Showing papers on "Wave propagation published in 1985"

Theory of microwave remote sensing

Abstract: Active and passive microwave remote sensing of earth terrains is studied. Electromagnetic wave scattering and emission from stratified media and rough surfaces are considered with particular application to the remote sensing of soil moisture. Radiative transfer theory for both the random and discrete scatterer models is examined. Vector radiative transfer equations for nonspherical particles are developed for both active and passive remote sensing. Single and multiple scattering solutions are illustrated with applications to remote sensing problems. Analytical wave theory using the Dyson and Bethe-Salpeter equations is employed to treat scattering by random media. The backscattering enhancement effects, strong permittivity fluctuation theory, and modified radiative transfer equations are addressed. The electromagnetic wave scattering from a dense distribution of discrete scatterers is studied. The effective propagation constants and backscattering coefficients are calculated and illustrated for dense media.

On the Three-Dimensional Propagation of Stationary Waves

Abstract: A locally applicable (nonzonally-averaged) conservation relation is derived for quasi-geostrophic stationary waves on a zonal flow, a generalization of the Eliassen-Palm relation. The flux which appears in this relation constitutes, it is argued, a useful diagnostic of the three-dimensional propagation of stationary wave activity. This is illustrated by application to a simple theoretical model of a forced Rossby wave train and to a Northern Hemisphere winter climatology. Results of the latter procedure suggest that the major forcing of the stationary wave field derives from the orographic effects of the Tibetan plateau and from nonorographic effects (diabatic heating and/or interaction with transient eddies) in the western North Atlantic and North Pacific Oceans and Siberia. No evidence is found in the data for wave trains of tropical origin; forcing by the orographic effects of the Rocky mountains seems to be of secondary importance.

Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux

Abstract: The artificial viscosity (Q) method of von Neumann and Richtmyer is a tremendously useful numerical technique for following shocks wherever and whenever they appear in the flow. We show that it must be used with some caution, however, as serious Q-induced errors (on the order of 100%) can occur in some strong shock calculations. We investigate three types of Q errors: 1. Excess Q heating, of which there are two types: (a) excess wall heating on shock formation and (b) shockless Q heating; 2. Q errors when shocks are propagated over a nonuniform mesh; and 3. Q errors in propagating shocks in spherical geometry. As a basis of comparison, we use as our standard the Lagrangian formulation with Q = C/sup 2//sub 0/rhol/sup 2/(u/sub x/)/sup 2/. This standard Q is compared with Noh's (Q and H) shock-following method, which employs an artificial heat flux (H) in addition to Q, and with the (non-Q) piecewise-parabolic method (PPM) of Colella and Woodward. Both the (Q and H) method and PPM (particularly when used with an adaptive shock-tracking mesh) give superior results for our test problems. In spherical geometry, Schulz's and Walen's tensor Q formulations of the hydrodynamic equations prove to be moremore » accurate than the standard Q formulation, and when Schulz's formulation is combined with Noh's (Q and H) method, superior results are achieved. Copyright 1987 Academic Press, Inc.« less

The question of classical localization A theory of white paint

Abstract: The expected behaviour of localizing media for classical wave propagation is analysed. Some possible examples in electromagnetic and acoustic phenomena are given.

Sound and structural vibration: Radiation, transmission and response

Abstract: Chapter 1. Wave in Fluids and Structures Chapter 2. Structural Mobility, Impedance, Vibrational Energy and Power Chapter 3. Sound Radiation by Vibrating Structures Chapter 4. Fluid Loading of Vibrating Structures Chapter 5. Transmission of Sound Through Partitions Chapter 6. Acoustically Induced Vibration of Structures Chapter 7. Acoustic Coupling between Structures and Enclosed Volumes of Fluid Chapter 8. Introduction to Numerically Based Analyses of Fluid-Structure Interaction Chapter 9. Introduction to Active Control of Sound Radiation and Transmission

Asymptotic Stability of Traveling Wave Solutions of Systems for One-dimensional Gas Motion

Abstract: The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet−γ (for someγ>0) ast→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.

