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Plane wave

About: Plane wave is a research topic. Over the lifetime, 24857 publications have been published within this topic receiving 461687 citations.


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BookDOI
01 Jan 1980
TL;DR: In this article, the authors used the Rayleigh expansion and the Grating Formula to determine the coefficients of a Grating function in the context of point-matching and point matching.
Abstract: 1 A Tutorial Introduction- 11 Preliminaries- 111 General Notations- 112 Time-Harmonic Maxwell Equations- 113 Boundary Conditions- 114 Electromagnetism and Distribution Theory- 115 Notations Used in the Description of a Grating- 12 The Perfectly Conducting Grating- 121 Generalities- 122 The Diffracted Field- 123 The Rayleigh Expansion and the Grating Formula- 124 An Important Lemma- 125 The Reciprocity Theorem- 126 The Conservation of Energy- 127 The Littrow Mounting- 128 The Determination of the Coefficients Bn by the Rayleigh Method- 129 An Integral Expression of ud in P Polarization- 1210 The Integral Method in P Polarization- 1211 The Integral Method in S Polarization- 1212 Modal Expansion Methods- 1213 Conical Diffraction- 13 The Dielectric or Metallic Grating- 131 General i ti es- 132 The Diffracted Field Outside the Groove Region- 133 Maxwell Equations and Distributions- 134 The Principle of the Differential Method (in P Polarization)- 14 Miscellaneous- References- Appendix A: The Distributions or Generalized Functions- AI Preliminaries- A2 The Function Space R- A3 The Space R1- A31 Definitions- A32 Examples of Distributions- A4 Derivative of a Distribution- A5 Expansion with Respect to the Basis ej(x) =exp [i (nK+k sine) x] = exp (i?n x)- A51 Theorem- A 52 Proof- A53 Application to deltaR- A6 Convolution- A61 Memoranda on the Product of Convolution in D'1- A62 Convolution in R1- 2 Some Mathematical Aspects of the Grating Theory- 21 Some Classical Properties of the Helmholtz Equation- 22 The Radiation Condition for the Grating Problem- 23 A Lemma- 24 Uniqueness Theorems- 241 Metallic Grating, with Infinite Conductivity- 242 Dielectric Grating- 25 Reciprocity Relations- 26 Foundation of the Yasuura Improved Point-Matching Method- 261 Definition of a Topological Basis- 262 The System of Rayleigh Functions is a Topological Basis- 263 The Convergence of the Rayleigh Series A Counterexample- References- 3 Integral Methods- 31 Development of the Integral Method- 32 Presentation of the Problem and Intuitive Description of an Integral Approach- 321 Presentation of the Problem- 322 Intuitive Description of an Integral Approach- 33 Notations, Mathematical Problem and Fundamental Formulae- 331 Notations and Mathematical Formulation- 332 Basic Formulae of the Integral Approach- 34 The Uncoated Perfectly Conducting Grating- 341 The TE Case of Polarization- 342 The TM Case of Polarization- 35 The Uncoated Dielectric or Metallic Grating- 351 The Mathematical Boundary Problem- 352 Vital Importance of the Choice of a Well-Adapted Unknown Function- 353 Mathematical Definition of the Unknown Function and Determination of the Field and Its Normal Derivative Above P- 354 Expression of the Field in M2 as a Function of ?- 355 Integral Equation- 356 Limit of the Equation when the Metal Becomes Perfectly Conducting- 36 The Multiprofile Grating- 37 The Grating in Conical Diffraction Mounting- 38 Numerical Application- 381 A Fundamental Preliminary Choice- 382 Study of the Kernels- 383 Integration of the Kernels- 384 Particular Difficulty Encountered with Materials of High Conductivity- 385 The Problem of Edges- 386 Precision on the Numerical Results- References- 4 Differential Methods- 41 Introductory Remarks- 411 Historical Survey- 412 Definition of Problem- 42 The E,, Case- 421 The Reflection and Transmission Matrices- 422 The Computation of Transmission and Reflection Matrices- 423 Numerical Algorithms- 424 Al ternative Matching Procedures for Some Grating Profiles- 425 Field of Application- 43 The H Case- 431 The Propagation Equation- 