Amplitude Of Wave Field Research Articles

Hybrid ray-mode analysis of acoustic scattering from a finite, fluid-loaded plate

In this paper we develop a hybrid ray-mode algorithm to model the acoustic scattering from a finite, fluid-loaded thin elastic plate excited by a time-harmonic line pressure force exterior to the plate. Emphasizing the mid-frequency regime where internal structural wave phenomena are important, the problem is formulated in terms of a self-consistent system parameterization based on the ray-mode phenomenology in a generalized version of the geometrical theory of diffraction (GTD). The GTD constituents involve specularly reflected, transmitted and edge-diffracted exterior ray fields, as well as subsonic and supersonic traveling and resonant internal mode fields, with exterior-interior coupling taking place through phase matching along the smooth portions of the plate and through the edges. The amplitudes of these GTD wavefields are determined by excitation, conversion and coupling coefficients, some of which can be extracted in analytical form from the canonical infinite plate geometry. Since the ray acoustic algorithm for a complex system is built around the interaction between localized events on constituent simpler canonical systems, a short-pulse-excited time domain (TD) reference base with its space-time resolved output response is convenient for sorting out the ray-mode phenomenology. This route is followed here for the finite plate, for which numerical TD reference data is furnished by a finite-difference time domain (FDTD) code. We confirm, either directly or via data processing, that all observed wave effects are accounted for in our frequency domain hybrid ray-mode algorithm. Extension into a TD wavefront-resonance algorithm is under consideration.

A wraparound elimination technique for the pseudospectral wave synthesis using an antiperiodic extension of the wavefield

The pseudospectral method (e.g., Kosloff and Baysal, 1982) is an attractive alternative to numerical modeling schemes such as finite difference or finite element because it requires several orders of magnitude less computer memory and computation time. In the pseudospectral method, the field variables are expanded in terms of Fourier interpolation polynomials which impose periodic boundary conditions on the equation of motion. Numerical differentiation is then implemented via the discrete Fourier transform. The periodicity condition causes the periodically extended wavefield on either side of the computational domain to propagate in from the sides, interfering with the actual solution. This phenomenon is called wraparound. In many current applications (e.g., Cerjan et al., 1985), an absorbing boundary condition has been used to damp the wraparound phases through a gradual reduction of the wavefield amplitude in the vicinity of the grid boundary. Such a method works effectively when the appropriate absorption coefficients are used. However, improper selection of absorption parameters can result in reflected waves of relatively large amplitude. Thus a major disadvantage of such methods is the difficulty of selecting appropriate damping parameters to suppress both the wraparound and the boundary reflected phases.

Numerical tests of 3D true‐amplitude zero‐offset migration1

AbstracrAn amplitude‐preserving migration aims at imaging compressional primary (zero‐or) non‐zero‐offset reflections into 3D time or depth‐migrated reflections so that the migrated wavefield amplitudes are a measure of angle‐dependent reflection coeffcients. The principal objective is the removal of the geometrical‐spreading factor of the primary reflections. Various migration/inversion algorithms involving weighted diffraction stacks proposed recently are based on Born or Kirchhoff approximations. Here, a 3D Kirchhoff‐type zero‐offset migration approach, also known as a diffraction‐stack migration, is implemented in the form of a time migration. The primary reflections of the wavefield to be imaged are described a priori by the zero‐order ray approximation. The aim of removing the geometrical‐ spreading loss can, in the zero‐offset case, be achieved by not applying weights to the data before stacking them. This case alone has been implemented in this work. Application of the method to 3D synthetic zero‐offset data proves that an amplitude‐preserving migration can be performed in this way. Various numerical aspects of the true‐amplitude zero‐offset migration are discussed.

