A time domain approach to SH waves on a piezoelectric layer

The propagation of transient shear horizontal waves in a piezoelectric layer with free boundaries is studied within a time domain approach. The layer is modeled as a dissipative electroelastic continuum via linear constitutive equations accounting for memory effects. A separation of space variables is employed to solve the problem for the Laplace transforms of the fields. The pertinent dispersion equations are derived in different cases where the boundary surfaces of the layer are matched with an external potential or are grounded. It is shown that transient wave solutions exist which are compatible with given time-dependent data at the surfaces. The wave amplitude decays along the layer according to the dissipative model and the potential field outside the layer, in the matched case, turns out to vanish as the reciprocal of the distance from the boundaries. Illustrative examples are given for a square pulse applied to the layer's surfaces.     

Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schrödinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed.     

Coupled electromagnetic wave propagation in a cylindrical dielectric waveguide filled with Kerr medium: nonlinear effects

Propagation of the coupled electromagnetic wave, which is a superposition of TE and TM waves, in a dielectric circular cylindrical waveguide filled with non-linear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Non-linear coupled TE–TM wave is characterized by two (independent) frequencies and two (coupled) propagation constants (PCs). The physical problem is reduced to a non-linear two-parameter transmission eigenvalue problem for Maxwell’s equations. The system of dispersion equations with respect to PCs is derived and solved numerically. Two types of coupled PCs and coupled guided modes are found: non-linear solutions of the first type become solutions of the corresponding linear problems as the nonlinearity coefficient tends to zero; solutions of the second type seem to be ’purely’ non-linear as they stay away from any linear solutions as coefficient of the nonlinearity tends to zero. Coupled PCs and coupled eigenmodes are calculated and plotted.     

The comparisons of thermo-elastic waves for Lord-Shulman mode and Green-Lindsay mode based on Guo��s eigen theory

The motion equations of anisotropic media, coupled to the heat conduction equations, are studied here based on the L-S model and the G-L model. The complete set of uncoupled elastic and heat wave equations for anisotropic media are deduced. The results show that the L-S model is suitable for elastic materials and the G-L model is more suitable for dissipative materials. Based on these laws, we discuss the propagation behaviors of heat wave and elastic waves for isotropic media.     

Physical variational principle in dissipative media

The physical variational principle (PVP) is a physical principle which is implied in the thermodynamiclaws. For a conservative system, the PVP is implied in the first thermodynamic law and gives the motionequation. But for a dissipative system, PVP is implied in the extended Gibbs equation, which is the result ofthe first and second thermodynamic laws. The precision of the PVP in a dissipative system is in the same orderof the Gibbs equation. The dissipative work and its converted internal irreversible heat are simultaneouslyincluded in the PVP to get the governing equation and the boundary condition of the dissipated variables.The “generalized motion equations” including governing equations of the mechanical momentum and thermoelastic,thermal viscoelastic, thermal elastoplastic, linear thermoelastic diffusive and linear electromagneticthermoelastic materials etc. can be derived by the PVP of dissipative media in this paper. The conservativesystem is the special case of the dissipative system. Other than the mathematical variational principle, whichis obtained by a known governing equation, the PVP is used to deduce the governing equation. The PVPsincluding the hyperbolic temperature wave equation with a finite phase speed are also discussed shortly.     

非线性梁压电分阶最优减振控制

提出非线性的分阶最优控制方法,并将其应用于梁的非线性振动压电减振控制。建立梁压电减振系统动力学模型,导出减振系统的非线性动力学运动微分方程,利用摄动法,实现非线性微分方程的线性化。将各阶线性方程解耦,化为状态空间方程。设计非线性分阶控制器,对减振系统进行分阶最优控制。仿真算例验证这种控制方法的有效性。     

Constitutive model and wave propagation in dissipative electroelastic crystals

We propose a constitutive model for dissipative ionic crystals within the classical continuum theory of thermoelectroelasticity which includes polarization gradients as independent variables. According to a previous work, dissipation is modeled by suitable evolution equations for a set of internal variables. The compatibility with the second law of thermodynamics is required to obtain the pertinent constitutive equations. The problem is then linearized about an unstrained and unpolarized state, and the nature of dissipative contributions is pointed out. We apply this model to the study of wave propagation by deriving the eigenvector equation for a two-dimensional problem in the quasi-static hypothesis. The dispersion relation of bulk waves is obtained and discussed for crystals with a centrosymmetric structure.     

