plane waves

The control of acoustic-frequency gaps by altering the geometry of the system is analyzed in the particular case of a set of parallel solid square-section columns distributed in air on a square lattice. This system is shown to be sensitive enough to the rotation of the columns to be considered for practical sonic band-gap-width engineering. For different geometric configurations, specific interpretation models are used, taking into account the important mismatch of the impedance between the compounds. The accuracy of the plane-wave calculation is discussed in the different cases.

The main research goal of the presented research has been to understand the changes in surface composition and chemistry at the nanoscopic temporal and spatial scales for long pulse Plasma Facing Components (PFCs) and link these to the overall machine performance of the National Spherical Torus Experiment Upgrade (NSTX-U). A study is presented of the lithium surface science, with atomic spatial and temporal resolutions. The dynamic surface responds and evolves in a mixed material environments (D, Li, C, B, O, Mo, W) with impingement of plasma particles in the energy range below 100 eV. The results, obtained by quantum-classical molecular dynamics, include microstructure changes, erosion, surface chemistry, deuterium implantation and permeation. Main objectives of the research are i) a comparison of Li and B deposition on carbon, ii) the role of oxygen and other impurities e.g. boron, carbon in the lithium performance, and iii) how this performance will change when lithium is applied to a high-Z refractory metal substrate (Mo, W). In addition to predicting and understanding the phenomenology of the processes, we will show plasma induced erosion of PFCs, including chemical and physical sputtering yields at various temperatures (300-700K) as well as deuterium uptake/recycling.

Ballistic phonon experiments, in the fractional quantum Hall regime, are described in which the interactions between the longitudinal (LA) and transverse acoustic (TA) modes and the two-dimensional electron system (2DES) can be resolved. The range of in-plane wave vectors of LA phonons that can interact with the 2DES is restricted and the measured fractional quantum Hall energy gap is found to depend on the in-plane wave vector. Whilst the energies found at Landau level filling factor n ¼ 2 5 are in good agreement with theory, those at n ¼ 2 3 and 1 3 are lower than theoretical predictions. Possible reasons for this discrepancy are discussed.

Holography can store full wide angle information about a registered object, since during registration process information about amplitude and phase of an optical wave scattered from an object is captured. Because of this unique feature people put hope in holography as the method which can be utilized in a 3D imaging display. In the paper we present the design of a wide viewing angle display system utilizing multiple Spatial Light Modulators (SLMs). The system is capable of displaying objects from both virtual and real worlds. In our system we are using phase only reflective SLMs based on liquid crystal on silicon (LCoS). There are designed to work with normal illumination. However in order to simplify an optomechanical system of the display here the SLMs are used with an inclined plane wave illumination. Therefore in the paper at first we focus on determination of a tilt depended SLM calibration, so thus SLM even with highly off axis inclined illumination is capable of an accurate wave reproduction. Then we focus on obtaining high quality reconstruction of objects from virtual world. We present an algorithm based on Gerchberg-Saxton scheme and diffraction computing between tilted and parallel planes. All of the paper discussions are accompanied with experimental results obtained in the multi SLMs display.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic (ae 2iωt ). The goal of this Note is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem. We show that the solution of the IBV problem has different asymptotic behaviors in different regions. In the region x > 4bt (b = (a 2ω)/2 > 0) the solution has the form of a Zakharov-Manakov vanishing asymptotics. In the region 4bt -1 2a N log t < x < 4bt, where N is an integer, the solution behaves as a finite train of asymptotic solitons. In the region 4(ba √ 2)t < x < 4bt the solution is a modulated elliptic wave. Finally, in the sector 0 < x < 4(ba √ 2)t the solution is a plane wave.