Surface wave scattering from sharp lateral discontinuities

The problem of surface wave scattering is re‐explored, with quasi‐degenerate normal mode coupling as the starting point. For coupling among specified spheroidal and toroidal mode dispersion branches, a set of coupled wave equations is derived in the frequency domain for first‐arriving Rayleigh and Love waves. The solutions to these coupled wave equations using linear perturbation theory are surface integrals over the unit sphere covering the lateral distribution of perturbations in Earth structure. For isotropic structural perturbations and surface topographic perturbations, these solutions agree with the Bom scattering theory previously obtained by Snieder and Romanowicz. By transforming these surface integrals into line integrals along the boundaries of the heterogeneous regions in the case of sharp discontinuities, and by using uniformly valid Green's functions, it is possible to extend the solution to the case of multiple scattering interactions. The proposed method allows the relatively rapid calculation of exact second order scattered wavefield potentials for scattering by sharp discontinuities, and it has many advantages not realized in earlier treatments. It employs a spherical Earth geometry, uses no far field approximation, and implicitly contains backward as well as forward scattering. Comparisons of asymptotic scattering and an exact solution with single scattering and multiple scattering integral formulations show that the phase perturbation predicted by geometrical optics breaks down for scatterers less than about six wavelengths in diameter, and second‐order scattering predicts well both the amplitude and phase pattern of the exact wavefield for sufficiently small scatterers, less than about three wavelengths in diameter for anomalies of a few percent.

Dynamical theory of X-ray diffraction by elastically bent crystals with microdefects. II. Diffraction intensity

The amplitudes of wave fields inside a macroscopically deformed crystal with defects are obtained in the two-beam approximation. The amplitudes of coherent (strong Bragg and “quasidiffuse”) and diffuse waves emerging from the crystal are determined by taking account of boundary conditions for both Bragg and Laue cases of diffraction. The general expressions for coherent and diffuse components of the differential reflection power of bent crystals with microdefects are obtained. [Russian Text Ignored].

Experimental verification of the statistical dynamical theories of diffraction

The influences of crystal imperfection on dynamical diffraction have been studied by measuring integrated intensities in the Laue case as a function of X-ray wavelength. SiO2-precipitate microdefects were introduced into Czochralski-grown silicon wafers by heating at 1223 K for five different periods of time varying from 25 to 145 h. The X-ray intensities were measured by the energy-dispersive diffraction method over the wavelength range 0.15 to 0.78 A. The measurements revealed that increases in both the integrated intensity and the Pendellosung beat spacing were accompanied by a decrease of the beat amplitude with increasing heating period. The data were compared quantitatively with those obtained using the theories of Kato [Acta Cryst. (1980), A36, 763–769, 770–778] and Becker & Al Haddad [Acta Cryst. (1992), A48, 121–134). The former theory, where the correlation length Γ for the wavefield amplitudes is assumed to be close to the extinction distance, did not describe the data. The latter theory fitted only the specimen heated for 25 h. A model assuming that Γ is independent of wavelength and varies with crystal perfection and reflection plane was proposed in lieu of Kato's assumption. The application of the model led to an excellent agreement for the specimens studied. The best fitted values of Γ, from 0.07 to 1.6 μm, were much the same as the correlation lengths of the phase factor, from 0.04 to 2.6 μm.

Complex wave-field reconstruction using phase-space tomography.

A method is proposed for the experimental determination of the amplitude and phase structure of a quasimonochromatic wave field in a plane normal to its propagation direction. The wave field may represent either a scalar electromagnetic (EM) field or the quantum mechanical (QM) wave function of a matter wave. For coherent EM fields or pure QM states, the method uniquely reconstructs the complex wave fields. For partially coherent EM fields or mixed QM states, it reconstructs the two-point correlation function or density matrix. The experiment uses only intensity measurements and refractive optics (lenses), and the data analysis algorithm is noniterative and requires no deconvolution.

3-D true‐amplitude finite‐offset migration

Compressional primary nonzero offset reflections can be imaged into three‐dimensional (3-D) time or depth‐migrated reflections so that the migrated wavefield amplitudes are a measure of angle‐dependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations. Here a 3-D Kirchhoff‐type prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zero‐order ray approximation. As a result, the principal issue in the attempt to recover angle‐dependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections. The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the source‐receiver configurations employed and the caustics occurring in the wavefield. Our weight function, which is computed using paraxial ray theory, is compared with the one of the inversion integral based on the Beylkin determinant. It differs by a factor that can be easily explained.

