Form Of Normal Modes Research Articles

Mean and variance of the forward field propagated through three-dimensional random internal waves in a continental-shelf waveguide

The mean and variance of the acoustic field forward propagated through a stratified ocean waveguide containing three-dimensional (3-D) random internal waves is modeled using an analytic normal mode formulation. The formulation accounts for the accumulated effects of multiple forward scattering. These lead to redistribution of both coherent and incoherent modal energies, including attenuation and dispersion. The inhomogeneous medium’s scatter function density is modeled using the Rayleigh-Born approximation to Green’s theorem to account for random fluctuations in both density and compressibility caused by internal waves. The generalized waveguide extinction theorem is applied to determine attenuation due to scattering from internal wave inhomogeneities. Simulations for typical continental-shelf environments show that when internal wave height exceeds the acoustic wavelength, the acoustic field becomes so randomized that the expected total intensity is dominated by the field variance beyond moderate ranges. This leads to an effectively saturated field that decays monotonically and no longer exhibits the periodic range-dependent modal interference structure present in nonrandom waveguides. Three-dimensional scattering effects can become important when the Fresnel width approaches and exceeds the cross-range coherence length of the internal wave field. Density fluctuations caused by internal waves are found to noticably affect acoustic transmission in certain Arctic environments.

Electronic spectra of molecules with two C3 v internal rotors: Torsional analysis of the [formula omitted] LIF spectrum of biacetyl

The laser-induced fluorescence excitation spectrum of the biacetyl A 1 A u ( S 1)– X 1 A g ( S 0) transition in the region from 22 182 to 22 800 cm −1 shows a complicated absorption line spectrum, which is believed to arise from a long progression in the torsional vibrations of the two equivalent methyl tops in this molecule. In this paper, we discuss three topics: (i) a numerical calculation of these energy levels using a kinetic and potential energy formalism and constants from the literature [M.L. Senent, D.C. Moule, Y.G. Smeyers, A. Toro-Labbé, F.J. Peqalver, J. Mol. Spectrosc. 164 (1994) 66], (ii) a qualitative description of the calculated energy level pattern using local-mode ideas applied to the two equivalent methyl rotors together with G 36 permutation-inversion group symmetry species, and (iii) a least-squares refinement of the torsional potential parameters based on results of some of our high-resolution rotational analyses, followed by a comparison of energy levels calculated from the refined parameters with a low-resolution spectrum taken in the region from 0 to 500 cm −1 above the A– X band origin. Concerning (ii), we find that the two vibrational levels with one quantum of torsional excitation are best described by a normal mode formulation, but that many levels with more than one quantum of torsional excitation are better described by a local-mode formulation. Concerning (iii), we obtain low-order molecular constants quite similar to those reported by Senent et al., but higher-order constants which are quite different. Our calculated spectrum agrees well with the low-resolution spectrum, but full confirmation of the present interpretation for levels with three or more quanta of torsion excited will require high-resolution studies of additional bands.

Nonlinear normal modes for vibratory systems under harmonic excitation

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to harmonic excitation. The approach is an extension of the nonlinear normal mode (NNM) formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree-of-freedom (dof), whose response is known. A reduced-order model for the forced system is then determined by the usual NNM procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple 2-dof mass–spring system with cubic nonlinearities, and for a discretized beam model with 12 dof. The results show that this method provides very accurate responses over a range of frequencies near resonances.

Nonlinear vibrations and chaos in gongs and cymbals

This paper summarizes some results obtained in the last few years for the modeling of nonlinear vibrating instruments such as gongs and cymbals. Linear, weakly nonlinear and chaotic regimes are successively examined. A theoretical mechanical model is presented, based on the nonlinear von Karman equations for thin shallow spherical shells. Modal projection and Nonlinear Normal Mode (NNM) formulation leads to a subset of coupled nonlinear oscillators. Current developments are aimed at using this subset for sound synthesis purpose.