432 Numerical Treatment- 433 Field of Application- 44 The General Case (Conical Diffraction Case)- 441 The Reflection and Transmission Matrices- 442 The Differential System- 443 Matching with Rayleigh Expansions- 444 Field of Application- 45 Stratified Media- 451 Stack of Gratings- 452 Plane Interfaces Between Homogeneous Media- 46 Infinitely Conducting Gratings: the Conformai Mapping Method- 461 Method- 462 Determination of the Conformai Mapping- 463 Field of Application- References- 5 The Homogeneous Problem- 51 Historical Summary- 52 Plasmon Anomalies of a Metallic Grating- 521 Reflection of a Plane Wave on a Plane Interface- 522 Reflection of a Plane Wave on a Grating- 53 Anomalies of Dielectric Coated Reflection Gratings Used in TE Polarization- 531 Determination of the Leaky Modes of a Dielectric Slab Bounded by Metal on One of Its Sides- 532 Reflection of a Plane Wave on a Dielectric Coated Reflection Grating Used in TE Polarization- 54 Extension of the Theory- 541 Anomalies of a Dielectric Coated Grating Used in TM Polarization- 542 Plasmon Anomalies of a Bare Grating Supporting Several Spectral Orders- 543 General Considerations on Anomalies of a Grating Supporting Several Spectral Orders- 55 Theory of the Grating Coupler- 551 Description of the Incident Beam- 552 Response of the Structure to a Plane Wave- 553 Response of the Structure to a Limited Beam- 554 Determination of the Coupling Coefficient- 555 Application to a Limited Incident Beam- References- 6 Experimental Verifications and Applications of the Theory- 61 Experimental Checking of Theoretical Results- 611 Generalities- 612 Microwave Region- 613 On the Determination of Groove Geometry and of the Refractive Index- 614 Infrared- 615 Visible Region- 616 Near and Vacuum UV- 617 XUV Domain- 618 X-Ray Domain- 62 Systematic Study of the Efficiency of Perfectly Conducting Gratings- 621 Systematic Study of Echelette Gratings in -1 Order Littrow Mount- 622 An Equivalence Rule Between Ruled, Holographic, and Lamel1ar Gratings- 623 Systematic Study of the Efficiency of Holographic Gratings in -1 Order Littrow Mount- 624 Systematic Study of the Efficiency of Symmetrical Lamellar Gratings in -1 Order Littrow Mount- 625 Influence of the Apex Angle- 626 Influence of a Departure from Littrow- 627 Higher Order Use of Gratings- 63 Finite Conductivity Gratings- 631 General Rules- 632 Typical Efficiency Curves in the Visible Region- 633 Influence of Dielectric Overcoatings in Vacuum UV- 634 The Use of Gratings in XUV and X-Ray Regions (?<1000 A)- 635 Conical Diffraction Mountings- 64 Some Particular Applications- 641 Simultaneous Blazing in Both Polarizations- 642 Spectrometers with Constant Efficiency- 643 Grating Bandpass Filter- 644 Reflection Grating Polarizer for the Infrared- 645 Transmission Gratings as Masks in Photolithography- 646 Gratings Used as Beam Sampling Mirrors for High Power Lasers- 647 Gratings as Wavelength Selectors in Tunable Lasers- 648 Transmission Dielectric Gratings used as Color Filters- Concluding Remarks- References- 7 Theory of Crossed Gratings- 71 Overview- 72 The Bigrating Equation and Rayleigh Expansions- 73 Inducti ve Gri ds- 731 Grids with Rectangular Apertures- 732 Numerical Tests and Applications- 733 Inductive Grids with Circular Apertures- 74 Capacitive and Other Grid Geometries- 741 High-Pass Filters- 742 Low-Pass Filters- 743 Bandpass Filters- 744 Bandstop Filters- 75 Spatially Separated Grids or Gratings- 751 The Crossed Lamellar Transmission Grating- 752 The Double Grating- 753 Symmetry Properties of Double Gratings- 754 Multielement Grating Interference Filters- 76 Finitely Conducting Bigratings- 761 A Short Description of the Method- 762 The Coordinate Transformation- 763 Integral Equation Form- 764 Iterative Solution of the Integral Equations- 765 Total Absorption of Unpolarized Monochromatic Light- 766 Reduction of Metallic Reflectivity: Plasmons and Moth-Eyes- 767 Equivalence Formulae Linking Crossed and Classical Gratings- 768 Coated Bigratings- References- Additional References with Titles

