Differential Wave Equation Research Articles

Threshold condition for traveling wave excitation in an annular prime-mover

The threshold condition for thermoacoustic instability in an annular prime-mover is derived. The region of spatially inhomogeneous temperature distribution (due to the presence of a stack and hot and cold heat exchangers) is assumed to be acoustically thin. The interaction of acoustic waves with the stack is assumed to be quasiadiabatic. Theoretical approach (based on the transformation of differential wave equation into Volterra integral equation, which is solved by iterative procedure) is valid for arbitrary spatial temperature distribution. Nevertheless, it was found that the threshold for the excitation of a traveling acoustic wave does not depend on the details of temperature spatial distribution but just on the ratio of the temperatures of hot and cold heat exchangers. Moreover, the critical temperature ratio does not depend on the thickness of the stack if the acoustic losses inside the heated region are negligible in comparison with acoustic losses in the rest part of the annular system. In this important limiting case critical temperature ratio diminishes when the dimensions of the system are scaled up. [Work supported by DGA.]

Drift Instability in a Non Uniform Electrical Field

The influence of a non-uniform electrical field, perpendicular magnetic on drift instability was studied by Sanuki et al.1,2 They have shown, that the drift instability is stabilized at a rather large gradient of an electrical field. This result was received by means of the analysis of an integral wave equation, which describes the plasma oscillations with Gaussian profile of density and linear profile of an electrical field at arbitrary Larmour radius of charged particles.We describe the drift oscillation by the differential wave equation. This equation can be used at any profiles of plasma density and electrical field, if Larmour radius of the charged particles is rather small. In case of linear profile of an electrical field, our results confirm those received in 1,2. We have also shown, that the drift instability is transformed to Kelvin-Helmholtz instability in case of an electrical field profile with an inflexion point (smooth step profile).

DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

An approximated 3-D model of cylinder-shaped piezoceramic elements for transducer design

In this paper an approximated 3-D model of cylinder shaped piezoceramics is described. In the hypothesis of axial symmetry, the element vibration in the extensional and radial directions is described by two coupled differential wave equations. The model is obtained choosing, as solution of these equations, two orthogonal wave functions, each depending only on one axis, corresponding to the propagation direction. The mechanical boundary conditions are applied imposing continuity between the stresses and the external forces on the surfaces of the element in an integral way, while, as far as the electrical boundary condition is concerned, two possibilities are explored: to neglect the piezoelectric constant in the transverse direction and to impose an integral condition also for the electric field. Comparisons with experimental results show this last approach to give better results. The model predicts with sufficient accuracy only the first radial and the first thickness modes of the cylinder-shaped piezoceramic element of arbitrary aspect ratio; but, for these modes, it is able to compute all the relations between the input applied voltage and the output forces and velocities on every external surface. Because only these two modes are of relevance in the practical applications of piezoceramic elements as ultrasonic transducers, the model can be used as a simple and useful tool in transducer design and optimization. Experimental validations of the model are also shown in the work.

Influence of the Kerr-medium type on the wave’s reflection states at a nonlinear interface for different angles of incidence

An analysis of a monochromatic plane wave’s reflection at a nonlinear interface is presented that constitutes a generalization and an extension of previous studies on this subject. We examine the influence of self-focusing and self-defocusing Kerr media on total and partial reflection states. We consider the whole range of angles of incidence and four types of media interfaces. In our model we do not apply the slowly varying amplitude approximation, and we give analytical real positive nonoscillating solutions to the nonlinear differential wave equations in total and partial reflection cases. For each case we also derive relations between the amplitude reflectance and input intensity, which indicate that in some situations the wave’s behavior at a nonlinear interface is governed by the nonlinear critical and the nonlinear characteristic angles. It is pointed out that the reflection states change bistably. We show, for different angles of incidence, the ranges in which the effective permittivity of nonlinear media can be changed via the incident intensity.