Experimental tests of the statistical dynamical theory

A homogeneous distribution of SiO2 precipitates in Czochralski-grown silicon containing different amounts of oxygen were produced by annealing the dislocation-free crystals at 1023 K. The resulting long-range strain field modifies the integrated reflecting power R of the Bragg reflections measured on an absolute scale with 316 keV γ-radiation. The thickness dependence of R has been modelled using the results of statistical dynamical theory. The assumption made in Kato's original theory, where the correlation length Γ for the wave-field amplitudes is proportional to the extinction length, has to be abandoned. Recent modifications to statistical dynamical theory by Becker & Al Haddad [Acta Cryst. (1990). A46, 123–139] lead to excellent agreement with the present experimental results. Furthermore, in the present case, the correlation length τ, describing short-range correlation in the fluctuations of the phase factor caused by the displacement field of the defects, turns out to be very small, so that the contribution of the mixed term to the integrated reflecting power could be neglected. Therefore, the defect scattering is characterized by the static Debye–Waller factor alone, which was determined accurately from the thickness dependence of the measured integrated reflecting power. From the ratio of the static Debye–Waller factors determined for different orders of reflection, the sizes of the SiO2 precipitates have been calculated and the results are in very good agreement with the values determined directly from small-angle neutron scattering on the same samples.

Correlation characteristics of waves in lossy media with smooth fluctuations of the refractive index with oblique incidence on boundary

Abstract The features of coherence function (and angular spectrum) and also the fluctuation of amplitude and phase of wavefield in a random strongly absorptive medium are investigated in the case of arbitrary angle of incidence at the surface. In this paper it is elucidated that with oblique incidence a dissipation in the random medium can accelerate the accumulation of wave fluctuations and its incoherence. This effect strongly depends on the rate of decrease of the ‘wings’ of the scattering indicatrix. An analytical theory (Rytov's approximation and modified method of parabolic equation) has been modelled by Monte Carlo simulation of wave propagation, and also by numerical solution of the model transfer equation. It is revealed that the width of an angle spectrum can nonmonotonically change with the immersion to the absorptive medium.

Excitation characteristics of sideband wave in a radio frequency stabilized simple mirror plasma

A nonlinearly excited sideband wave due to a magnetohydrodynamic interchange mode is observed in a radio frequency (rf)-produced simple mirror plasma, and its excitation is discussed. The measured amplitude of the sideband wave field is compared with the theoretical value showing the agreement in order of magnitude, but the radial distribution is different. As the rf current of the production source is changed, the sideband excitation level varies and this variation is explained by the density fluctuation and the difference of the dispersion relations between pump and sideband waves. Using the difference of the dispersion relations, it would be possible to suppress the sideband excitation and to reduce its destabilizing contribution.

Effects of wave-wave and wave-mean flow interactions on the evolution of a baroclinic wave

Abstract The weakly nonlinear evolution of a free baroclinic wave in the presence of slightly supercritical, vertically sheared zonal flow and a forced stationary wave field that consists of a single zonal scale and an arbitrary number of meridional harmonics is examined within the context of the conventional two-layer model. The presence of the (planetary-scale) stationary wave introduces zonal variations in the supercriticality and is shown to alter the growth rate and asymptotic equilibrium of the (synoptic-scale) baroclinic wave via two distinct mechanisms: The first is due to the direct interaction of the stationary wave with the shorter synoptic wave (wave-wave mechanism), and the second is due to the interaction of the synoptic wave with that portion of the mean field that is corrected by the zonally rectified stationary wave fluxes (wave-mean mechanism). These mechanisms can oppose or augment each other depending on the amplitude and spatial structure of the stationary wave field. If the stationary wave field is confined primarily to the upper (lower) layer and consists of only the gravest cross-stream mode, conditions are favorable (unfavorable) for nonzero equilibrium of the free wave. In addition to the time dependent heat flux generated by baroclinic growth of the free wave, its interaction with a stationary wave field consisting of two or more meridional harmonics generates time dependent heat fluxes that vary with period of the free wave. However, if the stationary wave field contains several meridional harmonics of sufficiently large amplitude, the free baroclinic wave is destroyed.