Scattering of a surface wave by a submerged sphere

Summary The problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential. These equations are easily solved numerically, so long as the sphere is not too close to the surface. The approach has been to formulate the problem as an integral equation, expand the Green's function, the velocity potential of the incident wave, and the total velocity potential in spherical harmonics, impose the boundary condition at the surface of the sphere, and carry out the integrations. The scattering cross section has been evaluated numerically and is shown to peak for values of the product of radius and wave number somewhat less than unity. Also, the Born approximation to the cross section is obtained in closed form.Supported by the Department of the Navy, Naval Sea Systems Command under Contract No. N00017-72-C-4401.     

Plane harmonic waves in an infinite piezoelectric plate with dissipation

In a previous paper, the three-dimensional equations of linear piezoelectricity with quasielectrostatic approximation were extended to include losses attributed to the mechanical damping in solid and the resistance in current conduction. These equations were used to investigate the plane wave propagation in an unbounded solid and forced thickness vibration of an infinite piezoelectric plate. In the present paper, these equations are used to obtain solutions of plane harmonic wave of arbitrary direction in an infinite and dissipative piezoelectric plate with general crystal symmetry. Dispersion curves are computed and plotted for real frequencies and complex wave numbers. All frequency branches are complex for dissipative plate. There are no longer any pure real or pure imaginary or complex conjugate frequency branches as those existing for nondissipative plates. Effects of dissipation on the wave propagation are examined in detail for AT-cut of quartz as well as barium titanate ceramic plate.     

Non-linear analysis of wave propagation using transform methods

Non-linear wave propagation/transient dynamics in lattice structures is modeled using a technique which combines the Laplace transform and the Finite element method. The first step in the technique is to apply the Laplace transform to the governing differential equations and boundary conditions of the structural model. The non-linear terms present in these equations are represented in the transform domain by making use of the complex convolution theorem. Then, a weak formulation of the transformed equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solutions of the linear parts of the transformed governing differential equations. Numerical results are presented for a viscoelastic rod and von Karman type beam.     

Propagation and amplification of gap waves between a piezoelectric half-space and a semiconductor film

Summary. We study wave propagation in a piezoelectric ceramic half-space with a thin semiconductor film and an air gap between the film and the half-space. Two-dimensional equations for a thin film are used to model the semiconductor film and the air gap. The half-space is governed by the three-dimensional equations of linear piezoelectricity. It is shown that an anti-plane wave can propagate in such a system. An equation that determines the dispersion relation of the wave is obtained. Solutions to the equation show that the wave has both dispersion and attenuation, and can be amplified by a biasing dc electric field.     

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.     

Propagation of electroacoustic axial shear waves in a piezoelectric medium reinforced by continuous fibers

The propagation of electroacoustic axial shear waves in a fiber reinforced piezocomposites is studied in which matrix and fibers consist of piezoelectric transversely isotropic materials with symmetry axes parallel to the fiber axes. The effective medium method self-consistent variant as developed by Sabina and Willis is used to obtain explicit equations for the complex wave vector and it is solved numerically. Its real part determines the effective wave velocity and the imaginary part the attenuation factor. Integral equations expressed via dynamic Green’s function kernels are set up. The central problem of the method is the axial shear electroacoustic wave scattering on one isolated fiber in the medium having the effective piezoelectric properties. It is solved approximately by the Galerkin type method. The obtained expressions for the effective wave velocity and attenuation factor cover not only the long-wave region but the intermediate wave and it is valid for long wavelenghts up to the diameter of the inclusion. Wave velocity and attenuation coefficient coincide with ones obtained earlier in some other way. Some numerical examples are presented for real materials.     

Water wave scattering by a pair of thin vertical barriers with submerged gaps

The scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory. Using Havelock’s expansion of water wave potential, the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall. These two pairs of integral equations can be solved approximately by employing a Galerkin single-term approximation technique to obtain numerical estimates for the reflection and transmission coefficients. These estimates for the reflection and transmission coefficients thus obtained are seen to satisfy the energy identity. The reflection coefficient is plotted against wave number in a number of figures for different values of various parameters involved in the problem. It is observed that the reflection coefficient vanishes at discrete frequencies when the vertical barriers are identical. For nonidentical vertical barriers the reflection coefficient never vanishes, though at some wave number it becomes close to zero. The results for a single barrier and fully submerged two barriers, and for a single barrier with a narrow gap, are also recovered as special cases.     