Semi-local functionals commonly used in density functional theory (DFT) studies of solids usually fail to reproduce localized states such as trapped holes, polarons, excitons, and solitons. This failure is ascribed to self-interaction which creates a Coulomb barrier to localization. Pragmatic approaches in which the exchange correlation functionals are augmented with small amount of exact exchange (hybrid-DFT, e.g. B3LYP and PBE0) have shown promise in rectifying this type of failure, as well as producing more accurate band gaps and reaction barriers. The evaluation of exact exchange is challenging for large, solid state systems with periodic boundary conditions, especially when plane-wave basis sets are used. We have developed parallel algorithms for implementing exact exchange into pseudopotential plane-wave DFT program and we have implemented them in the NWChem program package. The technique developed can readily be employed in Γ-point plane-wave DFT programs. Furthermore, atomic forces and stresses are straightforward to implement, making it applicable to both confined and extended systems, as well as to Car-Parrinello ab initio molecular dynamic simulations. This method has been applied to several systems for which conventional DFT methods do not work well, including calculations for band gaps in oxides and the electronic structure of a charge trapped state in the Fe(II) containing mica, annite.

The transmission line modeling method is used to get the time domain response of a dispersive cylindrical cloak to an electromagnetic (EM) plane wave that is slightly nonmonochromatic. Our objective is to numerically study two important phenomena derived from the dispersive nature of the invisibility shell: frequency shifts and time delays. On one hand, the frequency domain representation of the cloak's response shows that the frequency center is shifted once the EM wave has crossed the cloak; the shift intensity representation spans the entire rainbow spectrum depending on the observation angle. On the other hand, such a full-wave simulation constitutes tangible evidence of the existence of time delays when the EM wave passes through the device. We show that this phenomenon depends on the employed coordinate transformation.

The lattice parameter, bulk modulus, and cohesive energy of lithium hydride are calculated to very high accuracy through a combination of periodic and finite-cluster electronic structure calculations. The Hartree-Fock contributions are taken from earlier work in which plane-wave calculations were corrected for pseudopotential errors. Molecular electronic structure calculations on finite clusters are then used to compute the correlation contributions and finite-size effects are removed through the hierarchical scheme. The systematic improvability of the molecular electronic structure methods makes it possible to converge the static cohesive energy to within a few tenths of a millihartree. Zero-point energy contributions are determined from density functional theory phonon frequencies. All calculated properties of lithium hydride and deuteride agree with empirical observations to within experimental uncertainty.

The left-handed properties of metamaterials with saturable nonlinearity are analyzed with respect to their electromagnetic response as a function of externally varying parameters. We demonstrate that the response of the medium is strongly affected by the saturation of the nonlinear effects. The last can be exploited to modulate the amplitude or tune the frequency of the response. Moreover, the existence of bistability regions in large parts of the external parameter space allows for switching between different magnetization states, with either positive or negative response. The stability issue of multiple possible states is addressed through modulational instability analysis of plane wave envelopes in each of those states.

One of the most efficient approaches in computational cluster physics uses a plane-wave basis set and pseudopotentials to describe electron-ion interactions. This method -where the clusters are placed inside supercells -is restricted in its usual form to neutral systems because of the long-range interaction between a charged cluster and its periodic images. To eliminate this restriction, we propose to shield each charged cluster with a spherical shell having a symmetric charge that neutralizes the supercell. Furthermore, the shell is placed in such a way that it cancels the electric dipole of the charged cluster. We present relaxed geometries and cohesive energies of Na + N , N = 2 -9 and 21, obtained with Langevin quantum molecular dynamics. Our local density approximation structures are very similar to those found in other first principles calculations. Vertical and adiabatic ionization energies of Na N , N = 2, 3, 6, and 8 are displayed. We also show results for Na 2+ 8 , Na - 5 and Na - 7 .

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We study the quantum relativistic Coulomb-excitation process including recoil eAects in the plane- wave Born approximation. It is shown that there exists an interplay between quantum and relativistic recoil efI'ects in certain kinematical regions which cannot be accounted for in the conventional theories. This specific feature is shown to modify the angular distribution of Coulomb-induced fission fragmenta- tion in an essential manner. In contrast with usual treatments we find that our results compare favorably with recent data.

A uniform geometrical theory of diffraction (UTD) solution is developed for the problem of the diffraction by a thin dielectric/ferrite half plane when it is excited by a plane, cylindrical, or surface wave field. Both transverse electric and transverse magnetic cases are considered. The solution of this problem is synthesized from the solutions to the related problems of EM diffraction by configurations involving perfectly conducting electric and magnetic walls covered by a dielectric/ferrite half-plane of one half the thickness of the original half-plane.