MEAN AND VARIANCE OF THE FORWARD FIELDS PROPAGATED THROUGH 3D RANDOM INTERNAL WAVES WITH RAYLEIGHT-BORN SCATTERING

The mean and variance of the acoustic field forward propagated through an ocean waveguide containing random internal waves is modelled using a normal mode formulation [1] that takes into account the 3-D interaction of the acoustic field with medium inhomogeneities. The effects of dispersion, attenuation and redistribution of modal energy due to random multiple scattering in the forward azimuth are treated analytically. The inhomogeneous medium's scatter function density is modelled using the Rayleigh-Born approximation to Green's theorem to account for random fluctuations in density and compressibility caused by the internal waves. The generalized waveguide extinction theorem [2] is applied to determine the attenuation due to scattering from the internal wave inhomogeneities. Simulations for typical continental shelf environments quantitatively show how coherent and incoherent acoustic field intensities vary with range, depth and internal wave field properties

Spectral and modal formulations for the Doppler-shifted field scattered by an object moving in a stratified medium.

Spectral and normal mode formulations for the three-dimensional field scattered by an object moving in a stratified medium are derived using full-field wave theory. The derivations are based on Green's theorem for the time-domain scalar wave equation and account for Doppler effects induced by target motion as well as source and receiver motion. The formulations are valid when multiple scattering between the object and waveguide boundaries can be neglected, and the scattered field can be expressed as a linear function of the object's plane wave scattering function. The advantage of the spectral formulation is that it incorporates the entire wave number spectrum, including evanescent waves, and therefore can potentially be used at much closer ranges to the target than the modal formulation. The normal mode formulation is more computationally efficient but is limited to longer ranges. For a monochromatic source that excites N incident modes in the waveguide, there will be roughly N2 distinct harmonic components in the scattered field. The Doppler shifts in the scattered field are highly dependent upon the waveguide environment, target shape, and measurement geometry. The Doppler effects are illustrated through a number of canonical examples.

Influence of shallow water currents on the performance of a broadband time-reversing array

In recent shallow water experiments, the temporal and spatial focusing properties of time-reversing arrays (TRAs) were shown to be robust in a reciprocal medium and useful for underwater applications. The presence of oceanic currents in coastal environments leads to nonreciprocal acoustic propagation. In this case, time-reversal invariance is modified because the propagation speed inhomogeneity depends on the direction of acoustic propagation. Therefore, similarly to phase coherent reciprocal transmissions, a TRA will be influenced by the current-induced effects but not by the scalar contributions due to temperature or salinity. TRA performance, in the presence of steady currents, is investigated both theoretically using a simple first order normal mode formulation and numerically using a parabolic equation code for moving media, GCPEM [D. Mikhin, J. Acoust. Soc. Am. 105, 1362 (1999)]. In a multipath shallow ocean environment, the retrofocus field is shifted relative to its location in a nonmoving medium. This shift depends on the current speed and the range-depth dependency of the ocean current profile because each acoustic mode is influenced differently. The possibility of using TRAs for monitoring coastal currents will be discussed.

Receptivity and aeroacoustic resonance in channels with blowing walls

In this article, the creation of instability waves by acoustic waves in the two-dimensional Taylor flow developing in channels with blowing walls is studied. The receptivity of the flow to a localized defect of injection velocity is investigated by assuming normal mode form of the instabilities thanks to a theory of Hill [J. Fluid Mech. 292, 183 (1995)] based on adjoint methods. The results show good agreement with direct numerical simulations. A mechanism of aeroacoustic resonance is numerically put in light which involves the receptivity process together with the creation of acoustic waves by instability waves passing through a surface at constant pressure. Theoretical predictions of the temporal growth or damping rate of both acoustic and instability waves give satisfactory results. Discussion about experimental results available from the so-called VECLA facility is finally provided.

Nonuniform thermal magnetization noise in thin films: Application to GMR heads

A general scheme is developed to analyze the effect of nonuniform thermal magnetization fluctuations in a thin film. The normal mode formalism is utilized to calculate random magnetization fluctuations. The magnetization noise is proportional to the temperature and inversely proportional to the film volume. The total noise power is the sum of normal mode spectral noises and mainly determined by spin-wave standing modes with an odd number of oscillations. The effect rapidly decreases with increasing mode number. An exact analytical calculation is presented for a two-cell model.

