Extraction of Acoustic Normal Mode Depth Functions Using Range-Difference Method with Vertical Linear Array Data

Data-derived normal mode extraction is an effective method for extracting normal mode depth functions in the absence of marine environmental data.However,when the corresponding singular vectors become nonunique when two or more singular values obtained from the cross-spectral density matrix diagonalization are nearly equal,this results in unsatisfactory extraction outcomes for the normal mode depth functions.To address this issue,we introduced in this paper a range-difference singular value decomposition method for the extraction of normal mode depth functions.We performed the mode extraction by conducting singular value decomposition on the individual frequency components of the signal's cross-spectral density matrix.This was achieved by using pressure and its range-difference matrices constructed from vertical line array data.The proposed method was validated using simulated data.In addition,modes were successfully extracted from ambient noise.

Finite-Time Normal Mode Disturbances and Error Growth During Southern Hemisphere Blocking

The structural organization of initially random errors evolving in abarotropic tangent linear model, with time-dependent basic states taken from analyses, is examinedfor cases of block development, maturation and decay in the Southern Hemisphere atmosphere duringApril, November, and December 1989. The statistics of 100 evolved errors are studied for six-dayperiods and compared with the growth and structures of fast growing normal modes and finite-timenormal modes (FTNMs). The amplification factors of most initially random errors are slightly lessthan those of the fastest growing FTNM for the same time interval. During their evolution, thestandard deviations of the error fields become concentrated in the regions of rapid dynamicaldevelopment, particularly associated with developing and decaying blocks. We have calculatedprobability distributions and the mean and standard deviations of pattern correlations between eachof the 100 evolved error fields and the five fastest growing FTNMs for the same time interval. Themean of the largest pattern correlation, taken over the five fastest growing FTNMs, increases withincreasing time interval to a value close to 0.6 or larger after six days. FTNM 1 generally, but notalways, gives the largest mean pattern correlation with error fields. Corresponding patterncorrelations with the fast growing normal modes of the instantaneous basic state flow aresignificant' but lower than with FTNMs. Mean pattern correlations with fast growing FTNMs increasefurther when the time interval is increased beyond six days.

SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of system with 1 : 2 : 5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra, and some new bifurcation patterns are found. Different from the NNMs of systems with single internal resonance, the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degrees of freedom. At last, it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.

Normal Mode Analysis on Three Different Structures of a Duplex DNA d(CGCGAATTCGCG)

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An Efficient Three-Dimensional Coupled Normal Mode Model and Its Application to Internal Solitary Wave Problems

We present an efficient three-dimensional coupled-mode model based on the Fourier synthesis technique. In principle, this model is a one-way model, and hence provides satisfactory accuracy for problems where the forward scattering dominates. At the same time, this model provides an efficiency gain of an order of magnitude or more over two-way coupled-mode models. This model can be applied to three-dimensional range-dependent problems with a slowly varying bathymetry or internal waves. A numerical example of the latter is demonstrated in this work. Comparisons of both accuracy and efficiency between the present model and a benchmark model are also provided.

Normal mode splitting and ground state cooling in a Fabry-Perot optical cavity and transmission line resonator

Optomechanical dynamics in two systems which are a transmission line resonator and Fabrya-Perot optical cavity via radiation-pressure are investigated by linearized quantum Langevin equation. We work in the resolved sideband regime where the oscillator resonance frequency exceeds the cavity linewidth. Normal mode splittings of the mechanical resonator as a pure result of the coupling interaction in the two optomechanical systems is studied, and we make a comparison of normal mode splitting of mechanical resonator between the two systems. In the optical cavity, the normal mode splitting of the movable mirror approaches the latest experiment very well. In addition, an approximation scheme is introduced to demonstrate the ground state cooling, and we make a comparison of cooling between the two systems dominated by two key factors, which are the initial bath temperature and the mechanical quality factor. Since both the normal mode splitting and cooling require working in the resolved sideband regime, whether the normal mode splitting influences the cooling of the mirror is considered. Considering the size of the mechanical resonator and precooling the system, the mechanical resonator in the transmission line resonator system is easier to achieve the ground state cooling than in optical cavity.

