths of AE sources are changed in the vertical direction. After receiving AE signals, wavelet transform and Rayleigh–Lamb equations are utilized to extract time–frequency features and modes. Based on the wavelet transform with corresponding group-velocity curves, the influences of different propagation distances, the features of different source types and the rules of different source depths are examined. It is concluded that the features of AE sources with different propagation distances, types and depths can be obtained by AE technique for rail defect detection. It is very useful to analyze and detect defects in rail defect detection.

An extension of the Hamilton-Jacobi method

Abstract A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.

Invariant manifolds of Hamilton's equations

The invariance conditions of smooth manifolds of Hamilton's equations are represented in the form of multidimensional Lamb's equations from the dynamics of an ideal fluid. In the stationary case these conditions do not depend on the method used to parameterize the invariant manifold. One consequence of Lamb's equations is an equation of a vortex, which is invariant to replacements of the time-dependent variables. A proof of the periodicity conditions of solutions of autonomous Hamilton's equations with n degrees of freedom and compact energy manifolds that admit of 2n – 3 additional first integrals is given as an application of the theory developed.

Mean-field dynamics of a quantum dot-microcavity system

Mean-field evolution equations for the exciton and photon populations and polarizations (Bloch–Lamb equations) are written and numerically solved in order to describe the dynamics of electronic states in a quantum dot coupled to the photon field of a microcavity. The equations account for phase space filling effects and Coulomb interactions among carriers, and include also (in a phenomenological way) incoherent pumping of the quantum dot, photon losses through the microcavity mirrors, and electron–hole population decay due to spontaneous emission of the dot. When the dot may support more than one electron–hole pair, asymptotic oscillatory states, with periods between 0.5 and 1.5 ps, are found almost for any values of the system parameters.

Singular Stress Behavior in a Bonded Hereditarily-Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Stress singularity is investigated in a plane problem for a bonded isotropic hereditarily elastic (viscoelastic) aging infinite wedge. The general solution of the operator Lamb equations, which are partial differential equations in space co-ordinates and integral equations in time, respectively, is represented in terms of one-parametric holomorphic functions (the Kolosov-Muskhelishvili complex potentials depending on time) in weighted Hardy-type classes. After application of the Mellin transform with respect to the radial variable, the problem is reduced to a system of linear Volterra integral equations in time. By using the residue theory for the inverse Mellin transform, the stress asymptotics and strain estimates near the singular point are presented here for non-hereditary Dundurs parameters. The general case of the hereditary Dundurs operators is considered in Part 11 (see [21]).

An extension of the Hamilton-Jacobi method

A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.

Nonlinear Schr�dinger equations admitted by auto-B�cklund transformations

A generalized formulation of Backlund transformations that carry a nonlinear Schrodinger equation into itself is considered, and a class of potentials that satisfy these transformations is found. A set of operators that simplify the analysis of the Lamb equations is introduced.

Self-phase locking in lasers with homogeneously broadened emission lines

It is shown by numerical analysis based on Lamb's equations of motion, that standing-wave lasers with purely homogeneously broadened emission lines exhibit regular multimode oscillations. Specifically, modes lying far from the line centre are quenched due to mode competition, and the amplitudes of the oscillating modes approach steady-state values. The stabilization of the amplitudes is normally accompanied, or followed, by an evolution of the phases towards a phase-locked regime, where the relative phasesψ n = 2φ n −φn + 1 −φn − 1 [φ n phase in thenth mode, defined by (6)] attain either the value 0 or ± π. The build-up times for the relative phases are found to vary over a wide range.

Regular and chaotic behaviour of multimode gas lasers

The build-up of multimode gas laser spectra is studied on the basis of the full system of Lamb's equations of motion. Comparison is made with the results of the phase-averaged equations that constitute the so-called free-running approximation. It turns out that the range of validity of the latter is approximately given byr⪅0.5 wherer is the ratio of the homogeneous linewidth and the mode spacing (provided the former is mainly due to the decay of both the upper and the lower level). However, forr⪆2, the free-running approximation which generally predicts a steady-state spectrum to occur, breaks down. Actually, in this case multimode oscillation of a chaotic character takes place, and no steady-state regime is attained.

Structure and stability of warm cores in neutron stars

Relativistic equations of structure are solved using Lamb's equations of state for warm neutron degenerate matter. The stability of isothermal cores in neutron stars is discussed and also the possible compatibility of the results obtained with experimental evidence is shown.

Injection modulation in coupled laser oscillators

Injection modulation is the effect observed when laser oscillation is perturbed by an injected signal below the threshold of locking. In this regime the oscillation becomes a wave modulated in frequency as well as in amplitude. The modulation envelope has a characteristic waveform which depends on the amplitude and phase of the injected signal. Starting from the Lamb's equations for a dual-mode oscillator, we develop a theory of the injection modulation and calculate the waveforms in amplitude and frequency. The treatment applies both to the external injection into a laser and to the case of mutual coupling between two modes. Experimental results for a dual mode He-Ne laser are found to be in good agreement with the theory. It is pointed out how the injection modulation effects are appreciable even at very weak levels of injected amplitudes, e.g., down to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> with respect to the oscillation field amplitude.

Interaction of longitudinal oscillation modes emitted by a gas laser with a rapidly decaying lower level

A modification of the Lamb equations, suitable for fairly high saturation levels, is proposed for lasers with a rapidly decaying lower level. The interaction between modes is divided arbitrarily into competition and interaction proper. The latter governs the phase relationships between the fields of different modes and can be identified because of the smallness of the terms responsible for this interaction. If only the competition is allowed for, the system of equations for the field amplitudes always has a unique steady-state stable solution. However, there are conditions when the interaction must be allowed for in spite of its smallness. In this case we can have conditions when the laser action is accompanied by a continuous spontaneous pumping of the energy between different modes.