Simplified Wave Equation Research Articles

Analysis of flexural wave cloaks

This work presents a comprehensive study of the cloak for bending waves theoretically proposed by Farhat et al. [see Phys. Rev. Lett. 103, 024301 (2009)] and later on experimentally realized by Stenger et al. [see Phys. Rev. Lett. 108, 014301 (2012)]. This study uses a semi-analytical approach, the multilayer scattering method, which is based in the Kirchoff-Love wave equation for flexural waves in thin plates. Our approach was unable to reproduce the predicted behavior of the theoretically proposed cloak. This disagreement is here explained in terms of the simplified wave equation employed in the cloak design, which employed unusual boundary conditions for the cloaking shell. However, our approach reproduces fairly well the measured displacement maps for the fabricated cloak, indicating the validity of our approach. Also, the cloak quality has been here analyzed using the so called averaged visibility and the scattering cross section. The results obtained from both analysis let us to conclude that there is room for further improvements of this type of flexural wave cloak by using better design procedures.

Accelerating the convergence of a tokamak modeling code with Aitken’s method

Up till now, the approach to solve the coupled equations arising in modeling wave heating in tokamaks modeling has been to sequentially solve one equation after the other until convergence is reached, a method known as Iterative Substructuring. In this paper we use an elementary model of the physics inside a tokamak, consisting of a simplified wave equation, a simplified Fokker–Planck equation and a diffusion equation. Our aim is to accelerate the solution of the coupled equations using Aitken’s δ2 method.Results show that a substantial reduction in CPU time can be obtained with this approach. It is hoped that results obtained with the simplified model serve as a proof-of-principle and carry over to more complicated systems of coupled equations used to model tokamaks.

On the stability of Suydam modes in a nonuniformly rotating plasma

A simplified wave equation is derived that describes both Suydam modes in a nonuniformly rotating plasma column in a helical magnetic field and related flute modes. A study is made of a low-pressure plasma under the assumption that the azimuthal component of the magnetic field is much weaker than the axial component. It is shown that, when the monotonic radial variation of the plasma rotation velocity is sufficiently sharp, the plasma core becomes stable against short-wavelength Suydam modes. The instabilities that can develop in a nonuniformly rotating plasma are classified.

Wave-induced hydraulic forces on submerged aquatic plants in shallow lakes.

Hydraulic pulling forces arising from wave action are likely to limit the presence of freshwater macrophytes in shallow lakes, particularly those with soft sediments. The aim of this study was to develop and test experimentally simple models, based on linear wave theory for deep water, to predict such forces on individual shoots. Models were derived theoretically from the action of the vertical component of the orbital velocity of the waves on shoot size. Alternative shoot-size descriptors (plan-form area or dry mass) and alternative distributions of the shoot material along its length (cylinder or inverted cone) were examined. Models were tested experimentally in a flume that generated sinusoidal waves which lasted 1 s and were up to 0.2 m high. Hydraulic pulling forces were measured on plastic replicas of Elodea sp. and on six species of real plants with varying morphology (Ceratophyllum demersum, Chara intermedia, Elodea canadensis, Myriophyllum spicatum, Potamogeton natans and Potamogeton obtusifolius). Measurements on the plastic replicas confirmed predicted relationships between force and wave phase, wave height and plant submergence depth. Predicted and measured forces were linearly related over all combinations of wave height and submergence depth. Measured forces on real plants were linearly related to theoretically derived predictors of the hydraulic forces (integrals of the products of the vertical orbital velocity raised to the power 1.5 and shoot size). The general applicability of the simplified wave equations used was confirmed. Overall, dry mass and plan-form area performed similarly well as shoot-size descriptors, as did the conical or cylindrical models of shoot distribution. The utility of the modelling approach in predicting hydraulic pulling forces from relatively simple plant and environmental measurements was validated over a wide range of forces, plant sizes and species.