Both high-frequency acoustic modes and the boson peak related to SiO 4 librations are observed in a single inelastic x-ray scattering experiment. The experimental data are consistent with a picture where the acoustic modes experience a crossover at a frequency ⍀ co beyond which plane waves cease to exist. The spectra evolve with the scattering vector to merge into a broad boson peak at ⍀ BP Ϸ⍀ co . The latter, although essentially optic in nature, might hybridize with the resonant acousticlike modes, which can be crucial to their strong scattering.

The new nonlinear crystal for the mid-IR CdSiP 2 was discovered only very recently but the interest in this chalcopyrite is enormous because it possesses most of the attractive properties of the related ZnGeP 2 but allows in addition pumping at 1064 nm without two-photon absorption and uncritical phase-matching for 6 µm generation with maximized effective nonlinearity. The last feature is due to the fact that this crystal is negative uniaxial in contrast to ZnGeP 2 which shows positive birefringence. We now measured its nonlinear coefficient using SHG of femtosecond pulses generated near 4.6 µm from a seeded KNbO 3 optical parametric amplifier. The SHG efficiency was compared for uncoated samples of CdSiP 2 and ZnGeP 2 , both ≈0.5 mm thick, in the low conversion limit (<10% internal conversion efficiency) which justifies the use of the plane wave approximation. Taking into account the experimentally determined phase-matching angles for type-I SHG (oo-e type in CdSiP 2 and ee-o type in ZnGeP 2 ), which were in good agreement with the existing Sellmeier approximations, we arrived at d 36 (CdSiP 2 )~d 36 (ZnGeP 2 ) which is rather unexpected having in mind the larger band-gap of CdSiP 2 . The reliability of the measurement was tested at the same wavelength by comparing ZnGeP 2 with HgGa 2 S 4 which led to the result d 36 (ZnGeP 2 )~3d 36 (HgGa 2 S 4 ), in very good agreement with previous estimations.

We use complex semiclassical method to compute scattering amplitudes of a point particle in dilaton gravity with a boundary. This model has nonzero minimal black hole mass M_cr. We find that at energies below M_cr the particle trivially scatters off the boundary with unit probability. At higher energies the scattering amplitude is exponentially suppressed. The corresponding semiclassical solution is interpreted as formation of an intermediate black hole decaying into the final-state particle. Relating the suppression of the scattering probability to the number of the intermediate black hole states, we find an expression for the black hole entropy consistent with thermodynamics. In addition, we fix the constant part of the entropy which is left free by the thermodynamic arguments. We rederive this result by modifying the standard Euclidean entropy calculation.

Numerical examination of the solution of the boundary-value problem of the reflection and transmission of a plane wave due to a slab of an electro-optic structurally chiral material (SCM) indicates that the exhibition of the circular Bragg phenomenon by the SCM can be controlled not only by the sign and the magnitude of a dc electric field but also by its orientation in relation to axis of helicoidal nonhomogeneity of the SCM. Thereby, the possibility of electrical control of circular-polarization filters has been extended.

To see the similarities and differences with eletromagnetics, basic concepts and equations of elastodynamics are formulated in coordinate-free form applying concepts from Polyadic Algebra. Planewave propagation is studied for time-harmonic equations in isotropic and simple uniaxially anisotropic media and the Green dyadic is derived for the isotropic medium. As an extension, the Green dyadic for the anisotropic elastic medium is derived in perturbational approximation. In the Appendix A, some basic properties for tetradics, useful for practical analysis both in elastodynamics as well as in electromagnetics, are derived.

The problem of finding unique fields in perfect electromagnetic conductor (PEMC) is considered by analyzing transmission of an obliquely incident plane wave to PEMC half space and slab. It is shown that uniqueness depends on the exact definition of the PEMC medium conditions. Three possible medium conditions are tested, out of which only one, defining PEMC as the limiting case of a Tellegen medium, leads to uniqueness for fields inside the PEMC. Reflected fields are unique and the fields transmitted through the PEMC slab vanish for all cases.

