Showing papers on "Normal mode published in 1996"

Large Amplitude Elastic Motions in Proteins from a Single-Parameter, Atomic Analysis.

Abstract: Normal mode analysis (NMA) is a leading method for studying long-time dynamics and elasticity of biomolecules. The method proceeds from complex semiempirical potentials characterizing the covalent and noncovalent interactions between atoms. It is widely accepted that such detailed potentials are essential to the success of NMA’s. We show that a single-parameter potential is sufficient to reproduce the slow dynamics in good detail. Costly and inaccurate energy minimizations are eliminated, permitting direct analysis of crystal coordinates. The technique can be used for new applications, such as mapping of one crystal form to another by means of slow modes, and studying anomalous dynamics of large proteins and complexes. [S0031-9007(96)01063-0] PACS numbers: 87.15.By, 87.15.He Thermal equilibrium fluctuations of the x-ray crystal coordinates of proteins provide a basis for understanding the complex dynamics and elasticity of biological macromolecules [1]. Analysis of the normal modes of globular proteins shows an interesting anomaly. The density of the slow vibrational modes is proportional to their frequency, gsv d, v, rather than gsv d, v 2 as predicted by Debye’s theory [2]. Yet, the atoms in globular proteins are packed as tightly as in solids. We show that a single-parameter potential reproduces the slow elastic modes of proteins obtained with vastly more complex empirical potentials. The simplicity of the potential permits greater insight and understanding of the mechanisms that underlie the slow, anomalous motions in biological macromolecules such as proteins. To date, normal modes of globular proteins have been used to reproduce crystallographic temperature factors [3] and diffuse scatter [4]. Normal mode analyses (NMA’s) shed light on shear and hinge motions necessary for catalytic reactions, and have been used with some success to map one crystal form of a protein into another [5]. Finally, NMA’s yield macroscopic elastic moduli of large protein assemblies, based on their microscopic structure [6]. NMA studies of macromolecules are handicapped, however, by the complex phenomenological potentials used to model the covalent and nonbonded interactions between atom pairs. The necessary initial energy minimization requires a great deal of computer time and memory, and is virtually impossible for even moderately large proteins (with typically thousands of degrees of freedom) with a reasonable degree of accuracy. This inevitably leads to unstable modes which must be eliminated through elaborate methods, and which cast doubts on the validity of the analysis. Moreover, partly because the minimization is carried out in vacuo, the final configuration disagrees with the known crystallographic structure, complicating the interpretation of the results of NMA. A typical example of a semiempirical potential used in molecular dynamics studies and NMA’s has the form [7] Ep › 1 X bonds Kbsb 2 b0d 2 1 1 X angles Kus u2u 0 d 2

Vibrational wave functions and spectroscopy of (H2O)n, n=2,3,4,5: Vibrational self‐consistent field with correlation corrections

Abstract: Vibrational energy levels, wave functions, and ir absorption intensities are computed for (H2O)n clusters with n=2, 3, 4, and 5. The calculations were carried out by the vibrational self‐consistent field (VSCF) approximation, with corrections for correlations between the modes by perturbation theory. This correlation corrected VSCF (CC‐VSCF) is analogous to the familiar Moller–Plesset method in electronic structure theory. Test calculations indicate that this method is of very good accuracy also for very anharmonic systems. While the method is of highest relative accuracy for the stiffest modes, it works very well also for the soft ones. Some of the main results are (1) the frequencies calculated are in good but incomplete agreement with experimental data available for some of the intramolecular mode excitations. The deviations are attributed to the inaccuracy of the coupling between intramolecular and intermolecular modes for the potential function used. (2) Insight is gained into the pattern of blue‐ or redshifts from the corresponding harmonic excitation energies for the various modes. (3) Anharmonic coupling between the modes dominates in general over the intrinsic anharmonicity of individual modes in determining the spectrum. (4) The anharmonic corrections to the frequencies of some intermolecular modes (shearing, torsional) are extremely large, and exceed 100% or more in many cases. (5) An approximation of quartic potential field in the normal mode displacement is tested for the clusters. It works well for the high and intermediate frequency modes, but is in error for very soft shearing and torsional modes. (6) The relative errors of the VSCF approximation are found to decrease with the cluster size. This is extremely encouraging for calculations of large clusters, since the VSCF level is computationally simple.