Near-field and far-field propagation of beams and pulses in dispersive media

We formulate an efficient exact method of propagating optical wave packets (and cw beams) in isotropic and nonisotropic dispersive media. The method does not make the slowly varying envelope approximation in time or space and treats dispersion and diffraction exactly to all orders, even in the near field. It can also be used to determine the partial differential wave equation for pulses (and beams) to any order as a power series in the partial derivatives with respect to time and space. The method can treat extremely focused pulses and beams, e.g., from near-field scanning optical microscopy sources whose transverse spatial extent in smaller than a wavelength.

A general approximated two-dimensional model for piezoelectric array elements

In this paper we describe an approximate two-dimensional model of a single piezoelectric array element. Substantially, the element is a two-dimensional structure whose vibrations can be described by two coupled differential wave equations with coupled boundary conditions. We take as a solution of this system two orthogonal wave functions which depend only on one axis, corresponding to the propagation direction, and which satisfy the boundary conditions only in an integral form. This solution makes it necessary to neglect the piezoelectric coupling in the transverse direction; nevertheless this approximation does not substantially affect the computed results because the transverse elastic coupling is much stronger than the piezoelectric one. With our model, the external behavior of the element in the frequency domain can be described by a 5/spl times/5 matrix from which all the transfer functions of the element can be easily computed. We compared our results with those of the classical Mason-Sittig one dimensional model, finding, as expected, a small difference in the principal thickness resonance frequency and the presence of a lateral mode. To verify the results obtained with our model, we computed the frequency spectrum of the element varying the width/thickness ratio. We found a good agreement with the low branch of the spectrum computed by the coupled mode theory. Finally, we computed, in the frequency domain, the dynamic electromechanical coupling coefficient k/sub U/, evaluating the internal energy distribution of the element and applying the definition reported by Berlincourt. We compared the values at resonance with the appropriate quasistatic coupling factors k/sub 33/' and k/sub 31/' and with the effective coupling factor k/sub eff/ computed by the same model. >

On the Origin of Parasitic Modes in Numerical Modelling of Wave Propagation in Plasmas

The mode mixing encountered in solving the boundary value problem arising in studies of wave propagation in hot one-dimensional plasmas is investigated. The wave differential equations, second order in ion Larmor radius, are shown to contain close to the ion cyclotron resonance an unwanted coupling mechanism, closely related to mode conversion driven by the equilibrium gradients. Consequently, the solution mixes with parasitic modes which carry power out of the resonance zone. In the case of strong coupling, this may lead to a generation of excess energy flux for complex valued coefficients of the wave equations. The importance and effects of this phenomenon are demonstrated in the special case of simulations of minority ion cyclotron heating in tokamaks. It is shown that the mode mixing can be avoided by adjusting the elements of the coefficient matrix of the wave equations, which makes the second-order system amenable to the problems with high minority concentration which is important for studies of fusion physics.

A numerical technique for the determination of the propagation characteristics of nonlinear planar optical waveguides

A numerical method is proposed and implemented into computer code for the analysis of propagation characteristics of nonlinear optical waveguides. This technique follows the concept of the shooting method, and the fourth-order Runge-Kutta method is adapted to integrate the differential wave equation from one side to the other. An error function is defined to estimate the discrepancy between the computed and expected boundary values. A secant method is then used to evaluate the sign and magnitude of the correction term for the initial guesses of effective index. This procedure is performed in an iterative manner until the error reduces to a specified range. Experiments show that very accurate results can be obtained through this method, and only six to seven iterations are needed for solutions to converge. With this method we analyze nonlinear waveguides with varying guide parameters for realization of their behavior. All the findings and results can be used in further investigations of optical devices composed of waveguide structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The wave automaton for the time-dependent Schrödinger, classical wave and Klein-Gordon equations

The “wave automaton” model, recently studied by Vanneste et al. [Europhys. Lett. 17 (1992) 715], is a discrete in space and time S-matrix formulation of time-dependent wave propagation, in which the evolution operator is given explicitly. Here, we show how it can be related to various partial differential wave equations and demonstrate that, depending upon the parametrization of the S-matrices, it is equivalent to a finite-difference discretized version of the time-dependent Schrödinger, hyperbolic wave and Klein-Gordon equations. Compared to finite-difference versions of partial differential equations, the wave automaton has the advantage of dealing directly with measurable physical quantities, such as the local fluxes.

