Showing papers on "Amplitude published in 1983"
TL;DR: In this article, the authors proposed a frequency-domain scaling model for predicting seismic motions as a function of source strength, which can be applied to any time series having a stochastic character, including ground acceleration, velocity and the oscillator outputs on which response spectra and magnitude are based.
Abstract: Theoretical predictions of seismic motions as a function of source strength are often expressed as frequency-domain scaling models. The observations of interest to strong-motion seismology, however, are usually in the time domain (e.g., various peak motions, including magnitude). The method of simulation presented here makes use of both domains; its essence is to filter a suite of windowed, stochastic time series so that the amplitude spectra are equal, on the average, to the specified spectra. Because of its success in predicting peak and rms accelerations (Hanks and McGuire, 1981), an ω -squared spectrum with a high-frequency cutoff ( f m), in addition to the usual whole-path anelastic attenuation, and with a constant stress parameter (Δ σ ) has been used in the applications of the simulation method. With these assumptions, the model is particularly simple: the scaling with source size depends on only one parameter—seismic moment or, equivalently, moment magnitude. Besides peak acceleration, the model gives a good fit to a number of ground motion amplitude measures derived from previous analyses of hundreds of recordings from earthquakes in western North America, ranging from a moment magnitude of 5.0 to 7.7. These measures of ground motion include peak velocity, Wood-Anderson instrument response, and response spectra. The model also fits peak velocities and peak accelerations for South African earthquakes with moment magnitudes of 0.4 to 2.4 (with f m = 400 Hz and Δ σ = 50 bars, compared to f m = 15 Hz and Δ σ = 100 bars for the western North America data). Remarkably, the model seems to fit all essential aspects of high-frequency ground motions for earthquakes over a very large magnitude range .
Although the simulation method is useful for applications requiring one or more time series, a simpler, less costly method based on various formulas from random vibration theory will often suffice for applications requiring only peak motions. Hanks and McGuire (1981) used such an approach in their prediction of peak acceleration. This paper contains a generalization of their approach; the formulas used depend on the moments (in the statistical sense) of the squared amplitude spectra, and therefore can be applied to any time series having a stochastic character, including ground acceleration, velocity, and the oscillator outputs on which response spectra and magnitude are based .
1,708 citations
TL;DR: In this article, the authors examined the alternative of carrying out the migration through a reverse time extrapolation, which may offer improvements over existing migration methods, especially in cases of steeply dipping structures with strong velocity contrasts.
Abstract: Migration of stacked or zero-offset sections is based on deriving the wave amplitude in space from wave field observations at the surface. Conventionally this calculation has been carried out through a depth extrapolation. We examine the alternative of carrying out the migration through a reverse time extrapolation. This approach may offer improvements over existing migration methods, especially in cases of steeply dipping structures with strong velocity contrasts. This migration method is tested using appropriate synthetic data sets.
1,274 citations
01 Jan 1983
TL;DR: In the seismic reflection method, the seismic signal does vary in amplitude, shape, frequency and phase versus the propagation time as discussed by the authors, which is not the case in the seismic reflectance method.
Abstract: In the seismic reflection method, the seismic signal does vary in amplitude, shape, frequency and phase versus the propagation time
339 citations
TL;DR: In this paper, it was shown that the additional phase changes also give rise to line shape structure in the form of additional peaks separated from the main peak by multiples of the relaxation oscillation frequency.
Abstract: Spontaneous emission alters the phase and amplitude of the laser field. The amplitude changes induce relaxation oscillations, which cause additional phase changes while restoring the field amplitude to the steady state value. It was previously shown that the additional phase changes greatly enhance the linewidth. We show here that the additional phase changes also give rise to line shape structure in the form of additional peaks separated from the main peak by multiples of the relaxation oscillation frequency. The calculated mean square phase change and power spectrum are in good agreement with published observations.