Fault zone trapped seismic waves

AbstractTwo instances of fault zone trapped seismic waves have been observed. At an active normal fault in crystalline rock near Oroville in northern California, trapped waves were excited with a surface source and recorded at five near-fault borehole depths with an oriented three-component borehole seismic sonde. At Parkfield, California, a borehole seismometer at Middle Mountain recorded at least two instances of the fundamental and first higher mode seismic waves of the San Andreas fault zone. At Oroville recorded particle motions indicate the presence of both Love and Rayleigh normal modes. The Love-wave dispersion relation derived for an idealized wave guide with velocity structure determined by body-wave travel-time inversion yields seismograms of the fundamental mode that are consistent with the observed Love-wave amplitude and frequency. Applying a similar velocity model to the Parkfield observations, we obtain a good fit to the trapped wavefield amplitude, frequency, dispersion, and mode time separation for an asymmetric San Andreas fault zone structure—a high-velocity half-space to the southwest, a low-velocity fault zone, a transition zone containing the borehole seismometer, and an intermediate velocity half-space to the northeast. In the Parkfield borehole seismic data set, the locations (depth and offset normal to fault plane) of natural seismic events which do or do not excite trapped waves are roughly consistent with our model of the low velocity zone. We conclude that it is feasible to obtain near-surface borehole records of fault zone trapped waves. Because trapped modes are excited only by events close to the fault zone proper—thereby fixing these events in space relative to the fault—a wider investigation of possible trapped mode waveforms recorded by a borehole seismic network could lead to a much refined body-wave/tomographic velocity model of the fault and to a weighting of events as a function of offset from the fault plane. It is an open question whether near-surface sensors exist in a stable enough seismic environment to use trapped modes as an earth monitoring device.

Diffraction by a randomly distorted crystal. I. The case of short-range order

Kato's statistical theory of diffraction [Kato (1980). Acta Cryst. A36, 763-769, 770-778] is reformulated in a self-consistent manner. The local displacement field u(r) occurs through the phase factor φ(r) = exp [2πi h . u(r)]. The present paper is concerned with the limiting case where (φ~(r)) = E = 0: this corresponds to the situation where only secondary extinction is present. There are two correlation lengths in the problem, the first one τ for the phase factor φ, the second one Γ for the wave-field amplitudes. Kato assumed Γ » τ. It is shown in the present paper that Γ ≃ τ, a property which has important consequences for the general theory, where E ≠ 0, to be discussed in the second paper of this series.

A laboratory study of the electromagnetic bias of rough surface scattering by water waves

The design, development, and use of a focused-beam radar to measure the electromagnetic bias introduced by the scattering of radar waves by a roughened water surface are discussed. The bias measurements were made over wide ranges of environmental conditions in a wavetank laboratory. Wave-elevation data were provided by standard laboratory capacitance probes. Backscattered radar power measurements coincident in time and space with the elevation data were produced by the radar. The two data sets are histogrammed to produce probability density functions for elevation and radar reflectivity, from which the electromagnetic bias is computed. The experimental results demonstrate that the electromagnetic bias is quite variable over the wide range of environmental conditions that can be produced in the laboratory. The data suggest that the bias is dependent upon the local wind field and on the amplitude and frequency of any background wave field that is present. >

The nonlinear three-wave interaction with a finite spectral width

The nonlinear interaction of three waves propagating in an infinite plasma with a finite spectral bandwidth is studied. A Hamiltonian formulation of the interaction is used and, with the help of a projection operator method, generalized Langevin equations are derived for the wave field amplitudes. A simple form of the evolution equations, more complete than the usual fixed-phase equations, is obtained when the ballistic term and higher-order corrections of the memory effects in the Langevin equations are neglected. Approximate analytical solutions are derived using a multiple time scale method. These are compared with the results of numerical integration. A number of new qualitative features related to the finite spectral bandwidth are discussed in detail.

Recursive Linear Smoothing for the 2-D Helmholtz Equation

A fast algorithm for reconstructing images governed by a 2-D Helmholtz equation is presented. The computational complexity of the algorithm is 0(NMlogM) or O(NM2) depending on boundary conditions, where N and M are the number of spatial grid points in the x and y directions respectively. This problem arises when smoothing a large number of images governed by the 2-D wave equation, because a Fourier transform in time gives a new set of images governed by the Helmholtz equation. When the images come from a scattering process, we show that a linear least-squares Born inversion of the wave field amplitudes can be performed during the smoothing procedure without changing the computational complexity. We also show that the smoothing algorithm is well-posed, and t.hat the sample functions of the smoothed estimate possess smoothness properties consistent with the Helmholtz equation.