A method for treatment of the transverse or longitudinal shifts of multiply internally reflected light beams is developed in terms of eigenfunctions for the reflection processes. Localized wave packets composed of plane-wave eigenfunctions are used to discuss the detailed structure of the shifted intensity distributions for several experimental conditions. %'ith this approach image widths and over-all shifts, as well as splittings between images, can be calculated. The predictions are compared with the experimental results of Imbert and are found to agree with the observed transverse splitting between left and right circularly polarized light beams. Since the theory presented here disagrees with that proposed by Imbert in several respects, experiments are suggested which would distinguish between alternative theories.

Perfect electromagnetic conductor (PEMC) has been introduced as a generalization of both the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC). In the present paper, the basic problem of reflection and transmission of an obliquely incident plane wave at the interface of a PEMC half space or slab is considered. It is found that the field outside the PEMC medium can be uniquely determined but the interior field requires some assumptions to be unique. In particular, definition of the PEMC-medium condition as a limit of a Tellegen-medium condition and extraction of certain virtual fields (metafields) are shown to make the field inside the PEMC medium unique.

It has been known through some examples that parameters of an electromagnetic medium can be so defined that there is no dispersion equation (Fresnel equation) to restrict the choice of the wave vector of a plane wave in such a medium, i.e., that the dispersion equation is satisfied identically for any wave vector. In the present paper, a more systematic study to define classes of media with no dispersion equation is attempted. In addition to the previously known examples, a novel class of Case 1 media with no dispersion equation is seen to emerge through the analysis making use of coordinate-free four-dimensional formalism in terms of multivectors, multiforms and dyadics.

In differential-form representation, the Maxwell equations are represented by simple differential relations between the electromagnetic two-forms and source three-forms while the electromagnetic medium is defined through a constitutive relation between the twoforms. The simplest of such relations expresses the electromagnetic two-forms as scalar multiples of one another. Because of its strange properties, the corresponding medium has been considered as nonphysical. In this study such a medium is interpreted in terms of the classical Gibbsian vectors as a bi-isotropic medium with infinite values for its four medium parameters. It is shown that the medium is a generalization of both PEC (perfect electric conductor) and PMC (perfect magnetic conductor) media, with similar properties. This is why the medium is labeled as PEMC (perfect electromagnetic conductor). Defining a certain class of duality transformations, PEMC medium can be transformed to PEC or PMC media. As an application, plane-wave reflection from a planar interface of air and PEMC medium is studied. It is shown that, in general, the reflected wave has a cross-polarized component, which is a manifestly nonreciprocal effect.

A set of four scalar conditions involving normal components of the fields D and B and their normal derivatives at a planar surface is introduced, among which different pairs can be chosen to represent possible boundary conditions for the electromagnetic fields. Four such pairs turn out to yield meaningful boundary conditions and their responses for an incident plane wave at a planar boundary are studied. The theory is subsequently generalized to more general boundary surfaces defined by a coordinate function. It is found that two of the pairs correspond to the PEC and PMC conditions while the other two correspond to a mixture of PEC and PMC conditions for fields polarized TE or TM with respect to the coordinate defining the surface.

The well-known TE/TM decomposition of time-harmonic electromagnetic fields in uniaxial anisotropic media is generalized in terms of four-dimensional differentialform formalism by requiring that the field two-form satisfies an orthogonality condition with respect to two given bivectors. Conditions for the electromagnetic medium in which such a decomposition is possible are derived and found to define three subclasses of media. It is shown that the previously known classes of generalized Q-media and generalized P-media are particular cases of the proposed decomposable media (DCM) associated to a quadratic equation for the medium dyadic. As a novel solution, another class of special decomposable media (SDCM) is defined by a linear dyadic equation. The paper further discusses the properties of medium dyadics and plane-wave propagation in all the identified cases of DCM and SDCM.

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We compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules con- tained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two "rules of thumb" that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoreti- cal results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approxima- tion. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, pre- dicted by the theory and not observed in the experiment, are presumably washed out by the longitu- dinal and radial variation of the electric field in the cavity.