302 citations
TL;DR: In this paper, a theoretical probability density is derived for the joint distribution of wave periods and amplitudes which has the following properties: (1) the distribution is asymmetric, in accordance with observation; (2) it depends only on three lowest moments m 0, m 1, m 2 of the spectral density function.
Abstract: A theoretical probability density is derived for the joint distribution of wave periods and amplitudes which has the following properties: (1) the distribution is asymmetric, in accordance with observation; (2) it depends only on three lowest moments m 0 , m 1 , m 2 of the spectral density function. It is therefore independent of the fourth moment m 4 , which previously was used to define the spectral width (Cavanie et al . 1976). In the present model the width is defined by the lower-order parameter v = ( m 0 m 2 / m 2 1 - 1) ½ . The distribution agrees quite well with wave data taken in the North Atlantic (Chakrabarti & Cooley 1977) and with other data from the Sea of Japan (Goda 1978). Among the features predicted is that the total distribution of wave heights is slightly non-Rayleigh, and that the interquartile range of the conditional wave period distribution tends to zero as the wave amplitude diminishes. The analytic expressions are simpler than those derived previously, and may be useful in handling real statistical data.
286 citations
TL;DR: In this article, the authors report the results of some numerical experiments that bring out the lowpass characteristics of a purely elastic medium with a heterogeneous velocity structure and show that attenuation by intrinsic friction and scattering are approximately additive.
Abstract: We report the results of some numerical experiments that bring out the lowpass characteristics of a purely elastic medium with a heterogeneous velocity structure. Although the typical fluctuation is spatially confined within less than a wavelength, waves propagating over a sufficiently long path suffer major cumulative effects. We summarize the removal of high frequencies during transmission by a frequency-independent apparent Q, and show that attenuation by intrinsic friction and scattering are approximately additive. We propose some diagnostics that might help to distinguish the presence of velocity fluctuations and resultant scattering from the presence of anelasticity and true dissipation. When scattering dominates over intrinsic friction: (1) the coda of a transmitted wave contains relatively higher frequencies than the initial pulse; (2) the attenuation deduced from the power spectrum of the transmitted wave is greater than that deduced from the phase spectrum; (3) compressional and shear wave apparent Q's are approximately equal; and (4) estimates of apparent Q made from reflected coda vary with frequency, while estimates made from the transmitted waves do not. We also outline several topics in the theory of wave propagation that will be relevant in a satisfactory interpretation of short-period observations, if the amplitude of such signals is affected by scattering.
264 citations
TL;DR: In this article, moderate-amplitude axisymmetric oscillations of incompressible inviscid drops and bubbles are studied using a Poincare-Lindstedt expansion technique, and corrections to the drop shape and velocity potential caused by mode coupling at second order in amplitude are predicted for two-, three-and four-lobed motions.
Abstract: Moderate-amplitude axisymmetric oscillations of incompressible inviscid drops and bubbles are studied using a Poincare–Lindstedt expansion technique. The corrections to the drop shape and velocity potential caused by mode coupling at second order in amplitude are predicted for two-, three- and four-lobed motions. The frequency of oscillation is found to decrease with the square of the amplitude; this result compares well with experiments and numerical calculations for drops undergoing two-lobed oscillations.
237 citations
TL;DR: Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force as mentioned in this paper, where the beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value.
Abstract: Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value The stiffness of the beam has the properties of a bi-linear spring The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops Subharmonic oscillations are found to be characteristic of these types of motions For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations
225 citations
TL;DR: In this article, a parabolic equation governing the leading-order amplitude for a forward-scattered Stokes wave is derived using a multiple-scale perturbation method, and the connection between the linearized version and a previously derived approximation of the linear mild slope equation is investigated.
Abstract: A parabolic equation governing the leading-order amplitude for a forward-scattered Stokes wave is derived using a multiple-scale perturbation method, and the connection between the linearized version and a previously derived approximation of the linear mild slope equation is investigated. Two examples are studied numerically for the situation where linear refraction theory leads to caustics, and the nonlinear model is shown to predict the development of wave-jump conditions and significant reductions in amplitude in the vicinity of caustics.