A finite-difference method for the third-order simplified wave equation: assessment and application

A finite-difference method for coding the third-order simplified one-way wave equation is analyzed and assessed for application to two-dimensional waveguide structures. The general formulation for the simplified one-way wave equations based on the expansion of the eigenvalue equation is briefly discussed. The stability criteria of the finite-difference schemes are analyzed by applying the von Neumann method. Numerical dissipation is studied by calculating the power attenuation along the propagation direction. Finally, the EM wave propagation and scattering in the small and medium size open-ended parallel-plate waveguide cavities are calculated by using the method, and are compared with modal solutions. >

Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications

The finite-difference third-order simplified wave equation method is used to study the radiation losses of optical waveguides with step (abrupt) and parabolic tapers, including the radiation losses of the tapered ridge waveguides used as electro-optic phase modulators for tuning the frequencies of infrared lasers. The radiation losses of the parabolic taper given by Burns et al. is found to be under 5%. The results of a step taper are compared with Marcuse's; and found to be in good agreement. The numerical results are also compared with those from the coupled wave equations used by Sporloder and Unger. >

Investigation of microwave excited CO2 laser discharges

Microwave discharges for the excitation of CO2 lasers have been investigated. A waveguide structure with closed front and end, and with the discharge tube located parallel to the waveguide axis is used. The field distribution within the waveguide but outside the discharge tube is measured before and after ignition of the discharge. The wavelength of the microwave is reduced by the discharge plasma. The side-on light emission is detected. Light emission is detected over a distance within which the measured field strength in the waveguide varies fivefold. The experimental results are compared to the results of a self-consistent one-dimensional microwave discharge model. The model includes an equation for the electron density, the neutral gas density and a simplified wave equation for the electric field. The results show, in accordance with experimental results, a reduction of the wavelength due to the discharge plasma and the discharge excited over a range within which the electric field outside the plasma varies appreciably. In contrast to the electric field outside the plasma, the electric field inside is almost constant within the excited region. This is due to the high plasma conductivity which is a result of the high electron density which reaches values of up to 1012 cm-3. This leads to a power density up to several tens of W cm-3. In order not to overheat the gas the discharge has to be operated in a pulsed mode. Another method of limiting the gas temperature is to use a waveguide structure with a small electrode spacing or with a fast gas flow.

Diffraction in crystals at high energies

By means of several distinct stages of approximation, the way in which wave propagation in a lattice becomes classical at high energies is analysed. First, the principle that deflection angles (whether caused by 'quantum' or 'classical' processes) are small at high energies is used to derive a simplified wave equation involving the lattice potential averaged along the direction of the incident beam. Next, the many-beam solution of this equation for the case of systematic reflections is presented in a form which emphasises the spatial variation of the potential, rather than its Fourier components. Third, approximate analytical expressions for the Boch eigenvalues and eigenfunctions, and for the amplitudes of the diffracted beams, are derived by means of the WKB method; this leads to easily calculable expressions for the number of diffracted beams expected in a given situation, as well as for the number of Bloch waves contributing to these beams.

Perturbation Calculations ofF-Center Wave Functions with Point Ion and Pseudopotentials

The Hartree-Fock scheme and pseudopotential theory are used to derive a simplified wave equation for the calculation of $F$-center wave functions in a static lattice. The ions neighboring the vacancy are either represented by point charges or, for one to fourteen shells, by a localized pseudopotential of the form given by Austin and Heine. In both cases the potential is expanded in terms of Kubic harmonics about the center of the vacancy. The spherically symmetric part of the potential leads to a radial Schr\"odinger equation, which is solved numerically in integral equation form to give $1s$, $2p$, $3d$, $4f$, and $2s$ bound states. The first nonspherical terms in the potential is applied as a perturbation by means of a technique suggested by Dalgarno. Optical-absorption energies, oscillator strengths, and hyperfine interaction constants are computed for NaCl, KCl, and NaF and are compared with experiment and the previous theory of Gourary and Adrian. The results show that the charge distribution of the $F$ electron is concentrated too strongly at the nearest neighbor positive ions when the first nonspherical term is included in the point-ion potential. Since the pseudopotential for a positive ion has a partly repulsive core, the results obtained by including the pseudopotential are in better agreement with experiment.