Showing papers on "Normal mode published in 1979"

Some surface effects in the hydrodynamic model of metals

Abstract: This review aims mainly to disencumber the proper heuristic functions of the model from such misconceptions, avoidable in the light of the classic papers by Block (1933), Jensen (1937) and Samoilovich (1945). The boundary conditions and linearised differential equations are established without cutoff; they determine the normal modes and orthogonality relations. The model is quantised through its normal modes, and the equal-time commutation rules are discussed. The equations in Fourier space are found; it is argued that a cutoff, if required, should be imposed on the Hamiltonian in this representation and before diagonalisation, and the consequences are explored. With such a cutoff, surface though not bulk modes become dispersive even when beta =0. The formalism is applied briefly to image potentials, and in more detail to the attraction between two half-spaces; the role of bulk modes (when beta >0) is stressed; the asymptotics are discussed at long and short distances.

The Natural Modes of Graphite/Epoxy Cantilever Plates and Shells

Abstract: The natural frequencies and mode shapes of a number of Graphite/ Epoxy and Graphite/Epoxy-Aluminum plates and shells were experimen tally determined. The samples tested include 8 ply Graphite/Epoxy plates, cyclindrical shell sections, and Graphite/Epoxy-Aluminum hybrid plates of various laminates and aspect ratio. Fabrication and test procedures are described. Natural frequency and mode shape results are compared with those calculated by a finite element analysis. Agreement between calcu lated and observed mode shapes is excellent; while reasonable agreement is found for frequencies. Among the sources of this discrepancy in frequency results is the possibility of a difference between dynamic-flexural moduli and static in-plane moduli.

SNAP: The SACLANTCEN Normal-Mode Acoustic Propagation Model

Abstract: : A sound-propagation model based on normal mode theory is described. The model is designed to give a realistic treatment of the ocean environment, including arbitrary sound-speed profiles in both water column and bottom, compressional and shear wave attenuation, scattering at rough boundaries, and range dependence. Furthermore, the model has a flexible input/output structure that facilitates model handling and provides users with a wide choice of computational (output) options, ranging from plots of sound-speed profiles and individual mode functions to contoured transmission loss versus depth and range or versus frequency and range. The computer code is written in FORTRAN V with a few routines in NUALGOL. The version documented here runs on a UNIVAC 1106.

A normal mode theory of acoustic Doppler effects in the oceanic waveguide

Abstract: This paper considers the problem of computing the acoustic field generated by a moving point source. In particular, the acoustic field is obtained in terms of the normal modes of a horizontally stratified ocean. The source motion is assumed to be uniform (unaccelerated), but is not restricted to a path radial to the receiver. The structure of the Fourier inversion integral is carefully analyzed and an evaluation is carried out by the method of stationary phase. The stationary phase point is explicitly computed as an expansion in powers of the ratio of the source speed to the mode group velocity. The resulting expression for the velocity potential is examined for Doppler effects for both instantaneous (modal) Doppler as well as Doppler determined by a finite bandwidth Fourier transform.

Topics on Solitons in Plasmas

Abstract: Solitons can be regarded as nonlinear normal modes, in terms of which dynamical properties of a given physical system could be analyzed. The author, however, points out firstly that the cnoidal wave is also the genuine nonlinear wave playing important roles in nonlinear phenomena in plasmas and other dispersive media. Explicit analysis for the ion acoustic cnoidal wave is carried out up to the second order on the basis of Kodama-Taniuti's renormalized reductive perturbation theory. The nonlinear ion flux associated with the cnoidal wave is shown to have different dependence on the wave amplitude as compared with the quasi-linear ion flux. At the same time, it should be noticed that the nonlinear ion flux also exhibits profound frequency dependence, which is not predicted by the quasi-linear treatment. Secondly Bogoliubov-Mitropolsky perturbation analysis of the perturbed envelope soliton is briefly discussed referring to Karpman-Maslov's perturbation approach based on the inverse scattering method. Thirdly, brief summaries on plasma waves in magnetized plasmas are followed by a report on the new inverse scattering scheme for the derivative nonlinear Schr?dinger equation. In concluding remark, the potential importance of researches on solitons in strong plasma turbulence has been emphasized.