Normal Mode (NM) and Parabolic Equation (PE) have been used widely by Underwater Acoustic Community due to their effectiveness. In this paper we investigate NM and PE in term of their mathematical approach as well as their computation. Further, Tonkin Gulf has been modeled and simulated using both of NM and PE. The simulation results show that there are the agreement and the reliability between both methodologies.

A first-principles study of the relative stability of a few different structures, viz. AgCl, AgBr, and AgI, is undertaken. Modern pseudopotentials, the local-density approximation, and a plane-wave basis are used. The relative stability of the cubic zinc blende, rocksalt, and CsCl structures, as well as the wurtzite, ␤-Sn, NiAs, and cinnabar structures is investigated. Ground-state structural parameters are computed and compared with experiment. The optimization of the cinnabar structure leads to a simple rhombohedral phase, which is found to be the most stable among those considered in an intermediate range of pressures for AgCl and AgBr. According to these results, AgCl and AgBr prefer a rhombohedral route from the rocksalt structure to the CsCl structure instead of the usual discontinuous transition which is common in ionic compounds.

The edge-degree distance of a simple connected graph G is defined as the sum of the terms ( d(e |G ) + d(f |G ) ) d(e, f |G ) over all unordered pairs {e, f } of edges of G, where d(e |G ) and d(e, f |G ) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.

The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.

The Einstein-de Broglie sol#on concept is applied to shnulate stationary states of an electron hz a hydrogen atom. According to this concept, the electron is described by the localized regular sohttions to some nonlinear equations. It is shown that the electron-soliton center travels along some stationary orbit around the Coulomb center. The electromagnetic radiation is absent as the Poynting vector has non-wave asymptote O(r -3) after averag#~g over angles.

The present article deals with the investigation of the propagation of thermoelastic plane harmonic waves in a nonlocal thermoelastic medium. The Green and Naghdi theory II (without energy dissipation) of generalized thermoelasticity with elastic nonlocal effect is adopted to address this problem. The problem of reflection of thermoelastic waves due to an incident coupled longitudinal elastic wave from the rigid and thermally insulated boundary of a homogeneous, isotropic nonlocal thermoelastic halfspace is also studied. The amplitude ratios of the reflected waves and their respective energy ratios are determined analytically. For a particular model, the effect of elastic nonlocality parameter on the variations of phase speeds, attenuation coefficients, amplitude ratios and corresponding energy ratios of the reflected waves is presented graphically and analysis of these results is given.

We propose a preconditioning scheme for diagonalizing large Hamiltonian matrices arising in ®rst-principles planewave pseudopotential calculations. The new scheme is based on the Neumann expansion for the inverse, shifted Hamiltonian from which only a nonlocal part is omitted. The preconditioner is applied to a gradient vector using the fast Fourier transformation technique. In the framework of the Davidson-type diagonalization algorithm, we have found the present preconditioning scheme to be more ecient than widely accepted diagonal scaling methods.

An integral equation method employing complex images Green's functions is developed for analyzing different devices fabricated in 2-D dielectric photonic crystals. The integral equation is written in terms of the unknown equivalent current sources flowing on the surfaces of the periodic 2-D cylinders. The method of moments is then employed to solve for the unknown current distributions. The required Green's function of the problem is represented in terms of a finite summation of complex images instead of the conventional slowly converging infinite series. It is shown that when the field-point is far from the periodic sources, it is just sufficient to consider the contribution of the propagating poles in the structure. This will result in a summation of plane waves that has an even smaller size compared with the conventional complex images Green's function. This will enable us to analyze the dielectric periodic structures efficiently and accurately. The method is applied to a number of waveguide structures and its results are compared with the existing literature. Terms-Complex images, integral equation method, method of moments (MoM), periodic Green's function, photonic crystal devices. I. INTRODUCTION P HOTONIC crystals, proposed by Yablonovitch [1] and John [2], are now a popular subject of photonic researches. Photonic crystal, a periodic material with a photonic band gap, has a potential capability in manipulating and controlling light. It leads to design of applications to the bandgap devices [3], including optical waveguides, couplers, and high-quality resonant cavities [4]. Many numerical techniques have been developed and used for investigating the wave propagation behavior of different photonic crystal structures. Plane wave expansion method [5], transmission line method [6], finite difference [7] and finite-element [8] methods, the pseudospectral method [9], and time-domain techniques including the most popular finite-difference time domain [10], [11] and integral equation techniques [12] have been presented capable of analyzing these structures. The Floquet-Bloch theorem limits the analysis to a unit cell. When applying integral equation techniques to this cell, it is nec-