193 citations
TL;DR: In this paper, the authors work out the low energy expansion of ππ-scattering amplitude with a method proposed in the preceeding paper using the measured D-waves as an input.
173 citations
TL;DR: In this article, the amplitude, frequency, wavenumber and phase speed of an unstable deep-water wavetrain were measured using a Hilbert transform technique, showing that the modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or phase reversals.
Abstract: Time series of amplitude, frequency, wavenumber and phase speed are measured in an unstable deep-water wavetrain using a Hilbert-transform technique. The modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or ‘phase reversals’. These anomalies appear to correspond to the ‘crest pairing’ described by Ramamonjiarisoa & Mollo-Christensen (1979). The measurements offer a novel local description of the instability of deep-water waves which contrasts markedly with the description afforded by conventional Fourier decomposition. The measurements display very large local modulations in the phase speed, modulations that may not be anticipated from measurements of the phase speeds of individual Fourier components travelling (to leading order) at the linear phase speed (Lake & Yuen 1978).
TL;DR: In this article, a nonlinear "bend equation", based on a dynamical description of flow in bends and a kinematical description of bank erosion, is used to describe channel migration.
Abstract: Meander bends of high amplitude in alluvial rivers often display a high degree of coherency. The only analytical equation heretofore available to describe such bends is the sine‐generated curve, which has no rigorous derivation in the context of rivers. Herein a nonlinear “bend equation,” based on a dynamical description of flow in bends and a kinematical description of bank erosion, is used to describe channel migration. This equation admits solutions of constant amplitude that migrate downstream with constant speed. The solution at high amplitude displays a prominent skewing that reveals the direction of flow. At low amplitude, the solution reduces to the sine‐generated curve.
TL;DR: In this article, two-dimensional solitary and periodic waves in water of finite depth were considered and it was shown that elevation solitary waves cannot be obtained as the continuous limit of periodic waves as the wavelength tends to infinity.
Abstract: Two-dimensional solitary and periodic waves in water of finite depth are considered. The waves propagate under the combined influence of gravity and surface tension. The flow, the surface profile and the phase velocity are functions of the amplitude of the wave and the parameters l = λ/H and τ = T/ρgH2. Here λ is the wavelength, H the depth, T the surface tension, ρ the density and g the acceleration due to gravity. For . In addition, it is shown that elevation solitary waves cannot be obtained as the continuous limit of periodic waves as the wavelength tends to infinity. Graphs of the results are included.
TL;DR: In this article, time stationary solutions to the Vlasov-Poisson equations for ion holes and double layers are examined along with particle simulations that pertain to recent observations of small amplitude electric field structures on auroral field lines.
Abstract: Time stationary solutions to the Vlasov-Poisson equations for ion holes and double layers are examined along with particle simulations that pertain to recent observations of small amplitude electric field structures on auroral field lines. Both the time stationary analysis and the simulations suggest that the observed double layers evolve from holes in ion phase space. Multiple small amplitude double layers, as seen in long simulation systems, are observed to propagate past the spacecraft and may account for the acceleration of plasma sheet electrons to produce inverted-V precipitation.
TL;DR: In this paper, the Korteweg-de Vries equation was extended to second order in wave amplitude and the second order correction to the wave profile and the phase speed and the first order correction of the wavelength were determined.
Abstract: Solitary waves in density stratified fluids of shallow depth are described, to first order in wave amplitude, by the Korteweg–de Vries equation; the solution for a single solitary wave has the familiar ‘‘sech2’’ profile and a phase speed which varies linearly with the wave amplitude. This theory is here extended to second order in wave amplitude. The second‐order correction to the wave profile and the phase speed and the first‐order correction to the wavelength are all determined. Four special cases are discussed in detail. In certain special circumstances the first‐order theory may fail due to the vanishing of the nonlinear coefficient in the Korteweg–de Vries equation. When this occurs a different theory is required which leads to an equation with both quadratic and cubic nonlinearities.
TL;DR: Subsidiary maxima are observed in the field spectra of single mode semiconductor lasers as discussed by the authors, and they are linked to the relaxation resonance of the field spectrum of a single-mode semiconductor laser.
Abstract: Subsidiary maxima are observed in the field spectra of single mode semiconductor lasers Measurements of their power dependence show they are linked to the relaxation resonance We attribute these maxima to combined phase and amplitude fluctuations at the relaxation resonance A theoretical calculation of the field spectrum using the results of a noise analysisincorporating carrier dynamics agrees very well with observations
TL;DR: In this paper, the amplitude of the quark-quark scattering amplitude in non-Abelian gauge theories in the limit of very large center-of-mass energy is analyzed.
Abstract: In this paper we find the asymptotic behavior of the fixed-angle on-shell quark-quark scattering amplitude in non-Abelian gauge theories in the limit of very large center-of-mass energy ..sqrt..s . We sum the perturbation series to all orders in the coupling constant and all powers of lns, but ignore terms which are suppressed by a power of s order by order in perturbation theory. In the s..-->..infinity limit the amplitude vanishes as exp(-..cap alpha.. lns ln lns), where ..cap alpha.. is a constant. The phase of the amplitude is shown to be free from infrared divergences. Hence the phase is a perturbatively calculable function and may provide important tests of QCD.
TL;DR: In this article, the angular variation of diffracted signal amplitude when infinite plane P or SV waves are incident on a semi-infinite crack is derived for the case of a defect symmetrically placed with respect to transmitter and receiver.
TL;DR: In this article, a numerical solution of the unsteady boundary-layer equation is used to relate the amplitude of the asymptotic eigensolution, and consequently of the Tollmien-Schlichting wave, to that of the imposed free-stream disturbance for the special case of a uniformly pulsating stream.
Abstract: In the first part of this investigation, Goldstein (1983) has shown that the amplitude of the spatially growing Tollmien-Schlichting wave generated by a time-harmonic free-stream disturbance is related to the coefficient multiplying the lowest-order asymptotic eigensolution of the unsteady boundary-layer equation. In the present study, a numerical solution of the unsteady boundary-layer equation is used to relate the amplitude of the asymptotic eigensolution, and consequently of the Tollmien-Schlichting wave, to that of the imposed free-stream disturbance for the special case of a uniformly pulsating stream. It is pointed out that the ideas of this study can be extended to other, more complex bodies and free-stream oscillations.
TL;DR: In this article, the authors examined the local time variation of ssc amplitude at geostationary altitude and found that ssc amplitudes at synchronous altitude have a strong local time dependence in contrast to previous observations in the magnetosphere.
Abstract: Eighty-one storm sudden commencements (ssc) observed on board GOES satellites at geostationary orbit are used to examine the local time variation of ssc amplitude. It is found that ssc amplitudes at synchronous altitude have a strong local time dependence in contrast to previous observations in the magnetosphere. Amplitude ratios (satellite amplitude/ground amplitude) are larger than unity in most cases observed in the daytime during 0600–1500 LT. The ssc amplitude tends to become very small near midnight as compared with that on the ground. Even a decrease in total force sometimes occurs in the midnight region. This means that magnetic compression does not always occur in the magnetic equator near synchronous orbit. The implication of these ssc characteristics are discussed in relation to ssc-associated phenomena.
TL;DR: In this article, the authors compared line-of-sight measurements of the log-amplitude and phase fluctuations of pure tones between 250 and 4000 Hz propagated over distances between 2 and 300 m in the turbulent atmosphere close to the ground using simultaneously measured meteorological variables.