Shock viscosity and the prediction of shock wave rise times

Abstract: The present study is focused on viscouslike behavior of solids during large‐amplitude compressive stress‐wave propagation. Maximum strain rate in the plastic wave has been determined for 30 steady‐ or near steady‐wave profiles obtained with velocity interferometry methods. The materials include six metals, aluminum, beryllium, bismuth, copper, iron, and uranium, and two insulating solids, magnesium oxide and fused silica. A plot of Hugoniot stress versus maximum strain rate for each material is adequately described by η=aσmh. The exponent m is approximately 4 for all materials while the coefficient a is material dependent. A model is developed which incorporates the observed trends of the shock viscosity data in a three‐dimensional framework. Finite‐difference calculations using the model reproduce the experimental wave profile data.

Effective equations for wave propagation in bubbly liquids

Abstract: We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldy's approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

Plane waves in linear elastic materials with voids

Abstract: The behavior of plane harmonic waves in a linear elastic material with voids is analyzed. There are two dilational waves in this theory, one is predominantly the dilational wave of classical linear elasticity and the other is predominantly a wave carrying a change in the void volume fraction. Both waves are found to attenuate in their direction of propagation, to be dispersive and dissipative. At large frequencies the predominantly elastic wave propagates with the classical elastic dilational wave speed, but at low frequencies it propagates at a speed less than the classical speed. It makes a smooth but relatively distinct transition between these wave speeds in a relatively narrow range of frequency, the same range of frequency in which the specific loss has a relatively sharp peak. Dispersion curves and graphs of specific loss are given for four particular, but hypothetical, materials, corresponding to four cases of the solution.

Wave formation on a vertical falling liquid film

Abstract: The method of integral relations is used to derive a nonlinear two-wave equation for long waves on the surface of vertical falling liquid films. This equation is valid within a range of moderate Reynolds numbers and and be reduced in some cases to other well-known equations. The theoretical results for the fastest growing waves are compared with the experimental results concerning velocities, wave numbers, and growth rates of the waves in the inception region. The validity of the theoretical assumptions is also confirmed by direct measurements of instantaneous velocity profiles in a wave liquid film. The results of the experimental investigation concerning nonlinear stationary waves and the evolution of initial solitary disturbances are presented.

A full wave solution for propagation in multilayered viscoelastic media with application to Gaussian beam reflection at fluid–solid interfaces

Abstract: A new solution technique for wave propagation in horizontally stratified viscoelastic media is presented. The model provides a full wave solution for the field generated by a single source as well as for that generated by a vertical source array. It allows the spatial distribution of the acoustic field to be evaluated at least one order of magnitude faster than with existing models based on the Thomson–Haskell solution technique. The computational efficiency of the numerical code is demonstrated by providing exact numerical solutions for the reflectivity pattern associated with narrow ultrasonic beams incident on a fluid–solid interface near the Rayleigh angle.

Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media

Abstract: The dispersion relations for TE polarized waves guided by thin dielectric (planar integrated optics) films, surrounded on one or both sides by media with intensity-dependent refractive indexes, are solved numerically and interpreted. Comparisons are made between these nonlinear guided waves and the usual integrated optics modes associated with linear media. For media characterized by self-defocusing nonlinearities, increasing guided wave power leads to a decrease in the guided wave effective index and to mode cutoffs characterized either by finite or diverging total guided wave powers. For self-focusing media, the effective index increases with increasing power. New wave solutions are obtained with power thresholds and anomalous power-dependent field distributions. These characteristics are illustrated by numerical calculations for a specific material system and possible applications to optical devices are discussed.

Stochastic wave propagation

Abstract: Mathematical Preliminaries. Stochastic Media, Models and Analysis. Wave Propagation in Continuous Stochastic Media. Wave Propagation in Discrete Stochastic Media. Scattering of Waves at Stochastic Surfaces. Bibliography.