The MAX phase materials are a class of ternary compounds with formula unit M n+1 AX n (MAX), where n = 1, 2 or 3, M is an early transition metal, A is an A-group (mostly IIIA and IVA) element, and X is either C and/or N. These ternary carbides and nitrides have an unusual combination of the properties of both metals and ceramics. They exhibit high hardness, but fully reversible plasticity and negligible thermoelectric power. In this work, we report the electronic structure of nanolaminated Hf 2 AlX (X = C and N) by means of a first-principles method, the ''full-potential linearized plane-wave method'' (including spin-orbit interaction) based on the density functional theory. We have investigated the lattice parameters, bulk moduli, band structures, total and partial densities of states, and charge densities. We hope that this work will inspire future experimental research on these Hf-based ternary materials.

The last sentence of the paragraph following Eq. ( ), page 5249, should read as follows: "Hence, we decided to use the approximation C ~ -< Ytt, which may not be entirely invalid in view of our previous experience in electromagnetics. 15 • 20 ..

We compute the Yang-Mills vacuum wave functional in three dimensions at weak coupling with O(e 2 ) precision. We use two different methods to solve the functional Schroedinger equation. One of them generalizes to O(e 2 ) the method followed by Hatfield at O(e) . The other uses the weak coupling version of the gauge invariant formulation of the Schroedinger equation and the ground state wave functional followed by Karabali, Nair, and Yelnikov [2]. We compare both results and discuss the differences between them.

There are several kinds of experiments that can be done with multilayer stacks of dielectric media which require an understanding of light emission by sources within the stack for their analysis. These experiments may involve, for example, light-emitting tunnel junctions, Raman scattering in Kretschmann and other multilayered geometries, and Rayleigh scattering by small amounts of surface or interface roughness, either alone or in combination with other processes. A set of electromagnetic CJreen's functions for a multilayer stack of isotropic dielectric media [D. L. Mills and A. A. Maradudin, Phys. Rev. B 12, 2943] gives the electric fields produced every- where by a point source of current oscillating at a frequency f. These Green's functions can thus be used to solve this type of problem. In this paper we show how these Green's functions can be written in terms of 2)&2 transfer matrices of the type commonly used to find the fields in a dielec- tric stack due to an incident plane wave. With this simplification we can easily evaluate the Green's functions for a stack with an arbitrary number of layers. We further show that, when the electric fields generated by a point source within the stack are evaluated far away, they can be written directly in terms of the electric fields that would be generated at the location of the current source by plane waves incident from the direction of the observation point. We show that this follows from the Lorentz reciprocity theorem. Thus, in this case the formalism of Green's functions is not needed.

The quest for higher peak focused intensities and better temporal contrast drives one to improve the design of all possible stages of the chirped pulse amplification (CPA) system. In this paper, we have analyzed the role of dispersion and spectral profile on the temporal shape and contrast ratio of the output pulse of a CPA system. The simulations indicate that an initial sech 2 or a Gaussian pulse in the CPA system is best for a good contrast ratio. Incorporating a four-grating based pulse compressor in the CPA system improves the contrast as well as provides the flexibility to compensate the dispersion upto the fourth order. The simulations also detail the effect of spectral profile tailoring on the contrast ratio and peak power.