1,384 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a method for estimating the effective density and the bulk modulus of open cell foams and fibrous materials with cylindrical porous layers. But the authors do not consider the effect of noise on the propagation of sound.
Abstract: Preface to the second edition. 1 Plane waves in isotropic fluids and solids. 1.1 Introduction. 1.2 Notation - vector operators. 1.3 Strain in a deformable medium. 1.4 Stress in a deformable medium. 1.5 Stress-strain relations for an isotropic elastic medium. 1.6 Equations of motion. 1.7 Wave equation in a fluid. 1.8 Wave equations in an elastic solid. References. 2 Acoustic impedance at normal incidence of fluids. Substitution of a fluid layer for a porous layer. 2.1 Introduction. 2.2 Plane waves in unbounded fluids. 2.3 Main properties of impedance at normal incidence. 2.4 Reflection coefficient and absorption coefficient at normal incidence. 2.5 Fluids equivalent to porous materials: the laws of Delany and Bazley. 2.6 Examples. 2.7 The complex exponential representation. References. 3 Acoustic impedance at oblique incidence in fluids. Substitution of a fluid layer for a porous layer. 3.1 Introduction. 3.2 Inhomogeneous plane waves in isotropic fluids. 3.3 Reflection and refraction at oblique incidence. 3.4 Impedance at oblique incidence in isotropic fluids. 3.5 Reflection coefficient and absorption coefficient at oblique incidence. 3.6 Examples. 3.7 Plane waves in fluids equivalent to transversely isotropic porous media. 3.8 Impedance at oblique incidence at the surface of a fluid equivalent to an anisotropic porous material. 3.9 Example. References. 4 Sound propagation in cylindrical tubes and porous materials having cylindrical pores. 4.1 Introduction. 4.2 Viscosity effects. 4.3 Thermal effects. 4.4 Effective density and bulk modulus for cylindrical tubes having triangular, rectangular and hexagonal cross-sections. 4.5 High- and low-frequency approximation. 4.6 Evaluation of the effective density and the bulk modulus of the air in layers of porous materials with identical pores perpendicular to the surface. 4.7 The biot model for rigid framed materials. 4.8 Impedance of a layer with identical pores perpendicular to the surface. 4.9 Tortuosity and flow resistivity in a simple anisotropic material. 4.10 Impedance at normal incidence and sound propagation in oblique pores. Appendix 4.A Important expressions. Description on the microscopic scale. Effective density and bulk modulus. References. 5 Sound propagation in porous materials having a rigid frame. 5.1 Introduction. 5.2 Viscous and thermal dynamic and static permeability. 5.3 Classical tortuosity, characteristic dimensions, quasi-static tortuosity. 5.4 Models for the effective density and the bulk modulus of the saturating fluid. 5.5 Simpler models. 5.6 Prediction of the effective density and the bulk modulus of open cell foams and fibrous materials with the different models. 5.7 Fluid layer equivalent to a porous layer. 5.8 Summary of the semi-phenomenological models. 5.9 Homogenization. 5.10 Double porosity media. Appendix 5.A: Simplified calculation of the tortuosity for a porous material having pores made up of an alternating sequence of cylinders. Appendix 5.B: Calculation of the characteristic length LAMBDA'. Appendix 5.C: Calculation of the characteristic length LAMBDA for a cylinder perpendicular to the direction of propagation. References. 6 Biot theory of sound propagation in porous materials having an elastic frame. 6.1 Introduction. 6.2 Stress and strain in porous materials. 6.3 Inertial forces in the biot theory. 6.4 Wave equations. 6.5 The two compressional waves and the shear wave. 6.6 Prediction of surface impedance at normal incidence for a layer of porous material backed by an impervious rigid wall. Appendix 6.A: Other representations of the Biot theory. References. 7 Point source above rigid framed porous layers. 7.1 Introduction. 7.2 Sommerfeld representation of the monopole field over a plane reflecting surface. 7.3 The complex sin theta plane. 7.4 The method of steepest descent (passage path method). 7.5 Poles of the reflection coefficient. 7.6 The pole subtraction method. 7.7 Pole localization. 7.8 The modified version of the Chien and Soroka model. Appendix 7.A Evaluation of N. Appendix 7.B Evaluation of p r by the pole subtraction method. Appendix 7.C From the pole subtraction to the passage path: Locally reacting surface. References. 8 Porous frame excitation by point sources in air and by stress circular and line sources - modes of air saturated porous frames. 8.1 Introduction. 8.2 Prediction of the frame displacement. 8.3 Semi-infinite layer - Rayleigh wave. 8.4 Layer of finite thickness - modified Rayleigh wave. 8.5 Layer of finite thickness - modes and resonances. Appendix 8.A Coefficients r ij and M i,j. Appendix 8.B Double Fourier transform and Hankel transform. Appendix 8.B Appendix .C Rayleigh pole contribution. References. 9 Porous materials with perforated facings. 9.1 Introduction. 9.2 Inertial effect and flow resistance. 9.3 Impedance at normal incidence of a layered porous material covered by a perforated facing - Helmoltz resonator. 9.4 Impedance at oblique incidence of a layered porous material covered by a facing having cirular perforations. References. 10 Transversally isotropic poroelastic media. 10.1 Introduction. 10.2 Frame in vacuum. 10.3 Transversally isotropic poroelastic layer. 10.4 Waves with a given slowness component in the symmetry plane. 10.5 Sound source in air above a layer of finite thickness. 10.6 Mechanical excitation at the surface of the porous layer. 10.7 Symmetry axis different from the normal to the surface. 10.8 Rayleigh poles and Rayleigh waves. 10.9 Transfer matrix representation of transversally isotropic poroelastic media. Appendix 10.A: Coefficients T i in Equation (10.46). Appendix 10.B: Coefficients A i in Equation (10.97). References. 11 Modelling multilayered systems with porous materials using the transfer matrix method. 11.1 Introduction. 11.2 Transfer matrix method. 11.3 Matrix representation of classical media. 11.4 Coupling transfer matrices. 11.5 Assembling the global transfer matrix. 11.6 Calculation of the acoustic indicators. 11.7 Applications. Appendix 11.A The elements T ij of the Transfer Matrix T ]. References. 12 Extensions to the transfer matrix method. 12.1 Introduction. 12.2 Finite size correction for the transmission problem. 12.3 Finite size correction for the absorption problem. 12.4 Point load excitation. 12.5 Point source excitation. 12.6 Other applications. Appendix 12.A: An algorithm to evaluate the geometrical radiation impedance. References. 13 Finite element modelling of poroelastic materials. 13.1 Introduction. 13.2 Displacement based formulations. 13.3 The mixed displacement-pressure formulation. 13.4 Coupling conditions. 13.5 Other formulations in terms of mixed variables. 13.6 Numerical implementation. 13.7 Dissipated power within a porous medium. 13.8 Radiation conditions. 13.9 Examples. References. Index.

