Showing papers on "Longitudinal wave published in 1986"

Wave Interactions and Fluid Flows

Abstract: Part I. Introduction: 1. Introduction Part II. Linear Wave Interactions: 2. Flows with piecewise-constant density and velocity 3. Flows with constant density and continuous velocity profile 4. Flows with density stratification and piecewise-constant velocity 5. Flows with continuous profiles of density and velocity 6. Models of mode coupling 7. Eigenvalue spectra and localized disturbances Part III. Introduction to Nonlinear Theory: 8. Introduction to nonlinear theory Part IV. Waves and Mean Flows: 9. Spatially-periodic waves in channel flows 10. Spatially-periodic waves on deformable boundaries 11. Modulated wave-packets 12. Generalized Lagrangian mean (GLM) formulation 13. Spatially-periodic means flows Part V. Three-wave Resonance: 14. Conservative wave interactions 15. Solutions of the conservative interaction equations 16. Linearly damped waves 17. Non-conservative wave interactions Part VI. Evolution of a Nonlinear Wave-Train: 18. Heuristic derivation of the evolution equations 19. Weakly nonlinear waves in inviscid fluids 20. Weakly nonlinear waves in shear flows 21. Properties of the evolution equations 22. Waves of larger amplitude Part VII. Cubic Three- and Four-wave Interactions: 23. Conservative four-wave interactions 24. Mode interactions in Taylor-Couette flow 25. Rayleigh-Benard convection 26. Wave interactions in planar shear flows Part VIII. Strong Interactions, Local Instabilities and Turbulence: A Postscript: 27. Strong interactions, local instabilities and turbulence: A postscript References Index.

Dynamics of a fluid-driven crack in three dimensions by the finite difference method

Abstract: The finite difference method is applied to the study of the dynamics of a three-dimensional fluid-filled crack excited into resonance by the sudden failure of a small barrier of area ΔS on the crack surface. The impulse response of the crack is examined for various ratios of crack width to crack length and for several values of the crack stiffness C = (b/μ)(L/d), where b is the bulk modulus of the fluid, μ is the rigidity of the solid, and L and d are the crack length and crack thickness, respectively. The motion of the crack is characterized by distinct time scales representing the duration of brittle failure and the periods of acoustic resonance in the lateral and longitudinal dimensions of the source. The rupture has a duration proportional to the area of crack expansion and is the trigger responsible for the excitation of the crack into resonance; the resonant periods are proportional to the crack stiffness and to the width and length of the crack. The crack wave sustaining the resonance is analogous to the tube wave propagating in a fluid-filled borehole. It is dispersive, showing a phase velocity that decreases with increasing wavelength. Its wave speed is always lower than the acoustic velocity of the fluid and shows a strong dependence on the crack stiffness, decreasing as the stiffness increases. The initial motion of the crack surface is an opening, and the radiated far-field compressional wave starts with a strong but brief compression which has a duration proportional to the crack stiffness and size of the rupture area; the amplitude of this pulse increases with the area of rupture but decreases with increasing stiffness. Flow into the newly created cavity triggers a pressure drop in the fluid, which produces a partial collapse of the wall propagated over the crack surface at the speed of the crack wave. The collapse of the crack surface generates a weak long-period component of dilatation following the compressional first motion in the far-field P wave train; the dilatational component is clearer in the signal from stiffer cracks when seen in the direction of the rupture. The energy loss by radiation is stronger for high frequencies, resulting in a progressive enrichment of the crack response in lower frequencies over the duration of resonance. These source characteristics translate into a far-field signature that is marked by a high-frequency content near its onset and dominated by a longer-period component in its coda. The source duration shows a strong dependence on the fluid viscosity and associated viscous damping at the crack wall.

Decay instability of finite-amplitude circularly polarized Alfven waves - A numerical simulation of stimulated Brillouin scattering

Abstract: By means of a numerical simulation, nonlinear evolution of large amplitude dispersive Alfven waves is studied. An energy transfer from the parent wave to two daughter Alfven-like waves and a soundlike wave is observed (a stimulated Brillouin scattering process). The observed growth rates and propagation characteristics of these daughter waves agree with the analytical results, which we obtain by extending the previous treatments by Goldstein, Derby, Sakai, and Sonnerup. Ions are first trapped by the electrostatic potential of the daughter soundlike waves. Along with the eventual decay (ion Landau damping) of the soundlike waves, ions are phase-mixed and left heated in the parallel direction. The increased parallel energy of ions is transferred to the perpendicular thermal energy through the nonresonant scattering process in the colliding Alfven waves (parent and daughter waves). We further observe that the daughter Alfven waves, which still have a large amplitude, are also unstable for further decay, and that the wave energy is continuously transferred to the longer wavelength regime (inverse cascading process).