Mechanics of continua and wave dynamics

Abstract: I Theory of Elasticity.- 1. The Main Types of Strain in Elastic Solids.- 1.1 Equations of Linear Elasticity Theory.- 1.1.1 Hooke's Law.- 1.1.2 Differential Form of Hooke's Law. Principle of Superposition.- 1.2 Homogeneous Strains.- 1.2.1 An Elastic Body Under the Action of Hydrostatic Pressure.- 1.2.2 Longitudinal Strain with Lateral Displacements Forbidden.- 1.2.3 Pure Shear.- 1.3 Heterogeneous Strains.- 1.3.1 Torsion of a Rod.- 1.3.2 Bending of a Beam.- 1.3.3 Shape of a Beam Under Load.- 1.4 Exercises.- 2. Waves in Rods, Vibrations of Rods.- 2.1 Longitudinal Waves.- 2.1.1 Wave Equation.- 2.1.2 Harmonic Waves.- 2.2 Reflection of Longitudinal Waves.- 2.2.1 Boundary Conditions.- 2.2.2 Wave Reflection.- 2.3 Longitudinal Oscillations of Rods.- 2.4 Torsional Waves in a Rod. Torsional Vibrations.- 2.5 Bending Waves in Rods.- 2.5.1 The Equation for Bending Waves.- 2.5.2 Boundary Conditions. Harmonic Waves.- 2.5.3 Reflection of Waves. Bending Vibrations.- 2.6 Wave Dispersion and Group Velocity.- 2.6.1 Propagation of Nonharmonic Waves.- 2.6.2 Propagation of Narrow-Band Disturbances.- 2.7 Exercises.- 3. General Theory of Stress and Strain.- 3.1 Description of the State of a Deformed Solid.- 3.1.1 Stress Tensor.- 3.1.2 The Strain Tensor.- 3.1.3 The Physical Meaning of the Strain Tensor's Components.- 3.2 Equations of Motion for a Continuous Medium.- 3.2.1 Derivation of the Equation of Motion.- 3.2.2 Strain-Stress Relation. Elasticity Tensor.- 3.3 The Energy of a Deformed Body.- 3.3.1 The Energy Density.- 3.3.2 The Number of Independent Components of the Elasticity Tensor.- 3.4 The Elastic Behaviour of Isotropic Bodies.- 3.4.1 The Generalized Hooke's Law for an Isotropic Body.- 3.4.2 The Relationship Between Lame's Constants and E and v.- 3.4.3 The Equations of Motion for an Isotropic Medium.- 3.5 Exercises.- 4. Elastic Waves in Solids.- 4.1 Free Waves in a Homogeneous Isotropic Medium.- 4.1.1 Longitudinal and Transverse Waves.- 4.1.2 Boundary Conditions for Elastic Waves.- 4.2 Wave Reflection at a Stress-Free Boundary.- 4.2.1 Boundary Conditions.- 4.2.2 Reflection of a Horizontally Polarized Wave.- 4.2.3 The Reflection of Vertically Polarized Waves.- 4.2.4 Particular Cases of Reflection.- 4.2.5 Inhomogeneous Waves.- 4.3 Surface Waves.- 4.3.1 The Rayleigh Wave.- 4.3.2 The Surface Love Wave.- 4.3.3 Some Features of Love's Waves.- 4.4 Exercises.- 5. Waves in Plates.- 5.1 Classification of Waves.- 5.1.1 Dispersion Relations.- 5.1.2 Symmetric and Asymmetric Modes.- 5.1.3 Cut-Off Frequencies of the Modes.- 5.1.4 Some Special Cases.- 5.2 Normal Modes of the Lowest Order.- 5.2.1 Quasi-Rayleigh Waves at the Plate's Boundaries.- 5.2.2 The Young and Bending Waves.- 5.3 Equations Describing the Bending of a Thin Plate.- 5.3.1 Thin Plate Approximation.- 5.3.2 Sophie Germain Equation.- 5.3.3 Bending Waves in a Thin Plate.- 5.4 Exercises.- II Fluid Mechanics.- 6. Basic Laws of Ideal Fluid Dynamics.- 6.1 Kinematics of Fluids.- 6.1.1 Eulerian and Lagrangian Representations of Fluid Motion.- 6.1.2 Transition from One Representation to Another.- 6.1.3 Convected and Local Time Derivatives.- 6.2 System of Equations of Hydrodynamics.- 6.2.1 Equation of Continuity.- 6.2.2 The Euler Equation.- 6.2.3 Completeness of the System of Equations.- 6.3 The Statics of Fluids.- 6.3.1 Basic Equations.- 6.3.2 Hydrostatic Equilibrium. Vaisala Frequency.- 6.4 Bernoulli's Theorem and the Energy Conservation Law.- 6.4.1 Bernoulli's Theorem.- 6.4.2 Some Applications of Bernoulli's Theorem.- 6.4.3 The Bernoulli Theorem as a Consequence of the Energy-Conservation Law.- 6.4.4 Energy Conservation Law in the General Case of Unsteady Flow.- 6.5 Conservation of Momentum.- 6.5.1 The Specific Momentum Flux Tensor.- 6.5.2 Euler's Theorem.- 6.5.3 Some Applications of Euler's Theorem.- 6.6 Vortex Flows of Ideal Fluids.- 6.6.1 The Circulation of Velocity.- 6.6.2 Kelvin's Circulation Theorem.- 6.6.3 Helmholtz Theorems.- 6.7 Exercises.- 7. Potential Flow.- 7.1 Equations for a Potential Flow.- 7.1.1 Velocity Potential.- 7.1.2 Two-Dimensional Flow. Stream Function.- 7.2 Applications of Analytical Functions to Problems of Hydrodynamics.- 7.2.1 The Complex Flow Potential.- 7.2.2 Some Examples of Two-Dimensional Flows.- 7.2.3 Conformal Mapping.- 7.3 Steady Flow Around a Cylinder.- 7.3.1 Application of Conformal Mapping.- 7.3.2 The Pressure Coefficient.- 7.3.3 The Paradox of d'Alembert and Euler.- 7.3.4 The Flow Around a Cylinder with Circulation.- 7.4 Irrotational Flow Around a Sphere.- 7.4.1 The Flow Potential and the Particle Velocity.- 7.4.2 The Induced Mass.- 7.5 Exercises.- 8. Flows of Viscous Fluids.- 8.1 Equations of Flow of Viscous Fluid.- 8.1.1 Newtonian Viscosity and Viscous Stresses.- 8.1.2 The Navier-Stokes Equation.- 8.1.3 The Viscous Force.- 8.2 Some Examples of Viscous Fluid Flow.- 8.2.1 Couette Flow.- 8.2.2 Plane Poiseuille Flow.- 8.2.3 Poiseuille Flow in a Cylindrical Pipe.- 8.2.4 Viscous Fluid Flow Around a Sphere.- 8.2.5 Stokes' Formula for Drag.- 8.3 Boundary Layer.- 8.3.1 Viscous Waves.- 8.3.2 The Boundary Layer. Qualitative Considerations.- 8.3.3 Prandl's Equation for a Boundary Layer.- 8.3.4 Approximate Theory of a Boundary Layer in a Simple Case.- 8.4 Exercises.- 9. Elements of the Theory of Turbulence.- 9.1 Qualitative Considerations. Hydrodynamic Similarity.- 9.1.1 Transition from a Laminar to Turbulent Flow.- 9.1.2 Similar Flows.- 9.1.3 Dimensional Analysis and Similarity Principle.- 9.1.4 Flow Around a Cylinder at Different Re.- 9.2 Statistical Description of Turbulent Flows.- 9.2.1 Reynolds' Equation for Mean Flow.- 9.2.2 Turbulent Viscosity.- 9.2.3 Turbulent Boundary Layer.- 9.3 Locally Isotropic Turbulence.- 9.3.1 Properties of Developed Turbulence.- 9.3.2 Statistical Properties of Locally Isotropic Turbulence.- 9.3.3 Kolmogorov's Similarity Hypothesis.- 9.4 Exercises.- 10. Surface and Internal Waves in Fluids.- 10.1 Linear Equations for Waves in Stratified Fluids.- 10.1.1 Linearization of the Hydrodynamic Equations.