Thirteen measurements of the Compton profile of water carried out in five different laboratories are compared. The possible sources of error inherent in Compton measurements are analysed. Inconsistent normalization of the various profiles is found to be an important source of error. After renormalization of the profiles the results are found to be consistent to within + 2 % of the peak height of the profiles. It is shown that all probable sources of error can be analysed in terms of five general types of error and further that the effects of all of these types of error on the shape of the Compton profile are almost indistinguishable from one another. The determination of the position of the Compton peak is found to be another important potential source of error in experimental measurements. Certain assumptions relating to the conversion to a momentum scale were found to be mistaken and are corrected. A comparison of currently accepted relativistic and non-relativistic expressions for the cross section revealed that they are not in fact significantly different. Multiple scattering is found to be the only major source of error which is not yet well understood. Each experimental measurement is assessed in detail with particular regard to the differences between the techniques used by the contributing groups. Proposals are made relating to the accumulation, processing and presentation of Compton data. It is concluded that a reproducibility of + 1% of the peak height of the profile is quite feasible and that in a differential experiment + 0-5 % or even better is possible. All the research groups currently known to be involved in Compton profile measurements were invited to participate in the project. The following people agreed to carry out a measurement using the radiations indicated -

A numerical method is presented for the problem of transverse magnetic or transverse electric scattering from infinitely long, lossy dielectric cylinders of arbitrary cross section. The method is based on the solution of a system of linear ordinary differential equations in the statespace form. The solution is achieved by finding the state transition-matrix and the zero-state response of the respective system. Numerical results are presented for cylinders of various cross sections, and they are compared with the results obtained by either exact or approximate numerical methods.

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. it is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear SchrGdinger equation. The latter equation allows us to study the modulational instability conditions leading to diflerent zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.

Among a lot of fascinating nonlinear effects, there is a great deal of interest in self modulation of a plane wave, or modulational instability (MI), which occurs in nonlinear dispersive media. Qualitatively a MI is the tendency for amplitude of a modulated carrier wave to break into isolated structures or solitons. Energy originally resident in the carrier of wave packets or long pulse is gradually transferred by nonlinear interaction, in the medium, into the spectrum side bands. As the side-band energy grows in amplitude, the modulated wave breaks up into a series of localized objects or envelope solitons. Composite structures formed by a singly or doubly-curved shallow shell resting on a nonlinear elastic foundation are wave guides that enable one to focus a high energy density in order that nonlinearities be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional localized modes in elastodynamics. The purpose of the present work is to study the influences of the geometric dispersion, the prestress on the shallow shell and the material nonlinearity of the elastic foundation on the vibrations modes of the structure. The basic equations which govern the dynamics of the elastic structure are deduced from a variational principle. The analysis is restricted to signals which consist of a slowly varying envelope in space and time modulating a harmonic carrier wave. In the limit of low amplitude the coupled equations are solved by means of a reductive perturbation method. It is shown that the complex amplitude of the envelope is governed by a two-dimensional nonlinear Schrödinger equation. This equation allows us to study the modulational instability conditions leading to different zones of instability. The mechanism of the self generated nonlinear waves in the elastic structure beyond the birth of modulational instability is numerically investigated on the original equations.

The interpolation errors of the higher order bivariate Lagrange polynomial interpolation based on the rectangular, right and equilateral triangular interpolations are measured by using the maximum and root-mean-square (RMS) errors. The error distributions of above three kinds of interpolations are analyzed to find the regions having the smallest interpolation error. Both analytical and numerical results show that the right triangular interpolation is the most efficient interpolation method. Although both the maximum and RMS errors of the right triangular interpolation are larger than that of the rectangular interpolation, the number of data points used in the triangular interpolation is up to 50% less than that used in the rectangular one. On the other hand, the equilateral triangular interpolation using the regions inside a big triangle as interpolation area is proved to be the most accurate interpolation method. The interpolation errors of the equilateral triangular are almost half of that of the rectangular interpolation. In addition, the right triangular, equilateral triangular and rectangular interpolations for the third and fourth orders are applied to accelerate the calculation of doubly periodic Green's function (PGF). INDEX TERMS Doubly periodic Green's function (PGF), Lagrange polynomial interpolation, maximum error, plane waves, root-mean-square (RMS) error, triangular interpolations.