Abstract: Line‐of‐sight measurements of the log‐amplitude and phase fluctuations of pure tones between 250 and 4000 Hz propagated over distances between 2 and 300 m in the turbulent atmosphere close to the ground are compared quantitatively with simple theory using simultaneously measured meteorological variables. The theory is based on the assumption of homogeneous and isotropic turbulence and approximates the availability of eddy sizes in the source region of turbulence by a Gaussian spectrum. In particular the transverse or mutual coherence function (the coherence in a plane perpendicular to the direction of propagation) and the coherence in the direction of propagation which we call the longitudinal coherence, are also calculated and compared with the measurements. When the measured mean square phase fluctuations are compared with the theory using the meteorological measurements, good agreement is obtained. However the measured mean square log‐amplitude fluctuations are in general substantially smaller than predicted and, in addition, show clear evidence of saturation. The distance to saturation is shown to correspond to the longitudinal coherence length. Because of this behavior of the amplitude fluctuations both the transverse and longitudinal coherences are essentially a function of the phase variance only.
TL;DR: In this article, the authors derived an analytical expression of the minimum wave amplitude required to cause strong pitch angle diffusion for plasma sheet electrons and showed that most of the time (>91 %) this typical value of 2 mV m−1 is not reached.
Abstract: On the basis of theoretical calculations of electron diffusion coefficients and of OGO 5 data, Lyons [1974] suggested that electrostatic electron cyclotron harmonic waves had amplitudes large enough to cause the strong pitch angle diffusion of plasma sheet keV electrons and to be responsible for diffuse auroral precipitation. However, recent measurements of the wave location and amplitude performed aboard the GEOS spacecraft have brought new pieces of information challenging these conclusions. Our calculations are based on the theoretical tool developed by Lyons [1974] but take into account the recently observed wave confinement within a few degrees from the magnetic equator. Under these conditions, we avoid the numerical averaging of the pitch angle diffusion coefficient over the whole line of force, and we can derive an analytical expression of the minimum wave amplitude required to cause strong pitch angle diffusion for plasma sheet electrons. This expression has the advantage to be easily tractable in further calculations, and permits us to evaluate the dependence of the results on parameters that are not reported by observations, such as the wave number spectrum. Our results are found to differ from Lyons's [1974] predictions by a factor of about 2.5, and typically, a wave amplitude of more than 2 mV m−1 is required to put 1-keV electrons on strong diffusion. On the other hand, on the basis of a statistical analysis of electron cyclotron wave amplitudes measured in the nightside plasmasheet by the GEOS 2 spacecraft, we show that most of the time (>91 %), this typical value of 2 mV m−1 is not reached. This result appears inconsistent with the hypothesis that diffuse auroras, which are a permanent feature of the auroral zones, are due solely to electrostatic electron cyclotron waves. This leads us to the conclusion that these waves are not the only cause of diffuse electron precipitation. It is suggested that other mechanisms involving for instance the dynamics of the ions (such as field-aligned currents) could play an important role.
TL;DR: Forced oscillations of a two-degree-of-freedom autoparametric system with moderately high excitations were studied in this paper, and the approximate results obtained by the method of harmonic balance were found to be satisfactory by comparing with those obtained by numerical integration.
Abstract: Forced oscillations of a two degree-of-freedom autoparametric system are studied with moderately high excitations. The approximate results obtained by the method of harmonic balance are found to be satisfactory by comparing with those obtained by numerical integration. In the primary parametric instability zone, separate regions of stable and unstable harmonic solutions are obtained. In the regions of unstable harmonic solutions, depending on the forcing amplitude and frequency, the solutions may be amplitude modulated or completely nonperiodic. In the latter case the numerical integrations do not converge.
TL;DR: In this article, a probability density function of the surface elevation of a nonlinear random wave field is obtained for both deep water waves and waves in finite depth, where the amplitude and phase of the first-order component of the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively.