Analytical Model for Gravity and Rayleigh-Wave Investigation in the Layered Ocean-Earth Structure

In terms of the linear theory of elasticity we analyze the gravity and Rayleigh-wave excitation and propagation process in a three-layered ocean–Earth model, which includes a superficial liquid layer and two layers of sediments—soft and hard. To compute the oceanic surface-wave dispersion function, the Thomson-Haskell matrix method is used. The solution is obtained by applying the normal mode formalism to the flat, homogeneous, layered ocean–Earth structure. Based on this theory, the spectra of the excitation functions are investigated in detail for 45° thrust, dip-slip, and strike-slip point sources in the half-space. A numerical experiment reveals that the rigidity of the rock exerts a strong influence on both gravity and Rayleigh waves. In the case of Rayleigh waves, a layered structure of the bottom strongly affects the propagation process. The main focus of the present study is to model tsunami and tsunamigenic events. It is well known from observations that earthquakes that generate abnormally large tsunamis are mostly thrust events, characterized by long source duration (2–4 times longer than typical earthquakes of similar moment) and shallow source depth, and are located near the trench axis. Meanwhile, typical tsunamigenic earthquakes are generally dip-slip and thrust events with depths between 15 and 50 km. On this basis, and using the distributed model of seismic source, the influence of source mechanism, location, and source duration on the tsunami-wave amplitude are investigated here. The results of our model computations of the seismic sea waves from tsunami- and tsunamigenic earthquake are in good agreement with real observation data.

Excitation of Surface Waves by a Finite Seismic Source: The Case of Gravity Waves in the Liquid Layer-Elastic Half-Space

The excitation of gravity waves by both point and finite dimensioned line sources was considered in the flat elastic-gravitational model. The excitation functions were analyzed in terms of normal mode formalism in a far-field approximation of the surface wave. A line source was modeled by a set of point sources, uniformly distributed along a fault and moving along it with a constant velocity. A detailed analysis of the double-couple mechanism shows that a line source with a vertical dip-slip motion generates more intensive gravity waves than one with a strike-slip motion. The calculation of theoretical marigrams for gravity waves demonstrates that the amplitude of a wave generated by a line source is larger than that for a point source of the same seismic moment. A numerical experiment reveals that the rigidity of the rock and the source duration both exert a strong influence on the amplitude of the gravity waves generated by an underwater earthquake. A slow rupture with velocity 1-2 km/s occurring in low rigidity rock generates a more intensive gravity wave. It is suggested that the use of the more realistic finite dimensioned line source is appropriate to model ”tsunami-earthquake” events.

A spectral formulation for the Doppler-shifted field scattered by an object moving in a stratified medium

A spectral formulation for the 3-D field scattered by an object moving in a stratified medium is derived using full-field wave theory. The derivation is based on Green’s theorem for the time-domain scalar wave equation and accounts for Doppler effects induced by target motion as well as source and receiver motion. The formulation is valid when multiple scattering between the object and waveguide boundaries can be neglected and the scattered field can be expressed as a linear function of the object’s plane wave scattering function. A normal mode formulation that is more computationally efficient but less general is also derived from first principles. The advantage of the spectral representation is that it incorporates the entire wave number spectrum, including evanescent waves, and so can potentially be used at much closer ranges to the target than the modal formulation. The Doppler effects are illustrated through a number of canonical examples.

Source localization using the waveguide invariant approach and conventional plane wave beamforming with vertical hydrophone array data

Broadband source localization, both in range and depth, can be performed using conventional plane wave beamforming with vertical hydrophone array data. Little computation or prior knowledge of the environment is required. An analytical model, based on the application of the waveguide invariant approach to a normal mode formulation of the sound field, is developed to predict the behavior of striation patterns as a function of vertical wave number, range, and frequency. The approach is illustrated through numerical simulation and applied to data collected during the shallow-water (200-m) SWellEx-3 experiment in 1994. Extension to canonical deep-water environments also is discussed. [Work supported by ONR, Code 321US.]

