Showing papers on "Wave propagation published in 1993"

Photonic band-gap structures

Abstract: The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron-wave propagation in real crystals has proven to be a fruitful one. Initial efforts were motivated by the prospect of a photonic band gap. a frequency band in three-dimensional dielectric structures in which electromagnetic waves are forbidden irrespective of the propagation direction in space. Today many new ideas and applications are being pursued in two and three dimensions and in metallic, dielectric, and acoustic structures. We review the early motivations for this research, which were derived from the need for a photonic band gap in quantum optics. This need led to a series of experimental and theoretical searches for the elusive photonic band-gap structures, those three-dimensionally periodic dielectric structures that are to photon waves as semiconductor crystals are to electron waves. We describe how the photonic semiconductor can be doped, producing tiny electromagnetic cavities. Finally, we summarize some of the anticipated implications of photonic band structure for quantum electronics and for other areas of physics and electrical engineering.

Alternative form of Boussinesq equations for nearshore wave propagation

Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...

Boundary elements in dynamics

Abstract: Part 1 Basic equations and fundamentals of wave propagation: basic equations of linear elastodynamics formulation of the elastodynamic problem in terms of dilatation and rotation. Part 2 Harmonic problems - integral formulation and boundary elements for two dimensions: plane harmonic waves in elastic solids reflection and refraction of plane harmonic elastic waves. Part 3 Harmonic problems - boundary elements for three dimensions: scalar wave propagation computer code for three-dimensional scalar wave propagation using constant elements [CONDTHSH]. Part 4 Time domain formulation: fundamental solution and boundary integral formulation for transient scalar wave propagation two-dimensional formulation. Part 5 Dual reciprocity approach for transient elastodynamics: boundary integral formulation transformation of the domain integral. Part 6 Dynamic fracture mechanics: some basic fracture mechanics ideas dynamic stress intensity factors. Part 7 Dynamic soil-structure interaction: dynamic stiffness of foundations three-dimensional foundations. Part 8 Dynamic analysis of dam-soil-reservoir systems: some characteristics of the B.E. model gravity dams. Part 9 Dynamic analysis of poroelastic media: basic equations for dynamic poroelasticity boundary integral formulation. (Part Contents).

Jakes fading model revisited

Abstract: The Jakes fading model is a deterministic method for simulating time-correlated Rayleigh fading waveforms and is still widely used today. However, since it is difficult to create multiple uncorrelated fading waveforms with this model, the authors propose modifications to the model which solve this problem.< >

Accurate theoretical analysis of photonic band-gap materials.

Abstract: Two improvements for the solution of Maxwell's equations in periodic dielectric media are introduced, abandoning the plane-wave cutoff and interpolating the dielectric function. These improvements permit the accurate study of previously inaccessible systems. Example calculations are discussed, employing a basis of \ensuremath{\sim}${10}^{6}$ plane waves for which these two improvements reduce both the memory and central processing unit requirements by \ensuremath{\sim}${10}^{4}$.

Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge

Abstract: Light propagation in strongly scattering media can be described by the diffusion approximation to the Boltzmann transport equation. We have derived analytical expressions based on the diffusion approximation that describe the photon density in a uniform, infinite, strongly scattering medium that contains a sinusoidally intensity-modulated point source of light. These expressions predict that the photon density will propagate outward from the light source as a spherical wave of constant phase velocity with an amplitude that attenuates with distance r from the source as exp(-alpha r)/r. The properties of the photon-density wave are given in terms of the spectral properties of the scattering medium. We have used the Green's function obtained from the diffusion approximation to the Boltzmann transport equation with a sinusoidally modulated point source to derive analytic expressions describing the diffraction and the reflection of photon-density waves from an absorbing and/or reflecting semi-infinite plane bounded by a straight edge immersed in a strongly scattering medium. The analytic expressions given are in agreement with the results of frequency-domain experiments performed in skim-milk media and with Monte Carlo simulations. These studies provide a basis for the understanding of photon diffusion in strongly scattering media in the presence of absorbing and reflecting objects and allow for a determination of the conditions for obtaining maximum resolution and penetration for applications to optical tomography.