A normal mode model for acousto‐elastic ocean environments

Abstract: A normal mode method for propagation modeling in acousto‐elastic ocean waveguides is described. The compressional (p‐) and shear (s‐) wave propagation speeds in the multilayer environment may be constant or have a gradient (1/c2 linear) in each layer. Mode eigenvalues are found by analytically computing the downward‐ and upward‐looking plane wave reflection coefficients R1 and R2 at a reference depth in the fluid and searching the complex k plane for points where the product R1R2=1. The complex k‐plane search is greatly simplified by following the path along which |R1R2|=1. Modes are found as points on the path where the phase of R1R2 is a multiple of 2π. The direction of the path is found by computing the derivatives d(R1R2)/dk analytically. Leaky modes are found, allowing the mode solution to be accurate at short ranges. Seismic interface modes such as the Scholte and Stonely modes are also found. Multiple ducts in the sound speed profile are handled by employing multiple reference depths. Use of Airy function solutions to the wave equation in each layer when computing R1 and R2 results in computation times that increase only linearly with frequency.

Generation and detection of photons in a cavity with a resonantly oscillating boundary

Abstract: The problem of photon creation from vacuum in an ideal cavity with vibrating walls is studied in the resonance case, when the frequency of vibrations equals twice the frequency of some unperturbed electromagnetic mode. Analytical solutions are obtained in two cases: for the one-dimensional model (scalar electrodynamics) and for the three-dimensional (3D) cavity. In the first example, we have a strong intermode interaction; nonetheless, an explicit solution in terms of the complete elliptic integrals is found. The rate of photon generation in the principal mode rapidly assumes a constant value proportional to the product of the frequency by the dimensionless amplitude of oscillations. The total amount of photons created in all the modes increases in time as ${\mathit{t}}^{2}$. In the second example, the eigenmode spectrum is nonequidistant and the problem can be reduced to the problem of a single harmonic oscillator with a time-dependent frequency. The number of photons in the resonant mode of a 3D cavity increases exponentially in time and the field appears in a highly squeezed state with a strongly oscillating photon distribution function. The problem of detecting the created photons is analyzed in the framework of a simplified model, when a detector is replaced with a harmonic oscillator. It turns out that the presence of the detector changes the picture drastically: both the detector and the field mode occur in highly mixed (nonthermal) quantum states, with identical nonoscillating photon distribution functions. The detector gains exactly half of the total energy of excitation inside the cavity. The estimations show a possibility of creating up to several hundred or even thousand photons, provided that the cavity's Q factor exceeds ${10}^{10}$ and the amplitude of the wall's oscillations is greater than ${10}^{\mathrm{\ensuremath{-}}10}$ cm at a frequency of the order of 10 GHz. \textcopyright{} 1996 The American Physical Society.

A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation

Abstract: The propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition. The approach adopted is to construct two infinite first‐order differential equations for the components of the pressure and the velocity projected over the normal modes. From these an infinite matricial Riccati equation is derived for the impedance matrix. These equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes. The stiffness of the pressure‐velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide. The method is checked using different examples where the solutions of the plane‐wave approximation or the finite element method are known. Results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare.