Amplification of gravity and Rayleigh waves in a layered water-soil model

The coupled seismic to gravitational surface wave fields are analyzed in a liquid layer lying on the gravitating elastic, low-rigidity half-space. Solution is obtained within the framework of the normal mode formalism applied to the flat ocean-solid Earth model. From the theory of propagation of coupled surface waves (Rayleigh and Love) in layered media, we find the individual multipliers that determine the surface wave spectrum over the entire frequency range. Spectra of excitation functions are investigated for dip-slip point source in the half-space. Main results can be summarized as follows. When the half-space is filled with sediments, dip-slip excitation functions of gravity and Rayleigh waves are one order of magnitude larger than for the half-space composed of hard rocks. Including gravity in the elastic medium essentially changes the character of gravity wave spectrum, leading to an appearance of the third maximum. At the deepening of the source amplitude of this maximum increases. Theoretical marigrams show that including gravity in the half-space also increases period of the gravity wave excited by deep sources by a factor of two, up to 10 minutes. At the same time, presence of gravity force in the half-space has no effect on the spectrum of the Rayleigh wave.

On the Perturbations of Viscous Rotating Newtonian Fluids

The perturbations of weakly-viscous, barotropic, non-self-gravitating, Newtonian rotating fluids are analyzed via a single partial differential equation. The results are then used to find an expression for the viscosity-induced normal-mode complex eigenfrequency shift, with respect to the case of adiabatic perturbations. However, the effects of viscosity are assumed to have been incorporated in the unperturbed (equilibrium) model. This paper is an extension of the normal-mode formalism developed by Ipser & Lindblom for adiabatic pulsations of purely-rotating perfect fluids. The formulas derived are readily applicable to the perturbations of thin and thick accretion disks. We provide explicit expressions for thin disks, employing results from previous relativistic analyses of adiabatic normal modes of oscillation. In this case, we find that viscosity causes the fundamental p- and g- modes to grow while the fundamental c-mode could have either sign of the damping rate.

Low-frequency vibrational spectra of crystals of tutton salts

IR absorption spectra and polarized Raman spectra of crystals of Tutton salts K2M(SO4)26H2O and (NH4)2M(SO4)2·6H2O, where M=Co, Ni, Zn, have been obtained by experiment at 93 K and at room temperature. The frequencies and forms of normal modes of the [Zn(H2O)6]2+ octahedral complex have been calculated. The observed lines are assigned to the internal modes of the [M(H2O)6]2+ complex and external modes of the crystal lattice in accordance with the results of the calculations and factor-group analysis.

Surface loading of a viscoelastic planet--III. Aspherical models

Summary In Paper I of this series we developed a generalized normal-mode formalism for computing the response of an aspherical, self-gravitating, linear viscoelastic earth model to a surface load. In the present article we introduce an expansion for the normal modes of an aspherical earth model, using as basis functions the normal modes of a spherically symmetric reference model. This expansion leads to a non-linear eigenvalue problem for the expansion coefficients and decay rates. We develop a linearization of this problem using perturbation theory, which incorporates arbitrary levels of coupling between normal-mode multiplets. As an illustration, we consider the special case of radial perturbations to a spherically symmetric model, and compare predictions based upon perturbation theory with those based upon the usual non-perturbative forward theory. We demonstrate that including overtone coupling is necessary for the accurate prediction of perturbations to the normal-mode decay times and eigenfunctions. These calculations suggest that our theory can accurately accommodate order of magnitude lateral variations in mantle viscosity and significant changes in lithospheric thickness.