Abstract: Probability density function of the surface elevation of a nonlinear random wave field is obtained. The wave model is based on the Stokes expansion carried to the third order for both deep water waves and waves in finite depth. The amplitude and phase of the first-order component of the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively. The probability density function for the deep water case was found to depend on two parameters: the root-mean-square surface elevation and the significant slope. For water of finite depth, an additional parameter, the nondimensional depth, is also required. An important difference between the present result and the Gram-Charlier representation is that the present probability density functions are always nonnegative. It is also found that the 'constant' term in the Stokes expansion, usually neglected in deterministic studies, plays an important role in determining the details of the density function. The results compare well with laboratory and field experiment data.
TL;DR: In this article, a linear relation between the time-averaged amplitude of the envelope of the backscattered acoustic pulse at 192 kHz and the square root of suspended sediment concentration in the 10 to 103 mg 1−1 range was found.
Abstract: Experimental results are presented which indicate a linear relation between the time-averaged amplitude of the envelope of the backscattered acoustic pulse at 192 kHz and the square root of suspended sediment concentration in the 10 to 103 mg 1−1 range. Particle sizes ranged from 2 to 140 μm. The measurements were made in a negatively buoyant, mine-tailing discharge plume in a submarine channel at depths of 60 to 90 m in Rupert Inlet, British Columbia. From the theory of acoustic backscatter from a solid elastic sphere in the Rayleigh region it is shown that if the pressure amplitude of the backscattered wave is Rayleigh distributed, then such a linear relation is to be expected. Expressions for the optimum acoustic frequency for the detection of dilute suspensions at a given range and for the minimum detectable concentration are obtained assuming a thermal noise background. The possibility that bubbles contribute to the backscatter is considered and found to be unlikely on the basis of probable bubble lifetimes.
Book•
01 Jan 1983
TL;DR: In this paper, the authors evaluate the fatigue resistance of welded attachments subjected to variable amplitude fatigue loading and show that the results obtained from these variable amplitude tests are consistent with the previously reported constant amplitude test.
Abstract: The research described in this report is intended to provide information on evaluating the fatigue resistance of welded attachments subjected to variable amplitude fatigue loading. The research consisted of laboratory studies of welded attachments under random variable amplitude load spectra defined by a Rayleigh-type distribution with most stress-cycles below the constant amplitude fatigue limit. (Some stress cycles exceeded this limit.) Eight full-size beams with web attachments and cover plates were tested during the program. Fatigue crack growth data were generated using random block variable amplitude stress spectra defined by a Rayleigh-type distribution. Also, nonload-carrying fillet-welded cruciform-type specimens were tested under simple bending using a random variable amplitude block loading to supplement the existing shorter life studies carried out on stiffener details. The results obtained from these variable amplitude tests are consistent with the previously reported constant amplitude test. However, the existence of a fatigue limit below which no fatigue cracks propagate is assured only if none of the stress range cycles exceed this constant amplitude fatigue limit. If any of the stress range cycles (as few as one per thousand cycles) exceed the limit, fatigue crack propagation will likely occur. The random variable test data from the beam specimens generally fell between the upper and lower confidence limits projected from constant cycle data. The smaller simulated details generally resulted in the random variable test data falling near the upper confidence limit of constant amplitude test results.
TL;DR: In this paper, a stability analysis of model amplitude equations for rigid boundaries is presented for free-slip boundaries, where the Prandtl number P is finite and the flow is three-dimensional.
Abstract: Internally generated vertical vorticity enters the lowest‐order amplitude equations for free‐slip boundaries in an essential way when the Prandtl number P is finite and the flow three dimensional. For parallel rolls the band of stable wavenumbers is substantially modified from what was previously believed to be correct. In particular there are no stable states for P<0.301. Numerical simulations for free boundaries and larger P suggest a mechanism through which the box size determines the critical Rayleigh number for noisy time dependent convection. A stability analysis of model amplitude equations for rigid boundaries agrees qualitatively with the numerical results of Cleaver or Busse for P≲O(1). There is now considerable continuity between the stability diagrams for rigid and free boundaries.
TL;DR: In this article, the amplitude ratios of the ScS and S phases are used to place averages, and the degree of lateral variation in constraints on the shear-wave velocity gradient D" properties remains an open question.