Numerical simulation of nonoscillatory mirror waves at the Earth's magnetosheath

Abstract: The generation of nonoscillatory mirror waves is studied using a one-dimensional periodic hybrid electromagnetic simulation. The ion dynamics are treated exactly; the electrons are approximated as a finite pressure, massless fluid. Compression of the flux tubes in the magnetosheath causes a large pressure anisotropy, and it has been proposed that this anisotropy drives a mirror instability. The mirror waves have been identified by large amplitude fluctuations of the magnetic field, anticorrelated with pressure fluctuations. The simulations are initiated in a homogeneous high beta (beta = 2.5) plasma with the ambient magnetic field at various angles to the simulation axis. It is found that ion cyclotron waves are also driven by the pressure anisotropy, in competition with the nonoscillatory mirror waves. Simulations indicate that in a pure ¹H+ plasma the much faster growing ion cyclotron waves absorb the free energy in the anisotropy to the extent that mirror waves should not be observed. Analysis of the dispersion relations of mirror waves and ion cyclotron waves in the multi-component plasma indicates that 4He2+ and 16O6+ ions in the solar wind should stabilize the ion cyclotron waves sufficiently that the mirror waves become the dominant instability.

Nonlinear waves in compacting media

Abstract: An investigation of the mathematical model of a compacting medium proposed by McKenzie (1984) for the purpose of understanding the migration and segregation of melts in the Earth is presented. The numerical observation that the governing equations admit solutions in the form of nonlinear one-dimensional waves of permanent shape is confirmed analytically. The properties of these solitary waves are presented, namely phase speed as a function of melt content, nonlinear interaction and conservation quantities. The information at hand suggests that these waves are not solitons.

Analytical Studies of Body Wave Propagation and Attenuation

Abstract: : In situ crosshole and downhole seismic methods are becoming widely used as a means of nondestructively evaluating the elastic properties of geotechnical systems. The elastic constants are calculated from the records of body waves (longitudinal and transverse waves) traveling through the media. Measurements are made by generating a seismic disturbance at one point and measuring the time required for the disturbance to travel to one or more seismic receivers. Several simplifying assumptions are made in traditional analysis of seismic measurements for engineering purposes. These include assuming plane wave propagation, measurement of only far-field waves, and independence of the measurements on the source-receiver configuration and on the amount of material damping. An analytical study of the effects and the validity of the different assumptions is presented. It is found that, for the range of distances and frequencies typically used in engineering applications: body wave fronts generated by point source cannot be considered plane; and near-field effects associated with spherical wave fronts can be very important. The near-field effects are caused by coupling between waves which exhibit the same particle motion but which propagate at different velocities and attenuate at different rates. To minimize the detrimental effects of near-field waves in those methods based on spectral analysis techniques, it is recommended that, in the field setup, the ratio of distances from the source to the second and first receivers be of the order of two or greater.

Piezoelectric shear wave resonator and method of making same

Abstract: An acoustic shear wave resonator comprising a piezoelectric film having its C-axis substantially inclined from the film normal such that the shear wave coupling coefficient significantly exceeds the longitudinal wave coupling coefficient, whereby the film is capable of shear wave resonance, and means for exciting said film to resonate. The film is prepared by deposition in a dc planar magnetron sputtering system to which a supplemental electric field is applied. The resonator structure may also include a semiconductor material having a positive temperature coefficient of resonance such that the resonator has a temperature coefficient of resonance approaching 0 ppm/°C.

Optics of Floquet-Bloch waves in dielectric gratings

Abstract: The behavior of light in dielectric gratings is discussed in terms of the optical Floquet-Bloch waves (or modes). The emphasis is on the development of a good physical understanding of the nature of these waves, using the wavevector diagram to summarize their spatial dispersion and spectra. It is shown that Floquet-Bloch theory offers some advantages conceptually over the commonly used coupled-wave theory, because the rays of the Floquet-Bloch waves (given by their group velocities) play the same role in a periodic medium as do those of plane waves in isotropic or graded-index media. The effect on power conservation of truncating the Floquet expansions for the Floquet-Bloch waves is considered in detail. Using the greater intuitive power of Floquet-Bloch theory, it is shown (in contrast to recent claims to the contrary) how rigorous coupled-wave theory can be applied to symmetrical reflection gratings, and secondly how the light in these gratings can be viewed in terms of the multiple-beam interference of Floquet-Bloch waves, leading to behavior reminiscent of a low-finesse Fabry-Perot cavity.