Mechanisms of fast self pulsations in two-section DFB lasers

Abstract: We show theoretically, that the detuning between the resonance frequencies of differently pumped DFB sections gives rise to two different pulsation mechanisms, 1) dispersive self Q-switching of a single-mode and 2) beating oscillations between two modes of nearly equal threshold gain. Our analysis is based on the dynamic coupled wave equations accomplished with carrier rate equations. We demonstrate the existence of certain isolated values of the detuning between both sections, at which two longitudinal eigenmodes become degenerate. In the degeneration point, the longitudinal excess factor of spontaneous emission has a singularity and the system of eigenmodes becomes incomplete. We derive reduced equations governing the dynamics in the vicinity of degeneration points. For an example device, the numerical integration of these equations clearly demonstrates the two different self pulsations with repetition rates of more than 100 GHz.

Time resolved four‐ and six‐wave mixing in liquids. I. Theory

Abstract: Low‐frequency intermolecular dynamics in liquids is studied by ultrafast four‐ and six‐wave mixing. The theory of these nonlinear optical processes is given for electronically nonresonant optical interactions up to fifth order in the electric field. The Born–Oppenheimer approximation is used to separate the motional part of the response functions from coordinate independent electronic hyperpolarizabilities. A large variety of experiments, involving far‐infrared absorption, ordinary Rayleigh–Raman or hyper Rayleigh–Raman scattering is covered by this theory. The response in nonresonant six‐wave mixing comprises four dynamically different processes. It is shown that one of the terms contains information on the time scale(s) of intermolecular dynamics, that is not available from lower‐order nonresonant experiments. For instance, homogenous and inhomogeneous contributions to line broadening can be distinguished. The optical response of harmonic nuclear motion is calculated for nonlinear coordinate dependence of the polarizabilities. Results for level‐dependent and level‐independent damping of the motion are compared. It is shown that level‐dependent damping destroys the interference between different quantum mechanical pathways, yielding an extra contribution to the fifth‐order response that has not been discussed before. When two or more nuclear modes determine the optical response, their relative contributions to the four‐ and six‐wave mixing signals are in general different. These contributions are determined by the coordinate dependence of the electronic polarizability, which is usually not fully known. Model calculations are presented for the dynamic parameters of liquid CS2. The theory of this paper will be employed in Part II, to analyze experimental results on femtosecond four‐ and six‐wave mixing.

A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics

Abstract: A finite element for planar beams with active constrained layer (ACL) damping treatments is presented. Features of this non-shear locking element include a time-domain viscoelastic material model, and the ability to readily accommodate segmented (i.e. non-continuous) constraining layers. These features are potentially important in active control applications: the frequency-dependent stiffness and damping of the viscoelastic material directly affects system modal frequencies and damping; the high local damping of the viscoelastic layer can result in complex vibration modes and differences in the relative phase of vibration between points; and segmentation, an effective means of increasing passive damping in long- wavelength vibration modes, affords multiple control inputs and improved performance in an active constrained layer application. The anelastic displacement fields (ADF) method is used to implement the viscoelastic material model, enabling the straightforward development of time-domain finite elements. The performance of the finite element is verified through several sample modal analyses, including proportional-derivative control based on discrete strain sensing. Because of phasing associated with mode shapes, control using a single continuous ACL can be destabilizing. A segmented ACL is more robust than the continuous treatment, in that the damping of modes at least up to the number of independent patches is increased by control action.

The ‘‘Schroeder frequency’’ revisited

Abstract: It is noted that the cross‐over frequency that marks the transition from individual resonances of a multimode system to overlapping normal modes, when expressed as a cross‐over wavelength, equals—within a numerical constant—the diffuse‐field distance in both three‐ and two‐dimensional resonators.

Hydroelastic vibration of rectangular plates

Abstract: This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

Estimation of Mass, Stiffness and Damping Matrices from Frequency Response Functions

Abstract: A frequency-domain method for estimating the mass, stiffness and damping matrices of the model of a structure is presented. The developed method is based on our previous work on the extraction of normal modes from the complex modes of a structure. A transformation matrix is obtained from the relationship between the complex and the normal frequency response functions ofa structure. The transformation matrix is employed to calculate the damping matrix of the system. The mass and the stiffness matrices are identified from the normal frequency response functions by using the least squares method. Two simulated systems are employed to illustrate the applicability of the proposed method. The results indicate that the damping matrix can be identified accurately by the proposed method. The reason for the good results is that the damping matrix is identified independently from the mass and the stiffness matrices. In addition, the robustness of the new approach to uniformly distributed measurement noise is also addressed.