Abstract: The relative amplitudes and travel have been directed toward obtaining global times of ScS and S phases are utilized to place averages, and the degree of lateral variation in constraints on the shear-wave velocity gradient D" properties remains an open question. above the core-mantle boundary. A previously A conflicting result was found by Mitchell and reported long-period ScSH/SH amplitude ratio Helmberger (1973), who utilized the relative minimum in the distance range 65 o to 70 o is shown amplitudes and timing of long-period ScS and S to be a localized feature, apparently produced by phases to constrain the S-wave velocity gradient an amplitude anomaly in the direct S phase, and in D". They found a minimum in the ScSH/SH therefore need not reflect the velocity gradient amplitude ratio near 68 o , which was attributed to at the base of the mantle. The amplitude ratios low amplitudes of the ScS arrivals. Unable to that are free of this anomaly are consistent with explain this feature by models with negative or calculations for the JB model or models with mild near-zero shear velocity gradients in D", they positive or negative velocity gradients in the proposed models with positive S-wave velocity lowermost 200 km of the mantle. ScSV arrivals gradients above the CMB. These positive are particularly sensitive to the shear velocity gradients extended over 40 to 70 km above the structure just above the core-mantle boundary. core, reaching velocities at the CMB as high as The apparent arrival time of the peak of ScSV is 7.6 to 7.8 km/s. These models can explain the as much as 4 sreater than that of ScSH in the observed amplitude ratio behavior, as well as an distance range 75 v to 80 o for Sea of Okhotsk apparent difference observed in the arrival times events recorded in North America. This can be of transversely and radially polarized ScS. explained by interference effects produced by a Mitchell and Helmberger also proposed a low Q$ localized high velocity layer or strong positive zone in D", or finite outer core rigidity, to S wave velocity gradient in the lowermost 20 km explain the baseline of the ScSH/SH amplitude of the mantle. A velocity increase of about 5% ratios. While the majority of their data was for is required to explain the observed shift between deep South American events recorded in North ScSV and ScSH. This thin, high velocity layer America, they did analyze one deep Sea of Okhotsk varies laterally, as it is not observed in event for which the radial and transverse ScS similar data from Argentine events. Refined arrival times were not different, which suggested estimates of the outermost core P velocity lateral variations in the D" velocity structure. structure are obtained by modeling SKS signals in In this paper we extend the analysis of ScS the distance range 75 o to 85 o s
TL;DR: In this article, the radial order identifications of low-degree modes previously inferred from theory are confirmed and only marginal evidence of long-period, gravity-mode oscillations is found.
Abstract: Measurements are presented of solar velocity oscillations with spherical harmonic degree 1-139 and angular order approximately 0. With an amplitude sensitivity of approximately 2 cm/s, trapped acoustic wave modes of radial orders 2-26 are observed at frequencies between 1.7 and 5.5 mHz. The radial order identifications of low-degree modes previously inferred from theory are confirmed. Only marginal evidence of long-period, gravity-mode oscillations is found.
TL;DR: In this article, an efficient method for computing diffraction integrals is presented based on an idea put forward by Hopkins, where the integration domain is divided into subdomains, in each of which the phase and amplitude are approximated by simple functions which make it possible to evaluate the resulting integral in terms of known functions.
Abstract: An efficient method for computing diffraction integrals is presented. It is based on an idea put forward by Hopkins [1]. The integration domain is divided into subdomains, in each of which the phase and amplitude are approximated by simple functions which make it possible to evaluate the resulting integral in terms of known functions. While Hopkins employed a linear approximation to the phase and a constant approximation to the amplitude, we here approximate both the phase and the amplitude by parabolas. A comparison of the results of our method with those of Hopkins's method shows that our method requires fewer subdomains and less computation time to yield a desired accuracy. Another advantage of our method is that it can be applied to apertures of a general shape without significant loss of accuracy.