An easily implementable fourth-order method for the time integration of wave problems

We are concerned with the time-integration of systems of ordinary differential equations arising from the space discretization of partial differential wave equations with smooth solutions. A method is suggested that, while being as easily implementable as the standard implicit mid-point rule, is fourth-order accurate. The new method is symplectic so that it is very well suited for long-time integrations of problems with a Hamiltonian structure. Numerical experiments are reported that refer to a fourth-order Galerkin space discretization of the Korteweg-de Vries equation and to a pseudospectral space discretization of the same equation.

Photonic band gaps

Many major discoveries in physics this century originate from the study of waves in periodic structures. Examples include X-ray and electron diffraction by crystals, electronic band structure and holography. Analogies between disciplines have also led to fruitful new avenues of research. An exciting example is the recent discovery of three-dimensionally periodic dielectric structures that exhibit what is called a “photonic band gap” (PBG), by analogy with electronic band gaps in semiconductor crystals. Photons in the frequency range of the PBG are completely excluded so that atoms within such materials are unable to spontaneously absorb and re-emit light in this region; this has obvious beneficial implications for producing highly efficient lasers. Given that electrons and photons obey almost the same differential wave equation, the idea of a PBG is clearly far from crazy. It does, however, demand a radical rethink of how light behaves in periodic structures.

Waveguide coupling of ion Bernstein waves to tokamak plasmas

The authors propose a theoretical model of coupling of ion Bernstein (IB) waves to tokamak plasma using phased waveguides. To describe the propagation of these waves near higher harmonics of the ion cyclotron frequency, a set of differential wave equations is used, constructed by analogy to the familiar finite Larmor radius wave equations, but in such a way that its three independent WKB solutions correspond to the two cold plasma waves and the Bernstein wave to all orders in the ion Larmor radius. This system has a meaningful energy theorem and unambiguous boundary conditions at the plasma edge. It is integrated numerically to obtain the surface admittance matrix of the plasma. The efficiency of waveguide launching is evaluated by extending the theory developed for lower hybrid wave grills to take into account the finite height of the waveguides as well as the coupling between TE and TM modes at the waveguide mouths due to the fact that at these frequencies the admittance matrix is not diagonal

Vibration of surface foundations of arbitrary shapes

AbstractPresented is a systematic procedure for generating impedance (or compliance) matrices for foundations with arbitrary shapes, resting on an elastic half‐space medium. A technique to decompose prescribed harmonic tractions on the half‐space medium is employed to solve analytically the differential wave equations in cylindrical coordinates. However, the interaction stresses due to the vibration of a foundation with arbitrary shape are described in rectangular coordinates, and assumed to be piecewise constant in the region of the arbitrary shape. A coordinate transformation matrix is introduced for the piecewise constant tractions in order to use the solution of the differential wave equations in cylindrical coordinates. Finite element modelling is assumed in rectangular coordinates for the foundation itself. The impedance matrix is then obtained for the finite element model, using a variational principle and the reciprocal theorem. A simple example of a rigid square plate resting on a half‐space medium and subjected to vertical excitation is used to demonstrate the efficiency and effectiveness of the procedure. Some numerical aspects are investigated and some possible extensions of the procedure are also discussed.

Transverse electric field scattering by a Kerr media deposited on a conducting planar surface

The scattering of a transverse electric plane wave incident from a linear media by a nonlinear Kerr film deposited on a good conductor is studied. Self-focusing and self-defocusing are included both above and below the critical angle. The solution to the complex second-order nonlinear differential wave equation is constrained at the nonlinear/conductor planar interface by mixed boundary conditions and at the linear/nonlinear interface by the continuity of the tangential electric and magnetic fields. A potential-well approach is employed to classify and isolate the various solutions. Several examples illustrate the two nonlinear analogies of the evanescent and harmonic field solutions. Additionally, we obtain the surface-wave solutions and the transitional, hybrid, fields. Finally, we show that the reflectivity is multistable with respect to the incident intensity and that the reflected intensity is multistable as the incident angle changes for a constant input intensity.

Local realizations of kinematical groups with a constant electromagnetic field. I. The relativistic case

This paper is devoted to the study of the description of elementary physical systems interacting with an external constant electromagnetic field and the construction of their differential wave equations from a group-theoretical point of view. In this context certain local realizations of the Poincaré group are studied. The linearization of this problem is carried out by building the associated representation group that turns out to be the well-known Maxwell group. In this way the usual method (concerning local realizations) that has been employed in studying free systems to the interacting case is extended.

Construction of periodic solutions of second-order differential and integrodifferential wave equations

Construction of periodic solutions of second-order differential and integrodifferential wave equations

Microwave-induced auditory effect in a dielectric sphere

The acoustic pressure wave generation inside an electromagnetically lossy dielectric sphere from an incident microwave pulse is analyzed rigorously. The pressure wave equation derived using the first-order approximation of a thorough formulation on microwave-induced thermoacoustic effect in dielectrics, is used. The inhomogeneous hyperbolic-type pressure wave differential equation is solved by a Green's function theory approach. The boundary conditions on the dielectric sphere-air interface are taken into account. The power is computed by applying the exact Mie theory solution for the dielectric sphere. Two types of acoustic waves are derived inside the sphere: (a) a transient burst type pressure wave, corresponding to the free-space contribution of Green's function; and (b) an infinite set of damped oscillations related to the normal acoustic modes of the spherical resonator. Numerical results are computed and presented for several cases. >

Phase space methods and path integration: The analysis and computation of scalar wave equations

The scalar Helmholtz equation plays a significant role in studies of electromagnetic, seismic, and acoustic direct wave propagation. Phase space, or ‘microscopic’, methods and path (functional) integral representations provide the appropriate framework to extend homogeneous Fourier methods to inhomogeneous environments. The two complementary approaches to this analysis and computation of the n-dimensional Helmholtz propagator are reviewed. For the factorization/(one-way) path integration/invariant imbedding approach, the exact solution of the Helmholtz composition equation for the Weyl square root operator symbol is presented in the quadratic case. The filtered, one-way, phase space marching algorithm is examined in detail and compared numerically with wide-angle, one-way, partial differential wave equations formally derived from approximation theory. For the second approach, which directly constructs approximate two-way path functionals, the feasibility of a Monte Carlo (statistical) evaluation of the Feynman/Garrod propagator is discussed.

Application of the Boundary Element Method to Acoustic Cavity Response and Muffler Analysis

This paper is concerned with the application of the Boundary Element Method (BEM) to interior acoustics problems governed by the reduced wave (Helmholtz) differential equation. The development of an integral equation valid at the boundary of the interior region follows a similar formulation for exterior problems, except for interior problems the Sommerfeld radiation condition is not invoked. The boundary integral equation for interior problems does not suffer from the nonuniqueness difficulty associated with the boundary integral equation formulation for exterior problems. The boundary integral equation, once obtained, is solved for a specific geometry using quadratic isoparametric surface elements. A simplification for axisymmetric cavities and boundary conditions permits the solution to be obtained using line elements on the generator of the cavity. The present formulation includes the case where a node may be placed at a position on the boundary where there is not a unique tangent plane (e.g., at an edge or a corner point). The BEM capability is demonstrated for two types of classical interior axisymmetric problems: the acoustic response of a cavity and the transmission loss of a muffler. For the cavity response comparison data are provided by an analytical solution. For the muffler problem the BEM solution is compared to data obtained by a finite element method analysis.