The spectra of unresolved split normal mode multiplets

Abstract: Summary A seismic spectrum corresponding to an isolated multiplet nSl or nTl on a spherical earth is characterized fully by three parameters: an amplitude, a peak or central frequency and a half-width. The amplitude depends on the mechanism of the seismic source; it may be complex for single-station spectra, but it is real and positive for spectral stacks. The central frequency is the degenerate eigenfrequency of the multiplet, and the half-width is a measure of the decay rate due to the Earth's anelasticity. On an aspherical earth, the multiplet nSl or nTl is split into 2l+ 1 nearly degenerate singlets. In general this splitting cannot be resolved. If the apparent amplitude, central frequency and half-width of an unresolved multiplet are measured on an aspherical earth, and subsequently interpreted as if the Earth were spherical, there may be a bias introduced by the splitting. Perturbation theory is used here, correct to zeroth order in the eigenfunctions and first order in the eigenfrequencies, to investigate this bias. Correct to this order, single-station amplitudes, and therefore source mechanism determinations, are unaffected by asphericity. Measurements of Q made on spectral stacks are always biased toward low Q. The limiting case n≪l and s≪l, where s is the maximum significant degree in the spherical harmonic expansion of the asphericity, is examined in particular detail. Single-station spectra appear in this limit to consist of a single line broadened by attenuation alone; Q measurements made on these spectra, prior to stacking, are therefore unbiased. A travelling wave decomposition is considered in order to compare the results of normal mode perturbation theory in this limit with those obtained by applying the theory of geometrical optics to the equivalent surface waves. At the level of truncation considered in this paper, perturbation theory predicts that surface waves will propagate on an aspherical earth at a uniform velocity. This discrepancy with geometrical optics is attributed to the neglect of eigenfunction perturbations.

Erratum: A normal mode treatment of optical properties of a classical coupled dipole oscillator system with Lorentzian band shapes

Abstract: DeVoe’s classical coupled dipole oscillator model for molecular optical properties is specialized for the case where the individual oscillators have complex polarizabilities with Lorentzian band shapes. The optical properties of this system are derived in terms of a set of eigenvectors and eigenvalues which describe the normal modes of the system. When the real and imaginary parts of the interaction matrix can be diagonalized by the same transformation, the properties are expressed as explicit functions of frequency and do not require a matrix inversion at each frequency. This condition is met by a system in which all oscillators have the same bandwidth. For such a case the computation time is much less than that required by the more general method involving point‐by‐point matrix inversion. A further simplification is achieved for a system in which a subset of the oscillators have natural frequencies so high that their polarizabilities may be regarded as nondispersive. For this case the order of the eigenvalue problem is reduced to the number of dispersive oscillators. Sum rules are derived for the mixed dispersive/nondispersive system. The normal mode method is illustrated by calculations for a 12‐residue right‐handed α‐helix in which oscillators at 52 000 and 67 600 cm−1 are assigned to each residue. The predicted absorption and circular dichroic spectra are similar to those predicted using Gaussian band shapes for the individual oscillators by the matrix inversion method, while a 40‐fold saving in computer time was achieved by the normal mode method.

A Theory of Stationary Long Waves. Part III: Quasi-Normal Modes in a Singular Waveguide

Abstract: The present paper deals with the fundamental issue of whether one can treat waves as normal modes when critical surface, where the phase speed of the wave matches the zonal wind speed, are present. In particular the question of whether a Rossby critical level (such as the zero-wind line for stationary waves) is absorbing or reflecting is raised and subsequently addressed. It is found that the critical level is never totally absorbing; Rossby waves are partially reflected even if the critical layer is dominated by dissipative processes. The relevance of nonlinearity in planetary-scale Rossby wave critical layers is also discussed and it is found to be the dominant mechanism. With the relative magnitudes of nonlinearity versus viscosity relevant to the earth's atmosphere it is found that the steady-state critical level should be almost perfectly reflecting to incident Rossby waves. Consequently, normal-mode solutions can be found; the quantization condition for these waves is also derived.

Nonlinear excitations in classical ferromagnetic chains

Abstract: General single solitary-wave excitations are determined for the classical continuum Heisenberg chain in the presence of external magnetic and anisotropy fields. These include both domain walls and pure solitons as examples. Conditions for propagation are carefully analysed. The complete integrability of the zero-anisotropy limit is suggested as a basis for (i) controlled singular perturbation theory, and (ii) formulation of classical statistical mechanics in a natural configurational (nonlinear normal mode) representation.

Coupled mode tuning fork type quartz crystal vibrator

Abstract: Cutting a quartz tuning fork crystal vibrator at a preferred angle and with a thickness which establishes a close coupled relationship between the flexural and torsional modes of vibration of the arms or tines produces a vibrator having a highly favorable cubic frequency temperature characteristic at predetermined frequencies. Weight added to the ends of the vibrating tines reduces the frequency of both flexural and torsional vibration, whereas weight added at nodal points in the flexural vibration primarily reduces only the torsional vibrational mode. Accordingly, a wide range of frequency adjustments is possible. Deviations from the desired crystal operating frequency due to manufacturing variances are adjustable by sequential weight modifications while the desirable cubic temperature characteristic is retained. High accuracy over a wide temperature range permits operation in oscillator circuits at relatively low frequencies and with attendant low power consumption. Close coupling between vibrational modes occurs when the difference between the flexural and torsional frequencies is ≦0.15 of the flexural frequency. The principles for dimensioning the crystal, and adjusting thickness and weight, are applicable to both fundamental and overtone frequencies.