- 10.1.2 Linear Boundary Conditions.- 10.1.3 Equations for an Incompressible Fluid.- 10.2 Surface Gravity Waves.- 10.2.1 Basic Equations.- 10.2.2 Harmonic Waves.- 10.2.3 Shallow- and Deep-Water Approximations.- 10.2.4 Wave Energy.- 10.3 Capillary Waves.- 10.3.1 "Pure" Capillary Waves.- 10.3.2 Gravity-Capillary Surface Waves.- 10.4 Internal Gravity Waves.- 10.4.1 Introductory Remarks.- 10.4.2 Basic Equation for Internal Waves. Boussinesq Approximation.- 10.4.3 Waves in an Unlimited Medium.- 10.5 Guided Propagation of Internal Waves.- 10.5.1 Qualitative Analysis of Guided Propagation.- 10.5.2 Simple Model of an Oceanic Waveguide.- 10.5.3 Surface Mode. "Rigid Cover" Condition.- 10.5.4 Internal Modes.- 10.6 Exercises.- 11. Waves in Rotating Fluids.- 11.1 Inertial (Gyroscopic) Waves.- 11.1.1 The Equation for Waves in a Homogeneous Rotating Fluid.- 11.1.2 Plane Harmonic Inertial Waves.- 11.1.3 Waves in a Fluid Layer. Application to Geophysics.- 11.2 Gyroscopic-Gravity Waves.- 11.2.1 General Equations. The Simplest Model of a Medium.- 11.2.2 Classification of Wave Modes.- 11.2.3 Gyroscopic-Gravity Waves in the Ocean.- 11.3 The Rossby Waves.- 11.3.1 The Tangent of ?-Plane Approximation.- 11.3.2 The Barotropic Rossby Waves.- 11.3.3 Joint Discussion of Stratification and the ?-Effect.- 11.3.4 The Rossby Waves in the Ocean.- 11.4 Exercises.- 12. Sound Waves.- 12.1 Plane Waves in Static Fluids.- 12.1.1 The System of Linear Acoustic Equations.- 12.1.2 Plane Waves.- 12.1.3 Generation of Plane Waves. Inhomogeneous Waves.- 12.1.4 Sound Energy.- 12.2 Sound Propagation in Inhomogeneous Media.- 12.2.1 Plane Wave Reflection at the Interface of Two Homogeneous Media.- 12.2.2 Some Special Cases. Complete Transparency and Total Reflection.- 12.2.3 Energy and Symmetry Considerations.- 12.2.4 A Slowly-Varying Medium. Geometrical-Acoustics Approximation.- 12.2.5 Acoustics Equations for Moving Media.- 12.2.6 Guided Propagation of Sound.- 12.3 Spherical Waves.- 12.3.1 Spherically-Symmetric Solution of the Wave Equation.- 12.3.2 Volume Velocity or the Strength of the Source. Reaction of the Medium.- 12.3.3 Acoustic Dipole.- 12.4 Exercises.- 13. Magnetohydrodynamics.- 13.1 Basic Concepts of Magnetohydrodynamics.- 13.1.1 Fundamental Equations.- 13.1.2 The Magnetic Pressure. Freezing of the Magnetic Field in a Fluid.- 13.1.3 The Poiseuille (Hartmann) Flow.- 13.2 Magnetohydrodynamic Waves.- 13.2.1 Alfven Waves.- 13.2.2 Magnetoacoustic Waves.- 13.2.3 Fast and Slow Magnetoacoustical Waves.- 13.3 Exercises.- 14. Nonlinear Effects in Wave Propagation.- 14.1 One-Dimensional Nonlinear Waves.- 14.1.1 The Nonlinearity Parameter.- 14.1.2 Model Equation. Generation of Second Harmonics.- 14.1.3 The Riemann Solution. Shock Waves.- 14.1.4 Dispersive Media. Solitons.- 14.2 Resonance Wave Interaction.- 14.2.1 Conditions of Synchronism.- 14.2.2 The Method of Slowly-Varying Amplitudes.- 14.2.3 Multiwave Interaction.- 14.2.4 Nonlinear Dispersion.- 14.3 Exercises.- Appendix: Tensors.- Bibliographical Sketch.