Beyond Clausius-Mossotti: Wave Propagation on a Polarizable Point Lattice and the Discrete Dipole Approximation

Abstract: We derive the dispersion relation for electromagnetic waves propagating on a lattice of polarizable points. From this dispersion relation we obtain a prescription for choosing dipole polarizabilities so that an infinite lattice with finite lattice spacing will mimic a continuum with dielectric constant. The discrete dipole approximation is used to calculate scattering and absorption by a finite target by replacing the target with an array of point dipoles. We compare different prescriptions for determining the dipole polarizabilities. We show that the most accurate results are obtained when the lattice dispersion relation is used to set the polarizabilities.

Photonic band gaps and localization

Abstract: Localization, Diffusion, and Correlation: The Localization of Light S. John. The Speed of Diffusing Light B.A. van Tiggelen, A. Lagendijk. Diffusion of Classical Waves in Random Media C.M. Soukoulis, et al. Accurate Measurement of Backscattered Light from Random Media P.N. den Outer, et al. Photonic Band Gaps: Photonic Band Structure E. Yablonovitch. Photonic Gaps for Electromagnetic Waves in Periodic Dielectric Structures K.M. Ho, et al. Plane-Wave Calculation of Photonic Band Structure K.M. Leung. Measurements of Localization and Photonic Band Gap Systems in Two Dimensions S. Schultz, D.R. Smith. Wave Propagation in Random Media: Localization Transition in Anisotropic and Inhomogenous Systems P. Sheng, Z.Q. Zhang. Statistical Inversion of Stratified Media from Acoustic Pulses Scattered at Two Angles of Incidence B. White, et al. 30 additional articles. Index.

Wave Attenuation by Vegetation

Abstract: The vertically two-dimensional problem of small-amplitude waves propagating over submerged vegetation is formulated using the continuity and linearized momentum equations for the regions above and within the vegetation. The effects of the vegetation on the flow field are assumed to be expressible in terms of the drag force acting on the vegetation. An analytical solution is obtained for the monochromatic wave whose height decays exponentially. The expressions for the wave number and the exponential decay coefficient derived for arbitrary damping are shown to reduce to those based on linear wave theory and the conservation equation of energy if the damping is small. The analytical solution is compared with 60 test runs conducted using deeply submerged artificial kelp. The calibrated drag coefficients for these runs are found to vary in a wide range and appear to be affected by the kelp motion and viscous effects that are neglected in the analysis. The analytical solution is also shown to be applicable to subaerial vegetation, which is predicted to be much more effective in dissipating wave energy.

Simulation of three-dimensional optical waveguides by a full-vector beam propagation method

Abstract: Simulations of optical guided waves in three-dimensional waveguide structures by a full vector beam propagation method are described. Two sets of coupled equations governing the propagation of the transverse electric and magnetic fields are derived systematically. Polarization dependence and coupling due to the vectorial nature of the electromagnetic fields are considered in the formulations. The governing equations are solved subsequently by finite-difference schemes. The vector BPM is first assessed for a step-index circular fiber by comparing the numerical results with the exact analytical solutions. The guided modes of a rib waveguide are then investigated in detail. Comparisons among the scalar, semi-vector and full-vector simulations of the rib waveguide are made. Finally polarization rotation of a periodically loaded rib waveguide operated fully based on the vector nature of the electromagnetic waves is modeled and simulated. >

Twisted Gaussian Schell-model beams

Abstract: We introduce a new class of partially coherent axially symmetric Gaussian Schell-model (GSM) beams incorporating a new twist phase quadratic in configuration variables. This phase twists the beam about its axis during propagation and is shown to be bounded in strength because of the positive semidefiniteness of the cross-spectral density. Propagation characteristics and invariants for such beams are derived and interpreted, and two different geometric representations are developed. Direct effects of the twist phase on free propagation as well as on parabolic index fibers are demonstrated. Production of such twisted GSM beams, starting with Li-Wolf anisotropic GSM beams, is described.