Suspension Bridge Vibrations: Computed and Measured

Abstract: Measured frequencies of a suspension bridge were found to differ from those calculated by a wide margin. The discrepancies were found to be caused by sliding deck joints not sliding at small amplitudes, which affected the stiffness. When these and other minor effects were correctly modeled the agreement was excellent. Lateral-torsional coupled motion caused by a channel type cross-section was found to influence the position of the center of rotation, which has an effect on the aerodynamic behavior. Measured values of structural dampling are given, and a comparison is made between "approximate" and "exact" methods of computing natural frequencies and mode shapes. /Authors/

Stability of Bernstein–Greene–Kruskal equilibria

Abstract: The recent linear stability analysis of Lewis and Symon for spatially inhomogeneous Vlasov equilibria is illustrated with the case of an unstable periodic wavetrain of strongly inhomogeneous Bernstein–Greene–Kruskal equilibria. The stability formalism involves expanding an auxiliary function related to the perturbation distribution function in terms of the equilibrium Liouville eigenfunctions, and expanding the perturbation potential in terms of the eigenfunctions of an appropriately chosen field operator. The infinite‐dimensional dispersion matrix is truncated to M×M by assuming that the normal mode of interest of the perturbation potential can be adequately represented by M eigenfunctions of the field operator; the eigenfrequencies ω are the zeroes of the determinant of the dispersion matrix. A particular Bernstein–Greene–Kruskal equilibrium was chosen as a numerical example, and the growth rate and normal mode of the instability were determined by numerical simulation. The agreement of the theory with the simulation for the growth rate and normal mode of the instability was excellent, and it was possible to choose a field operator a priori such that a 1×1 dispersion matrix was sufficient for obtaining accurate results.

Comparison of normal mode and total field analysis techniques in planar integrated optics

Abstract: Normal mode and total field analysis techniques for calculating the fields generated by a polarization source are discussed. The amplitudes of the normal modes are evaluated using a Green’s-function approach and the results are compared to a previous treatment by Yariv. The generated fields are also calculated by solving the polarization driven wave equation subject to the usual electromagnetic boundary conditions at all of the pertinent interfaces. It is shown that these two approaches yield equivalent results for a simple case, and their relative merits for solving problems in different regions of a waveguide are discussed.

On the lattice dynamics of incommensurate crystal phases

Abstract: The lattice dynamics of incommensurate modulated crystals is discussed in the harmonic approximation. The study is based on the symmetry group of such a crystal, i.e. a superspace group. The normal modes may be characterised by irreducible representations of the symmetry group. It is shown that the relevant representations may be labelled by wave-vectors in the Brillouin zone of the basic space group. The reduction of the equations of motion is treated. Finally selection rules for infrared absorption are derived.

On the Solution of the Homogeneous Vertical Structure Problem for Long-Period Oscillations

Abstract: The problem of determining the long-period normal modes of a barotropic atmosphere with realistic temperature, distribution and in uniform motion is examined over the entire range of equivalent depth. It is demonstrated that there are only two solutions which satisfy a radiation/finite energy condition and approximately satisfy the homogeneous surface boundary condition. Outside of a particular interval of equivalent depth, it is shown that there exist no solutions. The problem is solved numerically over the restricted interval where two sharp dips in the surface error are found. The first of these corresponds to a 9.6 km equivalent depth and a Lamb structure. This mode, which is the counterpart of the thin film solutions on a sphere, is due to the hydrostatic nature of the basic state and exists despite the temperature variation. The second dip, corresponding to a 5.8 km equivalent depth here, is a result of buoyancy ducting and is a consequence of the temperature variation. The energy density o...

Field analysis of harmonic generation in thin-film integrated optics

Abstract: Harmonic generation of light in thin-film optical waveguides is analyzed theoretically by evaluating the total electromagnetic fields that satisfy the polarization driven wave equation and the electromagnetic boundary conditions at both interfaces. Analytical expressions are given for the amplitude of the growing wave component of the total fields. The results agree with those of normal mode analysis only in the limit of phase-matched harmonic generation. This analysis is believed to be applicable to a large range of wave interaction phenomena in integrated optics.

Thermal instabilities in magnetically confined plasmas - Solar coronal loops

Abstract: The thermal stability of confined solar coronal structures ('loops') is investigated, following both normal mode and a new, global instability analysis. It is demonstrated that: (1) normal mode analysis shows modes with size scales comparable to that of loops to be unstable, but to be strongly affected by the loop boundary conditions; (2) a global analysis, based upon variation of the total loop energy losses and gains, yields loop stability conditions for global modes dependent upon the coronal loop heating process, with magnetically coupled heating processes giving marginal stability. The connection between the present analysis and the minimum flux corona of Hearn is also discussed.