Elimination of branch cuts from the normal-mode solution using gradient half spaces

A method for eliminating the branch cuts and branch line integrals from the normal-mode solution to the range-invariant acoustic wave equation has been developed and implemented. In the usual normal-mode formulation, evaluation of the vertical wave number in the ocean waveguide’s lower half space involves a square root operation and gives rise to a branch point at the critical angle. When the Pekeris cut is chosen, energy steeper than the critical angle is represented by leaky modes, and lateral-wave energy near the critical angle is represented by the branch line integral. When a mode lies near the branch cut, the branch line integral can be significant at all ranges. Branch line integrals are cumbersome to compute, and, for range-dependent problems, cannot be included in adiabatic and coupled-mode algorithms. By inserting small attenuation and sound-speed gradients in the lower half space, the plane-wave solution is replaced by an Airy function solution, the branch point is eliminated, the Pekeris branch cut is replaced by a series of modes, and the leaky modes become bounded at infinite depth. Implementation of and example calculations using this approach using the normal-mode model ORCA are given.

A one-way coupled mode solution, Huygens’ principle and mode coupling coefficients

The normal mode formulation for waveguide propagation has physical appeal since sound propagates in discrete entities—namely modes—and an understanding of the progression of modes sheds light on arrival times of pulses and features of ocean ducting. In the development of the range-dependent version several researchers employ the conservation of normal pressure and normal particle velocity along the z axis in the mean as the conserved quantity that allows for an algorithm to couple the modes as sound progresses in range. The usual derivations are based on R. Evan’s seminal work on the two-way coupled mode formulation. Here it is shown that Huygens’ principle and an approximate form of the Green’s function in the waveguide leads to a suitable algorithm as long as reflected or backward going flux is small relative to forward flux. The derivation, moreover, leads to expressions for mode coupling amplitudes that allow one to follow the distribution of flux along specific modal tracts. [Work supported by the Office of Naval Research and Naval Research Laboratory.]

The effects of the theoretical formalism and data selection on mantle models derived from waveform tomography

Summary Most global tomographic models to date are derived using a combination of surface wave (or normal-mode) data and body wave traveltime data. The traveltime approach limits the number of phases available for inversion by requiring them to be isolated on the seismogram. This may ultimately result in limiting the resolution of 3-D structure, at least in some depth ranges in the mantle. In a previous study, we successfully derived a degree 12 whole-mantle SH-velocity tomographic model (SAW12D) using exclusively waveform data. In that inversion, a normal-mode formalism suitable for body waveforms, the non-linear asymptotic coupling theory (NACT), was combined with a body wave windowing scheme, referred to as the ‘individual wavepacket’ (IW) technique, which allows one to assign individual weights to different body wave energy packets. We here compare the relative merits of this choice of theoretical formalism and windowing scheme at different depth ranges in the mantle. Choosing as the reference a model obtained using 7500 transverse-component body wave and 8000 surface wave seismograms and the NACT and IW approaches, we discuss the relative performance of the path average approximation (PAVA), a zeroth-order theoretical approximation appropriate for single-mode surface waves, relative to NACT, and compare the IW windowing scheme with a more standard ‘full window’ (FW) approach, in which a single time window is considered from the first body wave arrival to the fundamental-mode surface waves. The combination PAVA/FW is often used in global tomography to supplement the traveltime data. We show that although the quality of the image derived under the PAVA/FW formalism is very similar to that derived under NACT/IW in the first 300 km of the upper mantle, where the resolution is dominated by surface waves, it deteriorates at greater depths. Images of the lower mantle are shown to be strongly sensitive to the theoretical formalism. In contrast, the resolution of structure near the core–mantle boundary depends mostly on the windowing scheme. This is because this resolution is controlled by low-amplitude phases such as Sdiff, which are downweighted in the FW scheme. Whilst the image obtained in D″ using the combination NACT/IW is in good agreement with images obtained by other authors using both waveforms and traveltimes, we show that, when using FW, uppermost mantle structure can be mapped into D″. This result is confirmed by synthetic tests performed on a composite of the upper-mantle geodynamic model 3SMAC. We also show, based on synthetic tests, that for structures in the upper mantle with sharp boundaries, differences are observed between NACT and PAVA. Whilst a combination of traveltimes and surface wave data is adequate for resolving relatively smooth features in the mantle, our results show that by potentially increasing the achievable sampling, the waveform approach shows great promise for future high-resolution tomographic modelling of mantle structure, if cast in an appropriate theoretical framework.