Photonic band structure and defects in one and two dimensions

Abstract: We present an experimental and numerical study of electromagnetic wave propagation in one-dimensional (1D) and two-dimensional (2D) systems composed of periodic arrays of dielectric scatterers. We demonstrate that there are regions of frequency for which the waves are exponentially attenuated for all propagation directions. These regions correspond to band gaps in the calculated band structure, and such systems are termed photonic band-gap (PBG) structures. Removal of a single scatterer from a PBG structure produces a highly localized defect mode, for which the energy density decays exponentially away from the defect origin. Energy-density measurements of defect modes are presented. The experiments were conducted at 6–20 GHz, but we suggest that they may be scaled to infrared frequencies. Analytic and numerical solutions for the band structure and the defect states in 1D structures are derived. Applications of 2D PBG structures are briefly discussed.

Properties of photon density waves in multiple-scattering media

Abstract: Amplitude-modulated light launched into multiple-scattering media, e.g., tissue, results in the propagation of density waves of diffuse photons. Photon density wave characteristics in turn depend on modulation frequency (ω) and media optical properties. The damped spherical wave solutions to the homogeneous form of the diffusion equation suggest two distinct regimes of behavior: (1) a high-frequency dispersion regime where density wave phase velocity Vp has a ω dependence and (2) a low-frequency domain where Vp is frequency independent. Optical properties are determined for various tissue phantoms by fitting the recorded phase (ϕ) and modulation (m) response to simple relations for the appropriate regime. Our results indicate that reliable estimates of tissuelike optical properties can be obtained, particularly when multiple modulation frequencies are employed.

Downstream Development of Baroclinic Waves As Inferred from Regression Analysis

Abstract: The structure and evolution of transient disturbances in the Northern Hemisphere winter season are examined using one-point regression maps and longitude-height sections derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) operational analyses for seven winter seasons. With the use of unfiltered time series of normalized 300-mb meridional wind perturbations at a grid point in the Pacific storm track as the reference time series, regression statistics for perturbations in the horizontal wind, geopotential height, temperature, and vertical velocity are derived. The resulting perturbation fields exhibit characteristics of midlatitude baroclinic waves, such as a westward tilt with height in the velocity and height fields and eastward tilt in the temperature field, with typical wavelengths of 4000 km and periods of around 4 days. The main difference between the results of this work and previous similar analyses is in the propagation characteristics of the baroclinic wave trains....

Propagation velocity of perturbations in turbulent channel flow

Abstract: A database obtained from direct numerical simulation of a turbulent channel flow is analyzed to extract the streamwise component of the propagation velocity V of velocity, vorticity, and pressure fluctuations from their space-time correlations. A surprising result is that V is approximately the same as the local mean velocity for most of the channel, except for the near-wall region. For y(+) less than 15, V is virtually constant, implying that perturbations of all flow variables propagate like waves near the wall. In this region V is 55 percent of the centerline velocity Uc for velocity and vorticity perturbations and 75 percent of U sub c for pressure perturbations. This is equal to U at y(+) = 15 for velocity and vorticity perturbations, and equal to U at y(+) = 20 for pressure perturbations, indicating that the dynamics of the nearwall turbulence is controlled by turbulence structures present near y(+) about 15-20. Scale dependence of V is also examined by analyzing the bandpass-filtered flow fields. This paper contains comprehensive documentation on the propagation velocities, which should prove useful in the evaluation of Taylor's hypothesis.

Flow in deformable porous media. Part 1 Simple analysis

Abstract: Many processes in the Earth, such as magma migration, can be described by the flow of a low-viscosity fluid in a viscously deformable, permeable matrix. The purpose of this and a companion paper is to develop a better physical understanding of the equations governing these two-phase flows. This paper presents a series of analytic approximate solutions to the governing equations to show that the equations describe two different modes of matrix deformation. Shear deformation of the matrix is governed by Stokes equation and can lead to porosity-driven convection. Volume changes of the matrix are governed by a nonlinear dispersive wave equation for porosity. Porosity waves exist because the fluid flux is an increasing function of porosity and the matrix can expand or compact in response to variations in the fluid flux. The speed and behaviour of the waves depend on the functional relationship between permeability and porosity. If the partial derivative of the permeability with respect to porosity, ∂kϕ/∂ϕ, is also an increasing function of porosity, then the waves travel faster than the fluid in the pores and can steepen into porosity shocks. The propagation of porosity waves, however, is resisted by the viscous resistance of the matrix to volume changes. Linear analysis shows that viscous stresses cause plane waves to disperse and provide additional pressure gradients that deflect the flow of fluid around obstacles. When viscous resistance is neglected in the nonlinear equations, porosity shock waves form from obstructions in the fluid flux. Using the method of characteristics, we quantify the specific criteria for shocks to develop in one and two dimensions. A companion paper uses numerical schemes to show that in the full equations, viscous resistance to volume changes causes simple shocks to disperse into trains of nonlinear solitary waves.