Dyson and Bethe‐Salpeter equations for dynamical X‐ray diffraction in crystals with randomly placed defects

AbstractA new formalism is found for the description of the dynamical X‐ray diffraction in statistically disturbed crystals both, in the coherent and in the incoherent approximations. The formalism is based on Dyson's equation for the averaged wavefield amplitude in the crystal and on Bethe‐Salpeter's equation for the averaged mutual coherence functions. The equations are simplified assuming the single‐group approximation. Dyson's equation is solved analytically by the Fourier method, and the first iteration of the solution of Bethe‐Salpeter's equation is found. On the basis of these solutions reflection curves as well as angular intensity distributions are computed for a crystal with randomly placed microdefects. The connection is shown of the theory presented with Kato's dynamical statistical theory.

Efficient global matrix approach to the computation of synthetic seismograms

Summary. A numerically efficient global matrix approach to the solution of the wave equation in horizontally stratified environments is presented. The field in each layer is expressed as a superposition of the field produced by the sources within the layer and an unknown field satisfying the homogeneous wave equations, both expressed as integral representations in the horizontal wavenumber. The boundary conditions to be satisfied at each interface then yield a linear system of equations in the unknown wavefield amplitudes, to be satisfied at each horizontal wavenumber. As an alternative to the traditional propagator matrix approaches, the solution technique presented here yields both improved efficiency and versatility. Its global nature makes it well suited to problems involving many receivers in range as well as depth and to calculations of both stresses and particle velocities. The global solution technique is developed in close analogy to the finite element method, thereby reducing the number of arithmetic operations to a minimum and making the resulting computer code very efficient in terms of computation time. These features are illustrated by a number of numerical examples from both crustal and exploration seismology.

Numerical comparison of two asymptotic methods for solving wave diffraction problems in smooth inhomogeneous media

UDC 534.1;621.371 The methods of Maslov's canonical operator (MCO) and Gaussian beam summation (GBS), which are used to describe the diffraction structures of wave fields in complex inhomogeneous media, are numerically compared. The analysis is performed on a sample calculation of the amplitude of an acoustic wave field propagating in an inhomogeneous sound channel. It is shown that, on the axis of the sound channel, the results of calculation by the MCO and GBS methods agree within graphical accuracy. However, far from the axis, application of GBS method encounters definite difficulties. On the basis of the results of the comparison, it is concluded that the MCO method better reveals the mechanism of the diffraction field formation in the focal region. In the short wave approximation of wave diffraction and propagation problems in complex inhomogenous media, the task often arises to describe the field in the focal region, where the traditional ray (geometric) approach is incorrect. As the practice of such calculations shows, focal regions of dissimilar form are a similar phenomenon. They really exist in the ionosphere and tropospheric channels of radio communication, in underwater acoustic channels, in fiber optics, etc. This makes the task of the detailed description of the field in them very topical. At the present time, the field in the focal region is most fully described in the framework of methods for plotting asymptotic solutions, using rapidly oscillating integrals, and by the methods of parabolic equations (PE). Among the first group of such methods, we shall examine the means of plotting solutions based on Maslov's canonical operstor (MCO) [i, 2] including the topological classification of the focal regions and the local description of the field in them [3-7], since the practice of solving similar problems recommends just this method to us. We will compare our numerical solution with a solution plotted by a new and promising method -- the method of Gaussian beam summation (GBS) [8-11]. The present work compares the methods of MCO and GBS with respect to the final result of their application in a concrete problem. This will allow a more complete answer to the question of how effective the examined methods are in the applied problems, since the theoretical evaluations of their effectiveness in the literature often have a rather abstract character and are uncorrelated. We made use of the results of [8, 9] as a completed and fully documented GBS solution of the problem, which studied the propagation of monochromatic sound waves, excited by a point source located on the axis of a symmetric acoustic wave guide in a plane-laminar medium. i. Statement of the Problem. Asymptotic Solution in the MCO Form. We shall examine Helmholtz's equation, which we shall write in the symbology of [8, 9]: