Showing papers on "Longitudinal wave published in 1987"

Elastic reverse-time migration

Abstract: Elastic, prestack, reverse-time, finite-difference migration of two-component seismic surface data requires data extrapolation and application of an imaging condition. Data extrapolation involves synchronous driving of the vertical-component and horizontal-component finite-difference meshes with the time reverse of the recorded vertical and horizontal traces, respectively. Extrapolation uses the coupled elastic wave equation for variable velocity solved with a second-order, explicit finite-difference scheme. The imaging condition at any point in the grid is the one-way traveltime from the source to that point.Elastic migrations of both synthetic test data and real two-component common-source gathers produce simpler images than acoustic migrations because of the coalescing of double reflections (compressional waves and shear waves) into single loci.

Perturbative inversion methods for obtaining bottom geoacoustic parameters in shallow water

Abstract: Three perturbative inverse methods for obtaining bottom geoacoustic parameters as a function of depth in shallow water are described. The required input data are the trapped mode eigenvalues for one or more frequencies, the group velocity dispersion curves for one or more modes, or the cw pressure field versus range. In each case, a Fredholm integral equation arises that can be solved using linear inverse theory, and for which resolution and variance estimates of the solution can be readily made. Attention is focused primarily on the modal eigenvalue inverse problem for which the theory for determining the compressional wave speed, compressional wave attenuation, and density is developed in detail. Properties of this technique are studied using synthetic data and include investigations of the dependence of the results on acoustic frequency, number of modes excited, and partial a priori knowledge of the bottom. The method is demonstrated on experimental data obtained in Nantucket Sound at 140 and 220 Hz. Directions of future research on these techniques are indicated.

Applied seismic wave theory

Abstract: Introduction. I. Capita Selecta from Vector Analysis. Scalar product, gradient, curl and divergence. Theorem of Stokes, theorem of Gauss and Green's theorem. II. One-Dimensional Discrete Spectral Analysis. The delta pulse and discrete functions. Fourier series of periodic time functions. Fourier integral of transients. Relationship between the discrete property and periodicity. Sampling and aliasing in time and frequency. Matrix formulations. Decomposition of a broad band experiment into monochromatic simulations. III. Multi-Dimensional Discrete Spectral Analysis. Basic theory. Spatial aliasing. Two-dimensional spectral analysis and plane wave decomposition. Extensions to three dimensions. IV. Vibrations. Basic concepts. Free vibrations. Forced vibrations. Coupled systems. From vibrations to waves. V. Acoustic Waves. Derivation of the acoustic wave equation. One-way versions of the acoustic wave equation. Plane waves. Spherical waves. Cylindrical waves. Principle of numerical modeling with the acoustic wave equation. VI. Elastic Waves. Compressional waves in homogeneous isotropic solids. Shear waves in homogeneous isotropic solids. Derivation of the elastic wave equation. One-way versions of the elastic wave equation. Principle of numerical modeling with the elastic wave equation. VII. Boundary Conditions. Reflection and transmission coefficients for acoustic boundaries. Reflection in terms of convolution. The fluid-solid boundary. Reflection and transmission coefficients for elastic boundaries. Summary. VIII. Kirchhoff and Rayleigh Integrals. Derivation of the Kirchhoff integral for homogeneous media. Derivation of the Rayleigh integrals for homogeneous media. Rayleigh integrals in terms of convolution. Transformation of Rayleigh integrals to the wave number domain. Kirchhoff and Rayleigh integrals for inhomogeneous fluid-like media. Rayleigh integrals as one-way versions of the Kirchhoff integral. Discrete version of the Kirchhoff integral. Discrete versions of the Rayleigh integrals. Summary. IX. Directivity Properties of Wave Fields. Fraunhofer approximations in terms of the Fourier integral. Directivity patterns. Far field expressions of scattered wave fields. Summary. X. Forward and Inverse Problems. Principle of one-way forward wave field extrapolation. Principle of one-way inverse wave field extrapolation. Principle of two-way techniques. Example. Summary. (Each chapter includes an introduction and references). Appendices. Subject Index.

Experimental Study of Instability Modes in a Three-dimensional Boundary Layer.

Abstract: Travelling waves in an unstable three-dimensional boundary layer are studied experimentally with the use of hot-wire anemometry. For the sake of realistic comparisons with stability theory, the tests were performed on a swept flat plate where infinite swept-wing conditions were approximated by means of contoured end plates. The required pressure gradient was imposed by a displacement body. The Reynolds number for the first appearance of travelling waves is roughly the same as that of stationary vortices. The frequencies of the most amplified waves depend on the Reynolds number. It is shown with the aid of a twin probe that in a more strongly amplified state, the travelling waves propagate in a direction different from that of the mean flow. Further upstream, where stationary waves first become visible in the oil-flow pattern, a uniform direction could not be identified. Under certain conditions, travelling waves of two frequency ranges are amplified that propagate in different directions. The present work is part of the transition experiment started at the DFVLR. It is closely connected to the theoretical work by Dallmann and Bieler.

Steepened magnetosonic waves at Comet Giacobini-Zinner

Abstract: Intense MHD waves at Comet Giacobini-Zinner were examined to investigate the mode and direction of wave propagation and thereby to provide important constraints on potential mechanisms for the wave origin in the vicinity of the comet. From observations of steepened wave forms, it is found that the waves must be propagating toward the sun but are blown back across the ICE spacecraft. The correlation between magnetic field magnitude and electron density enhancements indicates that these waves are fast magnetosonic mode emissions. The sense of rotation of the partial rotations are left-hand circularly polarized in the spacecraft frame, consistent with anomalously Doppler-shifted right-hand waves.

Singularities in the Transverse Fields of Electromagnetic Waves. I. Theory

Abstract: Electromagnetic waves generally contain three kinds of singularities called C lines, S surfaces and disclinations. These singularities are features of the transverse electric and transverse magnetic fields of the waves and all three kinds usually occur in any given wavefield. We show that in the case of nominally uniformly polarized waves, the simple line zeros predicted for interference fringes by scalar wave theory in fact have an underlying polarization structure consisting of two C lines and an S surface. In consequence, virtually all monochromatic electromagnetic waves contain polarization states ranging from right-hand circular, through linear to left-hand circular polarization. Singularities of the electric and magnetic fields are not generally coincident in space; in fact they can be separated by arbitrarily large distances. The separation of the electric and magnetic S surfaces means that there are regions where the transverse electric and transverse magnetic vectors counterrotate. C lines are probably the most significant of the singularities, since they are not only structural features of polarization, but also organize the time structure of electromagnetic waves. They play a crucial role in determining the topology of disclinations in paraxial wavefields. In pulsed electromagnetic waves all three singularities move through space. Their behaviour, including interactions between pairs of C lines, S surfaces or disclinations, which are likely to be frequent events in pulsed waves, is discussed.

The propagation of short surface waves on longer gravity waves

Abstract: To understand the imaging of the sea surface by radar, it is useful to know the theoretical variations in the wavelength and steepness of short gravity waves propagated over the surface of a train of longer gravity waves of finite amplitude. Such variations may be calculated once the orbital accelerations and surface velocities in the longer waves have been accurately determined – a non-trivial computational task.The results show that the linearized theory used previously for the longer waves is generally inadequate. The fully nonlinear theory used here indicates that for longer waves having a steepness parameter AK = 0.4, for example, the short-wave steepness can be increased at the crests of the longer waves by a factor of order 8, compared with its value at the mean level. (Linear theory gives a factor less than 2.)The calculations so far reported are for free, irrotational gravity waves travelling in the same or directly opposite sense to the longer waves. However, the method of calculation could be extended without essential difficulty so as to include effects of surface tension, energy dissipation due to short-wave breaking, surface wind-drift currents, and to arbitrary angles of wave propagation.

Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth

Abstract: A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetry breaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated.

Resonance theory of elastic waves ultrasonically scattered from an elastic sphere

Abstract: The interaction of elastic waves incident on an elastic spherical inhomogeneity is studied in detail, particularly in the resonance scattering regime. Incident and scattered compression and shear waves in lossless elastic media separate into three modes: a p mode for the compression wave, and s and t modes for the shear wave. A description of how the acoustic energy redistributes among these modes during the scattering process is contained in the scattering matrix that we separate here into background and resonance portions for the two extreme cases of a nearly soft and a nearly rigid elastic sphere. This produces farfield scattering amplitudes which are a superposition of a background contribution felt to contain reflected and Franz‐type circumferential waves and a resonance contribution that seems to contain refracted, Rayleigh, and whispering gallery waves. Limiting cases (a fluid sphere in an elastic medium, an elastic sphere in a liquid medium, and a fluid sphere in a fluid medium) are extracted from...

Excitation of compressional waves and the formation of shocklets in the earth's foreshock

Abstract: Large-amplitude waves in the earth's foreshock are sometimes observed in highly time-developed form, implying that nonlinearities are sufficiently strong to modify their waveforms before the solar wind carries them out of the foreshock. It is presently suggested that parallel propagating waves grow to finite amplitude in the reflected and intermediate ion zones of the earth's foreshock, and refract as they are carried by the solar wind into the 'diffuse' ion region, thereby becoming increasingly oblique and compressional. The time evolution of oblique, low-frequency compressive waves is simulated by means of a one-dimensional hybrid code in which main ions are treated as superparticles, while diffuse ions are seen as a double-adiabatic fluid and electrons as an isothermal fluid.

On waves in gases. Part II: Interaction of sound with magnetic and internal modes

Abstract: This work completes a two-part review on waves in gases, of which the first part [Rev. Mod. Phys. 58, 117 (1986)] dealt with the modern aspects of acoustics of jets, turbulence, and ducts; this second part extends the range of topics from sound to magnetic, internal, and (to a lesser extent) inertial waves, thus considering all four restoring forces (pressure, gravity, and Lorentz and Coriolis forces). The motivations for the study of these waves were outlined in the introduction to Part I. Part II reviews the coupling of acoustic, magnetic, and internal waves, in four stages: in Sec. I dispersion relations are used to study the propagation and radiation of magneto-acoustic-gravity-inertial waves in media for which the wave speeds and scattering scales are constant; in Sec. II the case of linear waves in stratified media, with nonuniform propagation velocity, is then discussed by means of special functions, appearing as exact solutions of second-order problems; in Sec. III the study of linear waves with variable propagation speeds is extended to certain classes of higher-order problems including a discussion of cutoff frequencies, critical levels, partition of energy, mode coupling and conversion, etc; in Sec. IV the preceding studies are extended to damped and nonlinear waves, to include dissipation with variable damping scales and large disturbances in media under nonuniform external forces, such as magnetic flux tubes. The conclusion (Sec. V) sums up both parts of the review, in the sense that it deals with all types of waves in fluids; it mentions a few currently controversial topics, points out some directions for future research, and indicates methods available to address these issues.

Analytical evaluation of variables affecting surface wave testing of pavements.

Abstract: Spectral-Analysis-of-Surface-Waves (SASW) is a promising nondestructive technique for evaluating the mechanical properties of pavement systems and soil deposits. In applying the technique, it is assumed that only plane Rayleigh waves are generated by the source. In reality, when an impulse is applied at the top of a layered system, body waves (shear and compression waves) and other types of surface waves are produced along with Rayleigh waves. In this paper, the dispersion curves (frequency or wavelength versus phase velocity) obtained by assuming only plane Rayleigh waves are compared with dispersion curves obtained when all types of waves are considered. Several cases with different types of layering are studied, and emphasis is placed on typical pavement systems. It is found that the receiver arrangement can significantly influence the dispersion curve and, hence, the resulting modulus profile. For a typical SASW setup in which the distance from the source to the first receiver is kept equal to the distance between the two receivers, wavelengths considered during analysis of the field data should be equal to or less than one-half of the distance between receivers. If this filtering of low frequencies is not performed, the assumption that only plane Rayleigh waves propagate through the medium can lead to errors when backcalculating physical properties from the dispersion curve.

Eigenmode analysis of compressional waves in the magnetosphere

Abstract: A field-aligned eigenode analysis of compressional Alfven instabilities has been performed for a two component anisotropic plasma in a dipole magnetic field. The eigenmode equations are derived from the gyrokinetic equations in the long wavelength (k rho < 1) and low frequency (..omega.. < ..omega../sub b/) limits, where rho is the hot particle gyroradius and ..omega../sub b/ is the hot particle bounce frequency. Two types of compressional instabilities are identified. One is the drift mirror mode which has an odd parity compressional magnetic component with respect to the magnetic equator. The other is the drift compressional mode with an even parity compressional magnetic component. For typical storm time plasma parameters neargeosynchronous orbit, the drift mirror mode is most unstable and the drift compressional mode is stable. The storm time compressional Pc 5 waves, observed by multiple satellites during November 14-15, 1979 (Takahashi et al., 1987), can be explained by the drift mirror instability.

Possible resonance effect in the distribution of earthquake damage in Mexico City

Abstract: The destruction caused by the earthquake of 19 September 1985 in Mexico City shows three remarkable features1. First, a (concentration of damage in the former lake bed; second, a peculiar distribution of high- and low-damage areas alternating within a few city blocks, and third, a selectivity for buildings between five and fifteen storeys high. While the last point is understood by civil engineers2 and the first confirms the relevance of the soft ground, the second feature is the most puzzling. To a physicist such a pattern evokes the idea of a standing wave, with low- and high-damage areas corresponding to nodes and antinodes. This suggests a resonance phenomenon of the waterlogged ground moving collectively. From the Navier equations of classical elasticity, it is known3 that a transfer of the incoming seismic-wave energy to longitudinal waves takes place if a sharp change from hard to soft ground exists. Solving then the two-dimensional Poisson equation for the longitudinal waves within the ancient lake boundaries, we show that for the observed seismic wave period of 2 s, a standing-wave solution exists which explains qualitatively the damage distribution pattern.

Novel Surface Transverse Wave Resonators with Low Loss and High Q

Abstract: Surface transverse wave (STW) resonator devices have been designed and fabricated on rotated Y cuts of quartz, with wave propagation perpendicular to the X axis. Packaged devices with center frequencies of 500 and 632 MHz show insertion loss less than 6 d B in a 50 ohm system. Unloaded Q is typically greater than 10,000. Devices at 1726 MHz show packaged insertion loss as low as 10 dB and unloaded Q as high as 5600. Temperature behavior was studied. STW resonators were found to exhibit significantly greater power-handling ability than SAW resonators with comparable design parameters and electrical performance in a comparison study. Device phase noise measurements indicate that STW resonators' noise improves when driven with high incident power. Single sideband device phase noise less than -135 dBc/Hz at 1 H z offset was achieved.

Weakly nonlinear non-symmetric gravity waves on water of finite depth

Abstract: A weakly nonlinear Hamiltonian model for two-dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that non-symmetric periodic waves exist, which appear via spontaneous symmetry-breaking bifurcations from symmetric waves.

Strong localization of classical waves: a numerical study

Abstract: We study numerically the propagation of classical waves through a network of wave guides, with a randomly varying index of refraction in two and three dimensions. We find evidence for the localization of these waves by disorder, although the localization lengths appear to be typically larger than for quantum waves.

Propagation of Compressive Waves through Fibril Magnetic Fields

Abstract: The surface effects of interactions between the solar 5-min p-modes and the large-scale fibril magnetic field are discussed using a multiple scattering approach. Attention is given to the propagation of linear disturbances in a two-dimensional, highly conducting magnetized plasma with many parallel flux tubes in pressure equilibrium with a surrounding stationary field-free plasma. Multiple scattering in the fibril half-space is shown to generate acoustic waves that cascade to ever-smaller length scales. The scale reduction, proportional to the depth into the fibril magnetic field, is responsible for the damping of p-mode oscillations observed in plages. 39 references.

Properties of short-crested waves in water of finite depth

Abstract: Short-crested waves are defined as propagating surface gravity waves which are doublyperiodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other. Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Pade approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy. The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawall occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.

Resonances and standing waves

Abstract: Recent investigations on aperiodic waves, which are generated by time-harmonic forces in waveguides, indicate a close relationship between resonances and certain time-harmonic solutions of homogeneous boundary value problems for the wave equation (“standing waves”). This paper is motivated by the observation that, in all known cases, standing waves are connected with resonances if and only if they are subject to suitable asymptotic restrictions as |x| ∞. These asymptotic properties are used to introduce a class of “admissible” standing waves. We prove that admissible standing waves do not exist in a certain class of local perturbations Ω of the n-dimensional domain Ω0 bounded by the hyperplanes xn = 0 and xn = π. This extends a result of M. Faulhaber on the absence of eigenvalues. As we shall show in a subsequent paper, absence of admissible standing waves implies absence of resonances.

The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds.

Abstract: Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus.

An eastward propagating compressional Pc 5 wave observed by AMPTE/CCE in the postmidnight sector

Abstract: This paper presents a detailed analysis of a compressional Pc 5 wave observed in the postmidnight sector on July 21, 1986, using data from the magnetometer, the charge-energy-mass spectrometer, and the medium-energy particle analyzer aboard the AMPTE/Charge Composition Explorer (CCE) spacecraft. The Pc 5 wave exhibited harmonically related transverse and compressional magnetic oscillations, modulation of the flux of medium energy protons, and a large azimuthal wave number, i.e., properties that are similar to those of compressional Pc5 waves observed previously at geostationary orbit. The unique observations recorded by the AMPTE/CCE included the occurrence of the wave in the postmidnight sector, its sunward propagation with respect to the spacecraft, and the left-handed polarization of the perturbed magnetic field. In spite of the morphological uniqueness observed, the excitation of the July 21 event is considered to be due to the same type of instability as operates at geostationary orbit.

On the behavior near the crest of waves of extreme form

Abstract: : This paper concerns waves of permanent form on the free surface of an ideal liquid which is in two-dimensional, irrotational motion under the action of gravity. We consider only extreme waves, often called 'waves of greatest height'; each of these is the end-member of a one-parameter family of waves, and is distinguished from other 'smaller' members of the family by a sharp crest. Although this corner is physically unrealistic, oceanographers have given such idealized, extreme waves a great deal of attention since Stokes postulated their existence in 1880. (One reason may be the physical importance of the smaller waves, and that scientists like to interpolate.) The present paper is a contribution to the strict mathematical theory of extreme waves, which was emerged only since 1978. We derive rigorously an asymptotic series that describes the flow near the crest. This confirms and sharpens certain earlier exploratory results due to Grant and Norman. The series should play a useful part in numerical computations of extreme waves. (Author)

A model for the harmonic of compressional Pc 5 waves

Abstract: Compressional Pc 5 magnetic waves in the magnetosphere are a unique phenomenon showing a nonsinusoidal waveform in spite of a well-defined period. Although the waveform can be Fourier-decomposed into the fundamental and the second harmonics, the phase between the two is kept constant from event to event, implying that the waveform is not the result of a chance superposition of two magnetospheric eigenmodes. A phenomenological explanation to this waveform is offered using a field-line configuration model that is a modified version of a previously proposed antisymmetric standing wave. In this model, the location of the equatorial node of field-line displacement is assumed to oscillate with the wave, with a peak-to-peak amplitude greater than 10 percent of the wavelength of the standing wave. The predicted waveform at various magnetic latitudes is found to be in excellent agreement with an observation taken near the magnetic equator by the Active Magnetospheric Particle Tracer Explorers/Charge Composition Explorer spacecraft.