Reaction diffusion patterns in the catalytic CO‐oxidation on Pt(110): Front propagation and spiral waves

Abstract: The dynamic behavior of elliptical front propagation and spiral‐shaped excitation concentration waves associated with the catalytic oxidation of CO on a Pt(110)‐surface was investigated by means of photoemission electron microscopy (PEEM) The properties of these patterns can be tuned through the control parameters, viz, the partial pressures of CO and O2 and the sample temperature Over a wide range of control parameters the transition between two metastable states (COad and Oad covered surface) proceeds via nucleation and growth of elliptical reaction‐diffusion (RD)‐fronts Front velocities and critical radii for nucleation are determined by the diffusion of adsorbed CO under reaction conditions If at constant pO2, T the CO partial pressure is increased beyond a critical value a transition to qualitatively different dynamic behavior takes place The elliptical fronts give way to oxygen spiral waves of excitation spreading across the CO‐covered areas For fixed experimental conditions a broad distribution of spatial wavelengths and temporal rotation periods was found This effect has to be attributed to the existence of surface defects of μm‐size to which the spiral tip is pinned These data lead to a dispersion relation between the front propagation velocity and the wavelength, respectively, period In addition, the dynamics of free spiral‐shaped excitation waves was investigated under the influence of externally modulated temperature Now the spiral starts to drift, resulting in distortion of the Archimedian shape and a pronounced Doppler effect

On the preferred source location for the convective amplification of ion cyclotron waves

Abstract: The propagation, growth and absorption of electromagnetic ion cyclotron waves in the Pc 1 frequency range is investigated using the HOTRAY ray tracing program for a realistic distribution of thermal plasma (H+, He+ and O+) that is assumed to be in diffusive equilibrium inside the plasmasphere and collisionless in the low-density region outside the plasmapause. Free energy for L- mode wave growth is provided by a bi- Maxwellian distribution of energetic H+ and O+ with a temperature and density modelled on satellite observations. Solutions to the hot plasma dispersion relation show that inside the plasmasphere the spatial growth rates are small whereas they increase outside the plasmapause with increasing L shell. Ray tracing shows that inside the plasmasphere guided L- mode waves only grow during one crossing of the magnetic equator and only achieve small path-integrated wave gain (≤ 2 e- foldings). At the plasmapause the density gradient enables guided mode waves to grow during several equatorial crossings and the net path-integrated gain is much larger (≃ 8.7 e-foldings). For the largest observed ring current densities of 4 × 106 m−3 at L = 4 the gain is above the critical level (10 e-foldings) for amplification to observable levels. Just outside the plasmapause the waves only grow during the first equatorial crossing and the gain is smaller. In the absence of nonconvective instabilities the path- integrated amplification of the guided mode tends to increase with L shell and reaches the critical level for observable waves only in the outer magnetosphere (L ≥ 7). Unguided L- mode waves have very small wave gain. For L ≥ 7 the plasma beta becomes large (β⊥ > 1) and should lead to the onset of nonconvective instabilities. However, we suggest that inhomogeneities in the medium and quasi-linear scattering will prevent absolute instabilities from occurring and that in reality the waves are propagating with very low group velocities. We suggest that the waves observed by Anderson et al. (1990, 1992a, b) beyond L = 7 near local noon are influenced by the enhanced wave gain due to these very low group velocity waves.