1,375 citations

Journal ArticleDOI
TL;DR: In this article, a new formulation of an approximate conservation relation of wave-activity pseudomentum is derived, which is applicable for either stationary or migratory quasigeostrophic (QG) eddies on a zonally varying basic flow.
Abstract: A new formulation of an approximate conservation relation of wave-activity pseudomomentum is derived, which is applicable for either stationary or migratory quasigeostrophic (QG) eddies on a zonally varying basic flow. The authors utilize a combination of a quantity A that is proportional to wave enstrophy and another quantity E that is proportional to wave energy. Both A and E are approximately related to the wave-activity pseudomomentum. It is shown for QG eddies on a slowly varying, unforced nonzonal flow that a particular linear combination of A and E, namely, M ≡ (A + E)/2, is independent of the wave phase, even if unaveraged, in the limit of a small-amplitude plane wave. In the same limit, a flux of M is also free from an oscillatory component on a scale of one-half wavelength even without any averaging. It is shown that M is conserved under steady, unforced, and nondissipative conditions and the flux of M is parallel to the local three-dimensional group velocity in the WKB limit. The autho...

1,353 citations

Book
01 Jan 1960

1,254 citations

Journal ArticleDOI
TL;DR: In this article, a density functional theory-based algorithm for periodic and non-periodic ab initio calculations is presented, which uses pseudopotentials in order to integrate out the core electrons from the problem.
Abstract: A density functional theory-based algorithm for periodic and non-periodic ab initio calculations is presented. This scheme uses pseudopotentials in order to integrate out the core electrons from the problem. The valence pseudo-wavefunctions are expanded in Gaussian-type orbitals and the density is represented in a plane wave auxiliary basis. The Gaussian basis functions make it possible to use the efficient analytical integration schemes and screening algorithms of quantum chemistry. Novel recursion relations are developed for the calculation of the matrix elements of the density-dependent Kohn-Sham self-consistent potential. At the same time the use of a plane wave basis for the electron density permits efficient calculation of the Hartree energy using fast Fourier transforms, thus circumventing one of the major bottlenecks of standard Gaussian based calculations. Furthermore, this algorithm avoids the fitting procedures that go along with intermediate basis sets for the charge density. The performance a...

1,150 citations


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No. of papers in the topic in previous years
YearPapers
2023147
2022340
2021490
2020646
2019642
2018654