Quasi-Modes as Dissipative Magnetohydrodynamic Eigenmodes: Results for One-Dimensional Equilibrium States

Abstract: Quasi-modes, which are important for understanding the MHD wave behavior of solar and astro-physical magnetic plasmas, are computed as eigenmodes of the linear dissipative MHD equations. This eigenmode computation is carried out with a simple numerical scheme, which is based on analytical solutions to the dissipative MHD equations in the quasi-singular resonance layer. Nonuniformity in magnetic field and plasma density gives rise to a continuous spectrum of resonant frequencies. Global discrete eigenmodes with characteristic frequencies lying within the range of the continuous spectrum may couple to localized resonant Alfven waves. In ideal MHD, these modes are not eigenmodes of the Hermitian ideal MHD operator, but are found as a temporal dominant, global, exponentially decaying response to an initial perturbation. In dissipative MHD, they are really eigenmodes with damping becoming independent of the dissipation mechanism in the limit of vanishing dissipation. An analytical solution of these global modes is found in the dissipative layer around the resonant Alfvenic position. Using the analytical solution to cross the quasi-singular resonance layer, the required numerical effort of the eigenvalue scheme is limited to the integration of the ideal MHD equations in regions away from any singularity. The presented scheme allows for a straightforward parametric study. The method is checked with known ideal quasi-mode frequencies found for a one-dimensional box model for the Earthis magnetosphere (Zhu & Kivelson). The agreement is excellent. The dependence of the oscillation frequency on the wavenumbers for a one-dimensional slab model for coronal loops found by Ofman, Davila, & Steinolfson is also easily recovered.

Normal-mode constraints on the structure of the Earth

Abstract: We have analyzed the splitting characteristics of 75 spheroidal free oscillations excited by the Great 1994 Bolivia and Kuril Islands earthquakes. These spheroidal modes may be roughly subdivided in terms of 8 radial modes, 40 mantle modes, and 27 core-sensitive modes. The splitting of each mode is corrected for the effects of rotation and hydrostatic ellipticity. The remaining signal is due to lateral variations in the mantle and core and may be expressed in terms of so-called splitting functions, which represent a local radial average of the Earth's even three-dimensional heterogeneity. In the surface-wave limit, splitting functions are the equivalent of an even-degree phase velocity map. Each mode is uniquely sensitive to the Earth's structure. Some modes are predominantly sensitive to compressional velocities in the upper mantle, others to shear velocity variations in the lowermost mantle, and some modes “see” the inner core. As part of our analysis, we determine the center frequency and quality factor of each individual mode; these observations constrain the terrestrial monopole. Collectively, the normal-mode splitting observations presented in this paper put constraints on the large-scale, even structure of the entire Earth. We compare the observed splitting functions with predictions from three recent Harvard models: SH12WM13, SKS12WM12, and PS12WM13. These models are constrained by traveltime and waveform data but contain no normal-mode information. We demonstrate that large-scale, even structure is quite accurately represented in current Earth models, but that the splitting of some predominantly compressional modes is not satisfactorily explained. Although a distinct mantle signal is observed in the splitting functions of core-sensitive modes, a characteristic zonal degree 2 pattern is missing. This missing signal is believed to be the result of inner core anisotropy.