Kinetic Theory of Plasma Waves

Abstract: In this paper, I outline the solution of Vlasov-Maxwell's equations with given initial conditions. When transients have died out, the temporal evolution of each spatial Fourier component is completely determined by a dispersion relation. I derive the electrostatic dispersion relation and discuss the resonant interaction between particles and electrostatic waves. A new derivation of the wave energy density in a plasma with arbitary dissipation is given. The numerical solution of the dispersion relation of waves in a Maxwellian plasma is discussed, and finally I show some examples of numerically evaluated dispersion surfaces.

Speed of sound in the solar interior

Abstract: The sound speed of the solar interior is directly determinable on the basis of the frequencies of solar 5-min oscillations, irrespective of solar model, and relying only on a simple asymptotic description of the oscillations in terms of trapped acoustic waves. It is plausible that, by using this asymptotic determination as an initial trial in a more accurate inversion, and imposing constraints of smoothness on the solution resulting from the iteration, a good model representing the large scale structure of the sun which satisfies the observed frequencies may be determined.

Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media

Abstract: A unified approach to the study of reflection and refraction of elastic waves in general anisotropic media is presented. Christoffel equations and boundary conditions for both anisotropic media in coordinate systems formed by incident and interface planes, rather than in crystallographic coordinates, are considered. Consideration of wave propagation in an acoustic‐axis direction is included in the general algorithm, so results can be obtained both generally and for planes of symmetry, including planes of isotropy. General features of the numerical results are discussed. Energy conversion coefficients are shown to satisfy reciprocity relations which are formulated. It is much more natural to consider intensity–conversion ratios, rather than amplitude–conversion ratios, showing the important role of ray (rather than wave‐vector) directions in describing phenomena such as grazing angles. In particular, it is shown that the incident wave vector for grazing incidence may be greater or less than 90°: The domain of incident wave‐vector angles can actually split into disjoint pieces. The reflection coefficient at grazing incidence is shown to be unity, as in the isotropic case. Critical‐angle phenomena are described naturally by this approach.

Traveling waves and chaos in convection in binary fluid mixtures.

Abstract: Rayleigh-B\'enard convection is studied in alcohol-water mixtures in which the diffusion of concentration opposes convection via the Soret effect Near onset, the convective rolls are found to move continuously as traveling waves, in contrast to the stationary roll patterns observed in homogeneous fluids Dependent upon the temperature difference across the fluid layer (ie, Rayleigh number), these traveling-wave states are either periodic or chaotic At larger Rayleigh numbers, time-independent flow is observed which is the same as that expected for the homogeneous fluid mixture

Excitation conditions of flexural traveling waves for a reversible ultrasonic linear motor

Abstract: This paper presents a theory and experiments on a reversible ultrasonic linear motor, consisting of a thin beam, two ultrasonic transducers, and a slider. The slider rides upon the crests of transverse traveling flexure waves propagating down the beam from one transducer to the other.

Electromagnetic wave theory

Abstract: Electrostatics and magnetostatics quasi-static current flow in heterogeneous conductors scalar waves in one, two, and three dimensions the electromagnetic field guided cylindrical waves basic radiation fields propagation and radiation in complicated media. Appendix: some essential results from vector analysis.

Thermal and plasma wave depth profiling in silicon

Abstract: We describe a depth‐profiling concept using the critically damped plasma wave corresponding to the propagation of the free‐carrier plasma density generated by a modulated laser in a semiconductor.

Nonlinear guided wave applications

Abstract: Potential device applications of third-order nonlinear phenomena in guided wave structures are discussed. One of the waveguiding media is assumed to exhibit an intensity-dependent refractive index that results in both an intensity-dependent propagation wavevector and field distribution for the guided waves. These two characteristics lead to many interesting all-optical devices whose operating principles and power levels are outlined here. For example, the intensity-dependent field patterns can be used for thresholding and switching operations in a waveguide context. The intensity-dependent wavevector, used in conjuction with a distributed input or output coupler, can lead to devices such as optical limiters and light-controlled spatial scanners. When gratings are used as distributed feedback elements within a nonlinear waveguide, a whole class of novel devices such as bistable switches and optically tunable optical filters should be feasible.

A general formulation of focus wave modes

Abstract: A general formulation is given for electromagnetic pulses which remain localized in a multidimensional space, and which propagate at the speed of light without dispersing (focus wave modes). It is shown that such modes necessarily have infinite electromagnetic energy in the source‐free, three‐dimensional space. Finite‐energy focus wave modes cannot exist without sources. A set of complete focus wave modes with Hermite–Gaussian transverse variation is derived. The relation of focus wave modes to the solutions of the paraxial wave equation is established.