A numerical method to obtain optimal quadrature formulas

This paper present a numerical method to obtain optimal quadrature formulas of Gauss type and Radau type in the sense of Sard. Using the relation between optimal quadrature formulas and nonospline functions, the optimal quadrature formula can be obtained by solving a set of no-linear simultaneous algebraic equations induced from the interpolatory conditions of the monospline. In attempting to solve this set of non-linear algebraic equations for numbers of knots and degrees of interpolution required in estimation problem applications insurmountable numerical errors were encountered. This paper solves the numerical problem by first reducing the number of unknowns and equations to approximately one half the original number. This is accomplished by showing and then using a symmetry property of the monospline. Second an iteration scheme which partitions the reduced order set of non-linear algebraic equations into a linear subsystem and a non-linear subsystem is developed to numerically solve the equations. This iteration algorithm provides the advantages of reducing the computational complexity, dynamically checking the convergence and explicitly evluating the resulting accuracy.     

Applications of a variational method for mixed differential equations

Summary Variational principles for elliptic boundary-value problems as well as linear initial-value problems have been derived by various investigators. For initial-value problems Tonti and Reddy have used a convolution type of bilinear form of the functional for the time-like coordinate. This introduces a certain amount of directionality thereby reflecting the initial-value nature of the problem. In the present investigation the methods of Tonti and Reddy are used to derive the appropriate variational formulation for the transonic flow problem. A number of linear and non-linear examples have been investigated. As a test for the existence of directionality, finite-differences are used to discretize the variational integral. For initial-value problems of wave equation and diffusion equation type, fully implicit finite-difference approximations are recovered. The small-disturbance transonic equation leads to the Murman and Cole differencing theory; when applied to the full potential-flow equations, the rotated difference scheme due to Jameson is obtained.An extended version of this paper was first presented at the Bat-Sheva International Seminar on Finite Elements for Non-Elliptic Problems, Tel-Aviv, Israel, July 1977.     

An efficient semi-analytic time integration method with application to non-linear rotordynamic systems

This paper presents a simple and efficient time-integration method for non-symmetric and non-linear equations of motion occurring in the analysis of rotating machines. The algorithm is based on a semi-analytic formulation combining powerful methods of linear structural dynamics applied to non-linear dynamic problems. To that purpose, the total solution is separated into a linear and a non-linear part, and a further partitioning into quasi-static and dynamic parts is performed. Modal analysis is applied to the undamped equations of the dynamic parts. The quasi-static parts contain all degrees of freedom, while a cost-saving modal reduction may be easily performed for the dynamic parts. Duhamel's integral is utilized for the modal equations. The time-evolution of the unknown modal excitations due to the dissipative, non-conservative, gyroscopic and non-linear effects entering Duhamel's integral is approximated during each time-step. The resulting time-stepping procedure is performed in an implicit manner, and the method is examined in some detail, in view of stability and accuracy characteristics. A rotordynamic system serves as a benchmark problem in order to demonstrate the computational advantages of the present method with respect to various other time-integration algorithms.     

Effects of viscosity on long waves

Summary A study of effects of viscosity on non-linear long waves is made. Beginning with the Navier-Stokes equations of motion, the long wave approximation is achieved by an expansion scheme similar to Friedrichs'. Non-linear solutions are obtained by applying the theory of relatively undistorted waves. It is found that shock formation is delayed by the viscous effect. Various conditions are obtained in determining the viscous, non-linear and radial decay effects on the solution for a shockless expansion wave-front propagating over large distances.     

New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics

The properties of Tzitzéica equations in non-linear optics have been the subject of many recent studies. In this article, a new and effective modification of Kudryashov method is adopted to study this class of non-linear evolution equations. As an achievement, new exact traveling wave solutions of Tzitzéica, Dodd–Bullough–Mikhailov (DBM) and Tzitzéica–Dodd–Bullough (TDB) equations are formally extracted. It is believed that the modified Kudryashov method along with the symbolic computation package suggests a promising technique to handle non-linear evolution equations in non-linear optics.