Modulation instability of circularly polarized Alfvén waves

Abstract: The stability of a finite amplitude circularly polarized Alfven wave of wave number k0 is studied by using the two-fluid isentropic equations. Linear perturbation analysis, involving two sideband transverse waves having wave numbers k0 ± k and a longitudinal wave with wave number k, is used to find the exact sixth-order dispersion relation. The analysis is then limited to the case where k ≪ k0. The resulting fourth-order dispersion relation is examined analytically and numerically, and a surface is found that separates stable and unstable regions in parameter space. This surface describes the boundary between stable and unstable regions not only for k ≪ k0 but for the entire branch of the dispersion relation which extends to k = 0. We refer to this branch as the modulation branch and the corresponding instability as a modulation instability. A sufficient condition for modulation stability is found to be υϕ0 cs for right-hand polarized waves, where υϕ0 and cs are the phase velocity of the unperturbed wave and the unperturbed sound speed, respectively. Modulation wave amplitudes and growth rates are given.

Traveling and standing waves in binary-fluid convection in finite geometries.

Abstract: The effect of finite geometry on the competition between traveling waves and standing waves in systems with a Hopf bifurcation to a state with spatial structure is considered in the linear and weakly nonlinear regimes. The spatial structure observed by Kolodner et al. in binary-fluid convection is explained in terms of the reflection of the linear traveling waves. The reflection coefficient is calculated, and is found to go to zero as the frequency of the waves becomes small. The pattern expected in a saturated nonlinear state is discussed.

A global theory of internal solitary waves in two-fluid systems

Abstract: : The study of single-crested progressing gravity waves was initiated over a century ago with the observations by Russell of what he termed solitary waves, which progressed without change of form over a considerable distance on the Glasgow-Edinburgh Canal. The mathematical analysis of this wave motion on the surface of water, begun in the nineteenth century, has undergone a rapid development in the last three decades, due to the scattering theory for the Korteweg-de Vries equation, which models the motion of long waves due to the development of techniques in nonlinear analysis allowing for the analysis of finite amplitude motions. The work on surfce waves has many parallels in the study of waves in fluids with variable density. In the case of a heterogeneous fluid with a free upper surface, gravity waves still occur, in analogy with surface waves in a fluid of constant density. What is distinctive about a fluid with density stratification, however, is the presence of waves which are predominantly due to the stratification and not to the free surface. These waves, called internal waves, exist in a heterogeneous fluid even when it is confined between horizontal boundaries, a configuration which precludes gravity waves in a fluid of constant density. This paper is concerned with progressing solitary gravity waves in a system consisting of two fluids of differing densities confined in a channel of unit depth and infinite horizontal extent.

Generation mechanism of internal waves by tidal flow over a sill

Abstract: The generation mechanism of internal waves by relatively strong tidal flow over a sill is clarified analytically. Special attention is directed to the role of the tidal advection effect, which is examined by use of characteristics. An internal wave which propagates upstream is gradually formed over the sill through the interference among infinitesimal amplitude internal waves (elementary waves) emanated from the sill at each instant of time. In the accelerating (decelerating) stage of tidal flow, the effective amplification of an internal wave takes place as the Froude number exceeds (falls below) unity, because during this period the internal wave slowly travels downstream (upstream), crossing over the sill, where elementary waves are efficiently superimposed. When the strength of the tidal advection effect is appropriate, the internal wave formed in the accelerating stage (Ac wave) and that formed in the decelerating stage (Dc wave) overlap, so that the resultant wave height becomes very large. Since the relative position of Ac and Dc waves varies depending on the strength of the tidal advection effect, the resultant internal wave form is strongly affected by it. When the tidal advection effect is strong, the Ac wave is carried too far downstream to interfere with the Dc wave, and hence the resultant wave form consists of two crests and two troughs. When the tidal advection effect is moderate, the optimum interference between Ac and Dc waves occurs, and the resultant wave form consists of one crest and one trough, with its horizontal scale approaching that of the sill. When the tidal advection effect is weak, the internal wave height becomes very small, since elementary waves propagate almost freely in the tidal flow without coming close together, and the resultant wave form approaches a sinusoidal wave of tidal frequency.

Longitudinal wave propagation in a generalized thermoelastic cylinder

Abstract: In this paper the longitudinal wave propagation in a circular infinite cylinder is studied. The infinite circular cylinder is assumed to be made of a generalized thermoelastic material. The dispersion relation is obtained for the case in which the temperature is kept constant on the surface of the cylinder. Because of the complexity of the dispersion relation, the numerical solutions are given. For various values of parameters appearing in the field equations, some dispersion, attenuation, and phase velocity diagrams are presented

Shock waves for compressible navier‐stokes equations are stable

Abstract: It is shown that shock waves for the compressible Navier-Stokes equations are nonlinearly stable. A perturbation of a shock wave tends to the shock wave, properly translated in phase, as time tends to infinity. Through the consideration of conservation of mass, momentum and energy we obtain an a priori estimate of the amount of translation of the shock wave and the strength of the linear and nonlinear diffusion waves that arise due to the perturbation. Our techniques include the energy method for parabolic-hyperbolic systems, the decomposition of waves, and the energy-characteristic method for viscous conservation laws introduced earlier by the author.

Compressional wave hyperthermia treating method and apparatus

Abstract: A wide bandwidth compressional wave transducer array obtains information concerning tissue of a subject being treated and supplies hyperthermia compressional wave treating energy to a treated region of the subject. The array transducers are pulsed on and off with an on duty cycle portion of less than one. An array of compressional wave imaging transducers is in the center of the array of compressional wave hyperthermia focused far field transducers used to analyze and treat. The power and duty cycle of the compressional wave hyperthermia focused far field are varied to control the energy incident on certain regions in a treated subject. The ultrasonic frequencies of hyperthermia beams derived from individual transducers are randomly angle modulated so the energy of adjacent far field focused beams overlaps to a greater extent than focused coherent beams.

Reflection and transmission of an obliquely incident wave by an array of spherical cavities

Abstract: A plane wave is incident on a doubly periodic array of spherical cavities in an elastic solid. The cavities are of equal radius d, and their centers are located in a single plane, the x1x2 plane, at positions x1=ma, x2=nb. The propagation vector of a plane, time‐harmonic, incident longitudinal wave is located in the x1x3 plane. The scattering problem is formulated rigorously by taking advantage of the geometrical periodicity. The reflected and transmitted longitudinal and transverse wave motions may be expressed as superpositions of an infinite number of wave modes, each with its own cutoff frequency. Reflection and transmission coefficients have been defined as integrals over a single cavity in terms of the displacement components and auxiliary surface traction terms on the surface of the cavity. The system of singular integral equations for the displacement components has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the re...

Compressional wave propagation in liquid and/or gas saturated elastic porous media

Abstract: Concepts from the theory of interacting continua are employed to develop constitutive relations for liquid and/or gas saturated elastic porous media. The model is formulated by defining intrinsic stress tensors and densities in terms of the partial stress tensors, partial densities, and actual volume fractions occupied by each component. It is assumed that the constitutive law for each component as a single continuum relates intrinsic pressure to intrinsic deformation. Relative motion between the constituents is allowed through simple Darcy‐type expressions. The governing equations together with the constitutive relations are used to investigate the propagation of both harmonic and transient pulses. In general three modes of wave propagation exist. In the case of a transient pulse, these modes lead to a three‐wave structure. Laplace transform techniques are used to derive closed‐form solutions for transient loading for two limiting values of viscous coupling (i.e., weak viscous coupling, strong viscous co...

Negative energy waves in hydrodynamics

Abstract: The utility of the concept of negative energy waves (NEW) in hydrodynamics is discussed. Examples are given of the excitation of waves by flow past elastic membranes, and of the amplification and generation of capillary-gravity and internal waves in liquids in the presence of vertically inhomogeneous flows. The concepts of "linear" and "nonlinear" energy are introduced, and it is shown that energy defined as the first integral of the equations of motion linearized against the flow background can be negative, whereas the inclusion of all the quadratic terms in the expression for the energy can give a positive value. Nonlinear processes associated with NEW are also discussed, as is the radiation instability of oscillators in hydrodynamics. The review is largely based on the authors' own work.

Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves

Abstract: INTRODUCTION TO IMAGE FORMATION AND FOCUSING Brief history Applications Combined method of ray tracing and diffraction DIFFRACTION OF SCALAR WAVES Diffraction of three-dimensional waves Diffraction of two-dimensional waves Ray connection of Huygens principle Numerical methods for evaluating diffraction integrals ASYMPTOTIC THEORIES OF DIFFRACTION Method of stationary phase for single integrals Method of stationary phase for double integrals Asymptotic theory of diffraction for two-dimensional waves Asymptotic theory of diffraction for three-dimensional waves FOCUSING OF SCALAR WAVES Three-dimensional waves Two-dimensional waves Computation of focused fields DIFFRACTION AND FOCUSING OF ELECTROMAGNETIC WAVES Diffraction theory Focusing problems ZONE-PLATE LENSES OF SOUND Zone-plate lenses for sound waves in water Zone-plate lens for focusing of ocean swells FOCUSING OF WATER WAVES Linear waves Nonlinear waves REFERENCES INDEX

Biot‐Gardner theory of extensional waves in porous rods

Abstract: Many measurements have been made on fluid‐saturated porous rods executing extensional, flexural, and torsional motion. Measurements for extensional and flexural motion yield a loss parameter for Young's modulus waves QY, and the measurement for torsional motion yields QS for shear waves. QP has then been calculated for compressional waves in bulk rock, on the assumption that the fluid‐saturated rock is an isotropic solid. I point out the fallacy of computing Qp from these measurements and also urge workers to recognize the losses due to simple fluid viscosity in interpreting their data on extensional waves in rods. By application of published theory, I show that peaks in attenuation of extensional waves are to be expected at frequencies of several hertz to several kilohertz, depending upon rod radius. Computed curves are compared with published measurements on Navajo sandstone saturated with water, ethanol, and n‐decane. In each case, computed peak frequency agrees with published measurements. Shift of th...

Symmetry and the bifurcation of capillary-gravity waves

Abstract: Chen & Saffman [4] have given a weakly nonlinear theory of steady periodic two-dimensional irrotational waves on the surface of a perfect fluid of infinite depth which takes account of the effects of gravity and surface tension. They remark that the question of existence of such waves is “a non-trivial and still incompletely solved problem”. It is our purpose in the present paper to give a rigorous mathematical account of the existence theory for such waves of small amplitude. In so doing we are able to give a firm basis for the more formal aspects of Chen & Saffman’s work, and to vindicate many of their main conclusions. Our work will be couched throughout in the language of modern bifurcation theory, and is a consequence of the Lyapunov-Schmidt reduction procedure in the presence of certain symmetry considerations. Other studies have contributed to the rigorous theory, but it is only possible to comment further once the problem has been described in detail.

Flow patterns and nonlinear behavior of traveling waves in a convective binary fluid

Abstract: Flow visualization, heat-transprot measurements, and the light-intensity profile as a function of time have been used to study nonlinear propagating waves in ethanol-water mixtures heated from below. The experimental results reveal the main features of the travelling waves predicted by recent theory.

The limiting configuration of interfacial gravity waves

Abstract: Progressive gravity waves at the interface between two unbounded fluids are considered. The flow in each fluid is taken to be potential flow. The problem is converted into a set of integrodifferential equations, reduced to a set of algebraic equations by discretization, and solved by Newton’s method together with parameter variation. Meiron and Saffman’s [J. Fluid Mech. 129, 213 (1983)] calculations showing the existence of overhanging waves are confirmed. However, the present calculations do not support Saffman and Yuen’s [J. Fluid Mech. 123, 459 (1982)] conjecture that the waves are geometrically limited (i.e., that solutions exist until the interface intersects itself). It is proposed that along a solution branch starting with sinusoidal waves of small amplitude, one reaches solutions with vertical streamlines and then overhanging waves. Continuing on this branch, one returns to nonoverhanging waves and then back toward a wave with vertical streamlines. It is suggested that this succession of patterns ...

A plane-wave decomposition for elastic wave fields applied to the separation of P-waves and S-waves In vector seismic data

Abstract: We describe a method to decompose a two‐dimensional (2-D) elastic wave field recorded along a line into its longitudinal and transverse parts, that is, into compressional (P) waves and shear (S) waves. Separation of the data into P-waves and S-waves is useful when analyzing vector seismic measurements along surface lines or in boreholes. The method described is based on a plane‐wave expansion for elastic wave fields and is illustrated with a synthetic example of an offset vertical seismic profile (VSP) in a layered elastic medium.