Low-frequency atomic motion in a model glass

Abstract: A molecular-dynamics simulation is presented which explores the microscopic dynamics of a monatomic model glass. The investigated systems consist of up to 32000 atoms interacting via a Lennard-Jones potential. The normal-modes analysis has been used to determine the pattern of atomic displacements. Except for the highest frequencies, all the vibrational modes are found to be delocalized. In the lowest-frequency region the pattern of atomic displacements associated with a given eigenmode is composed by an uncorrelated random component plus well-defined sinusoidal-like waves. The two components are of comparable amplitudes.

Inertia and ion Landau damping of low‐frequency magnetohydrodynamical modes in tokamaks

Abstract: The inertia and Landau damping of low‐frequency magnetohydrodynamical modes are investigated using the drift‐kinetic energy principle for the motion along the magnetic field. Toroidal trapping of the ions decreases the Landau damping and increases the inertia for frequencies below (r/R)1/2vthi/qR. The theory is applied to toroidicity‐induced Alfven eigenmodes and to resistive wall modes in rotating plasmas. An explanation of the beta‐induced Alfven eigenmode is given in terms of the Pfirsch–Schluter‐like enhancement of inertia at low frequency. The toroidal inertia enhancement also increases the effects of plasma rotation on resistive wall modes.

A simple and accurate added mass model for hydrodynamic fluid—Structure interaction analysis

Abstract: A theoretical model of an added mass representation for a flexible cylinder vibrating in a fluid medium is presented. To accomplish this, the fluid-structure interaction problem under the influence of harmonic ground and inertia dominated hydrodynamic loading, is first studied by solving the coupled differential equations exactly. Explicit expressions for computing the hydrodynamic interaction pressure and eigenquantities like natural frequencies and mode shapes are given here. However, this analytical model, as in many other mathematical models, suffers from a severe handicap; its expressions are too complicated and require the use of a computer program to generate the results. One solution which is of particular interest, is the computation of natural frequencies. Using the added mass representation, a simple formula for evaluating the natural frequency is proposed. The formula is very simple to use, requiring only a minimal computational effort on a standard calculator. Comparison with the analytical solutions shows that the formula is extremely accurate, with errors under 0.5% or less, in nearly all the cases tested. Also, more importantly, this accuracy does not appear to deteriorate in the computation of higher natural frequencies, and thus should be very useful for designers working in the dynamics of submerged structures, taking into account their hydrodynamic interactions.

Structure and vibrations of catechol and catechol⋅H2O(D2O) in the S0 and S1 state

Abstract: The inter‐ and intramolecular vibrations in the S0 and S1 state of catechol, d2‐catechol, catechol(H2O)1, and d2‐catechol (D2O)1 have been investigated experimentally by resonant two photon ionization (R2PI), spectral hole burning (SHB), and dispersed fluorescence spectroscopy (DF). The experimental frequencies are compared to the vibrational frequencies obtained from ab initio normal mode calculations using the 6‐31G(d,p) basis set. In order to get a complete interpretation of the S0 state spectra of d2‐catechol the strong coupling of the two OD torsional motions has been taken into account. A two‐dimensional calculation of the torsional eigenvalues based on an ab initio potential [6‐31G(d,p) basis] obtained from single point calculations is presented. Due to these calculations all vibrations in the S0 state can be assigned. Furthermore a new assignment of the vibrations in the S1 state of d2‐catechol is given. In the case of catechol (H2O)1 [d2‐catechol(D2O)1] different structural isomers are discussed....

Modal analysis of spontaneous emission in a planar microcavity.

Abstract: A complete set of cavity modes in planar dielectric microcavities is presented which naturally includes guided modes. We show that most of these orthonormal fields can be derived from a coherent superposition of plane waves incoming on the stack from the air and from the substrate. Spontaneous emission of a dipole located inside the microcavity is analyzed, in terms of cavity modes. Derivation of the radiation pattern in the air and in the substrate is presented. The power emitted into the guided modes is also determined. Finally, a numerical analysis of the radiative properties of an erbium atom located in a Fabry-P\'erot multilayer dielectric microcavity is investigated. We show that a large amount of light is emitted into the guided modes of the structure, in spite of the Fabry-P\'erot resonance, which increases the spontaneous emission rate in a normal direction. \textcopyright{} 1996 The American Physical Society.

Scintillating shallow‐water waveguides

Abstract: An analysis of acoustic wave propagation in a random shallow‐water waveguide with an energy absorbing sub‐bottom is presented, in which deviations of the index of refraction are a stochastic process. The specific model studied is motivated by the oceanic waveguide in shallow waters, in which the sub‐bottom sediment leads to energy loss from the acoustic field, and the stochastic process results from internal (i.e., density) waves. In terms of the normal modes of the waveguide, the randomness leads to mode coupling while the energy loss results from different attenuation rates for the various modes (i.e., mode stripping). The distinction in shallow water is that there exists a competition between the mode‐coupling terms, which redistribute the modal energies, and mode stripping, which results in an irreversible loss of energy. Theoretically, averaged equations are formulated for both the modal intensities and fluctuations (the second and fourth moments of acoustic pressure, respectively), similar to previous formulations which, however, did not include the effects of sub‐bottom absorption of acoustic energy. The theory developed here predicts that there is a mismatch between decay rates between the second and fourth moments, implying that the scintillation index (which is a measure of the strength of the random scattering) grows exponentially in range. Thus the usual concept of equilibrium or saturated statistics must be modified. This theoretical prediction is generally valid and depends only on assuming the forward scattering approximation, the Markov approximation (i.e., the short‐range nature of the correlations between sound‐speed fluctuations) and neglecting the cross‐modal coherences. In order to assess the importance of these assumptions Monte Carlo simulations of stochastic coupled‐mode equations are presented. For these simulations, models of internal‐wave processes, deterministic shallow‐water acoustic environments, and sub‐bottom attenuation that simplify the numerical computation were chosen. While these models are unrealistic they illustrate the theoretically predicted behavior.

The Dynamics of Ultrasonically Levitated Drops in an Electric Field

Abstract: Ultrasonic and electrostatic levitation techniques have allowed the experimental investigation of the nonlinear oscillatory dynamics of free droplets with diameter between 0.1 and 0.4 cm. The measurement of the resonance frequencies of the first three normal modes of large amplitude shape oscillations in an electric field of varying magnitude has been carried out with and without surface charges for weakly conducting liquids in air. These oscillations of nonspherical levitated drops have been driven by either modulating the ultrasonic field or by using a time‐varying electric field, and the free decay from the oscillatory state has been recorded. A decrease in the resonance frequency of the driven fundamental quadrupole mode has been measured for increasing oblate deformation in the absence of an electric field. Similarly, a decrease in this frequency has also been found for increasing DC electric field magnitude. A soft nonlinearity exists in the amplitude dependence of the resonant mode frequencies for freely decaying as well as ultrasonically and electrically driven uncharged drops. This decrease in resonance frequency is accentuated by the presence of free surface charge on the drop. Subharmonic resonance excitation has been observed for drops in a time‐varying electric field, and hysteresis exists for resonant modes driven to large amplitude. Mode coupling from lower‐order resonances to higher‐order modes has been found to be very weak, even for fairly large amplitude shape oscillations. Most of these results are in general agreement with predictions from recent analytical and numerical investigations.

Solvation and solvent effects on the short‐time photodissociation dynamics of CH2I2 from resonance Raman spectroscopy

Abstract: Resonance Raman spectra of CH2I2 have been obtained at excitation wavelengths of 369, 355, and 342 nm in cyclohexane solution and in methanol solution at excitation wavelengths of 355 and 342 nm. Resonance Raman spectra were also measured for CH2I2 in the vapor phase with an excitation wavelength of 355 nm. The resonance Raman spectra of CH2I2 exhibit most of their intensity in fundamentals, overtones, and combination bands of modes nominally assigned as the I–C–I symmetric stretch, the I–C–I bend, and the I–C–I antisymmetric stretch vibrations. The absorption spectra and resonance Raman intensities of the gas phase and methanol solution phase diiodomethane spectra were simulated using a simple model and time‐dependent wave packet calculations. Normal mode coefficients from normal coordinate calculations were used to convert the motion of the wave packet on the excited electronic state surface from dimensionless normal coordinates into internal coordinates of the molecule. The short‐time photodissociation...