Photonic band-gap crystals

Abstract: The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron wave propagation in real crystals has proven to be a very fruitful one Initial efforts were motivated by the prospect of a photonic band gap, a frequency band in three-dimensional dielectric structures in which electromagnetic waves are forbidden, irrespective of propagation direction in space Today many new ideas and applications are being pursued in two and three dimensions, and in metallic, dielectric and acoustic structures, etc The author reviews the early motivations for this work, which were derived from the need for a photonic band gap in quantum optics This led to a series of experimental and theoretical searches for the elusive photonic band-gap structures, those three-dimensionally periodic dielectric structures which are to photon waves what semiconductor crystals are to electron waves Then he describes how the photonic semiconductor can be 'doped', producing tiny electromagnetic cavities Finally he summarizes some of the anticipated implications of photonic band structure for quantum electronics and the prospects for the creation of photonic crystals in the optical domain

Concepts and results for 3D digital terrain-based wave propagation models: an overview

Abstract: Mobile communication links are severely influenced by propagation effects. Wave propagation in the VHF/UHF frequency range over natural and man-made terrain is strongly dependent on topography and morphography. Propagation modeling is based on a ray-optical approach. Wave interactions, like diffraction and scattering, over the propagation path are described by the uniform theory of diffraction (UTD) and physical optics (PO). Propagation models for rural and urban areas are presented for 2-D and 3-D ray tracing. Near-range models apply to the corresponding areas in forest and urban sites. The field-strength delay spectrum describes ray contributions with deterministic amplitudes but statistical phases are used to derive time-and frequency-domain channel characteristics. Comparisons between measured and predicted data are presented. >

Instabilities of a propagating pulse in a ring of excitable media.

Abstract: Instabilities in the circulation of a pulse in a ring of excitable cardiac tissue are analyzed using two different formulations: (1) a reaction-diffusion partial differential equation (PDE) model for cardiac electrical activity using the Beeler-Reuter equations to represent ionic currents in the cardiac cells; (2) a neutral delay-differential equation that we propose as a model for the PDE. Stability analysis and numerical simulation of the delay equation agree with results from simulations of the PDE model.

Spectral Estimates of Gravity Wave Energy and Momentum Fluxes. Part I: Energy Dissipation, Acceleration, and Constraints

Abstract: The spectral characteristics of atmospheric gravity wave motions are remarkably uniform in frequency and wavenumber despite widely disparate sources, filtering environments, and altitudes of observation. This permits a convenient and useful means of describing mean spectral parameters, including energy density, anisotropy, energy and momentum fluxes, and wave influences on their environment. The purpose here is to provide a general formulation of the mean energy spectrum as well as estimates of the wave energy and momentum fluxes and the flux divergences expressed as the energy dissipation rate and the induced accelerations in the lower and middle atmosphere. These results show spectral observations to be consistent with independent estimates of energy dissipation rates and to suggest a high degree of anisotropy of the gravity wave field under conditions of strong wave filtering by large-scale, low-frequency motions. In two companion papers, these results are employed to construct a parameterizat...

Physical Modeling with the 2-D Digital Waveguide Mesh

Abstract: An extremely efficient method for modeling wave propagation in a membrane is provided by the multidimensional extension of the digital waveguide. The 2-D digital waveguide mesh is constructed out of bidirectional delay units and scattering junctions. We show that it coincides with the standard finite difference approximation scheme for the 2-D wave equation, and we derive the dispersion error. Applications may be found in physical models of drums, soundboards, cymbals, gongs, small-box reverberators, and other acoustic constructs where a one-dimensional model is less desirable.

Leaky-wave propagation and radiation for a narrow-beam multiple-layer dielectric structure

Abstract: Previous work has demonstrated that very narrow beam radiation patterns can be obtained from a simple source embedded within multiple dielectric layers of appropriate thicknesses above a ground plane. The configuration consists of dielectric layers having permittivities epsilon /sub 1/ and epsilon /sub 2/ stacked in an alternating arrangement, with epsilon /sub 2/> epsilon /sub 1/. This narrow-beam effect can be attributed to weakly attenuated leaky waves that exist on the structure. Simple asymptotic formulas for the propagation and attenuation constants are derived. The formulas show how the beamwidth varies with the number of layers and the material constants. The exact radiation pattern is compared with the leaky-wave pattern for a specific case to demonstrate the role of the leaky waves in determining the total pattern. >

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.