Canonical model of volcano acoustics

Abstract: A full wave-theoretical model is developed for the acoustic field generated by an explosive point source embedded in a magma column that is open to the atmosphere. The Green's functions for the field in the magma and the atmosphere are derived on the basis of several simplifying assumptions concerning the geometry of the conduit, the boundary conditions, and the geoacoustic properties of the magma. A kinematic model for the acoustic signature of the explosive source is proposed, which, when combined with the Green's functions, provides full analytical expressions for the acoustic field in the magma and in the atmosphere as (complex) functions of frequency. By Fourier inversion the airborne pressure spectrum is transformed into a pressure time series. The predicted sound pulse in the atmosphere and its energy spectrum are highly dispersive, showing complicated structure that arises from the coherent addition of many normal modes of oscillation (i.e. depth and radial resonances) in the magma column. At high frequencies, for which the aperture of the vent is many wavelengths across, each mode is launched into the atmosphere as a parallel-sided beam of sound with a characteristic angle of elevation, which, through Snell's law, is determined by the speed of sound in the magma relative to that in air. At somewhat lower frequencies, the modal beams of sound undergo angular spreading due to diffraction at the edge of the vent. In the lowest-frequency regime, where the wavelength is comparable with the aperture, the airborne field shows little angular structure. A comparison between airborne acoustic data that we recorded in July 1994 at the western vent of Stromboli Volcano and the predictions of the theory, using parameters that are characteristic of Stromboli, show compelling agreement. The theoretical and observed power spectra both display the following features: (1) a concentration of energy below 20 Hz, associated with the first four longitudinal resonances; (2) radial resonances between 35 and 65 Hz; and (3) a broad minimum around 30 Hz, arising because the source lies near nulls in longitudinal modes that would otherwise be excited. The conclusion is that the airborne sound signature from an explosive volcanic event may be inverted to provide estimates of the depth and radius of the magma conduit, the depth, spectral shape and peak shock-wave pressure of the source, and the viscosity of the magma.

Vibrational wave functions and energy levels of large anharmonic clusters: A vibrational SCF study of (Ar) 13

Abstract: The vibrational ground state and the fundamental excited states of (Ar)13 were studied by vibrational self‐consistent field (VSCF) calculations. These calculations treat the interaction between different modes through a mean potential approximation, and incorporate anharmonicity in full. The good accuracy of VSCF for such systems was demonstrated by test calculations for (Ar)3 and other clusters. The study of (Ar)13 focused on the properties of the wave functions and the excitation energies, on the role of the coupling between the modes and on the deviation from the harmonic approximation. It was found that SCF excitation energies for the fundamental transitions differ from the harmonic values by about 25% for the softest modes, and by about 10% for the stiffest modes. Coupling between the modes, treated by SCF, was found to be much more important than the intrinsic anharmonicity of the individual modes. For the ground state, the harmonic wave function compares well with VSCF, but for the fundamental excited states appreciable differences were found. The results for a potential field expanded to fourth‐order polynomial in the normal mode displacements are found to be valid, almost indentical with those for a more elaborate sixth‐order polynomial expansion. The fundamental excitation frequencies computed using the Aziz–Slaman Ar–Ar pair potential are very similar, with some quantitative deviations, to the values obtained with a Lennard‐Jones potential. The differences are larger for certain specific modes, and very small for the others. These calculations demonstrate the computational power of VSCF as a tool for quantum‐mechanical calculations for large clusters, at the level of specific wave functions.