Determining the long living quasi-normal modes of relativistic stars

Methods of finding quasi-normal modes of non-rotating relativistic stars have been well established, however, none of the existing treatments which take spacetime and fluid oscillations fully into account can determine modes of long decay time, e.g., the p and g mode series, or the f modes for stars with low compactness ratio （M/R）. In this paper we show how the quasi-normal modes of long lifetime can be determined through refinements of a treatment originally due to Detweiler and Lindblom. The determination of the p mode series has been argued in the literature to have implication on the life time of gravitational wave sources and stellar stability. In this paper we 1） provide detailed steps in our treatment to facilitate future effort in this direction; 2） correct mistakes in the literature on the formulation; and 3） analyse the accuracy of the quasi-normal mode frequencies obtained and the limitations of the treatment.

Gravitational quasi-normal modes of static R^2 Anti-de Sitter black holes

In this paper, we study the gravitational quasi-normal modes（QNMs） for a static R^2 black hole（BH） in Anti-de Sitter（AdS） spacetime. The corresponding master equation of odd parity is derived and the QNMs are evaluated by the Horowitz and Hubeny method. Meanwhile the stability of such BH is also discussed through the temporal evolution of the perturbation field. Here we mainly consider the coefficient λ, which is related to the radius of AdS black hole, on the QNMs of the R^2 AdS BH. The results show that the Re（ω） and ｜Im（ω）｜ of the QNMs increase together as ｜λ｜ increases for a given angular momentum number l. That indicates with a larger value of ｜λ｜ the corresponding R^2 AdS BH returns to stable much more quickly. The dynamic evolution of the perturbation field is consistent with the results derived by the Horowitz and Hubeny method. Since in the conformal field theory the QNMs can reflect its approach to equilibrium, so our related results could be referential to studies of the AdS/CFT conjecture. The relationship between λ and the properties of the static R^2 BH might be helpful for the development of R^2 gravitational theory.

A Very Compact Normal Mode Multiloop Helical Antenna with Enhanced Bandwidth

The physical size of an antenna becomes an important characteristic when receiving signals in bands with long wavelengths. Size determines two important aspects of antenna performance;impedance and efficiency. For example, the VHF antennas installed on radio sets that intended to receive FM or the latest technology Digital Audio Broadcasting (DAB) radio signals in Bands II, III respectively. Antennas that are installed on mobile platforms (i.e. portable receivers) require a receiver that utilizes a whip telescopic antenna with adjustable length which can operate as a λ/4 monopole antenna. Whereas, non-portable applications like a deck commercial receiver has no built in antenna due to the large size of the radiator needed and so must be connected with an external antenna. This paper presents a new design of a very small size Normal Mode Multiloop Helical Antenna (NMMHA) with superior performance developed for commercial receivers operate in band II, III. The major drawback which has been overcome with this design is the very narrow bandwidth of the Normal Mode Helical Antenna, which originally was optimized to provide the minimum Voltage Standing Wave Ratio VSWR response across Band II (87.5 - 108 MHz). The NMMHA’s size allows it to be a build in block of a deck commercial receiver.

Vibrational Normal Modes of an Equilateral Triangular Mechanical Molecule

Three point-like massive particles/atoms are connected with three springs forming an equilateral triangle replicating a prototype triatomic molecule. The triangle is inscribed within a stationary frame via three additional springs confining the vibrations of the molecule to a 2D space. It is the objective of this research flavored investigation to seek the normal vibrational modes for this three-body six-spring structure. The entire analysis including symbolic, numeric, and graphics is carried out by adapting a suitable Computer Algebra System (CAS), Mathematica. For a comprehensive understanding, the frequency of the normal mode is used for a visual animation;an actual mechanical replica of the “molecule” for the scenario on hand is fabricated.

A NEW MOMENT METHOD FOR THE FAST AND ACCURATE ANALYSIS OF NORMAL MODE HELICAL ANTENNAS

In this letter, a new moment method using helical segments is presented to model Normal Mode Helical Antenna (NMHA). Using this method, the NMHA can be modeled by a few segments. The current distributions and radiation patterns of some NMHAs are calculated.A comparison is made between results obtained using this helical segment algorithm and a linear segment algorithm, and the results of the two algorithms agree fairly well. When calculating the impedance matrix [Z], all the elements of the matrix can be obtained by only calculating a few elements with the application of the symmetric and periodic characteristics of the NMHA.Therefore, the CPU time and the memory storage are significantly reduced, with the accuracy and speed enhanced.

Numerical solutions of rotational normal modes of a triaxial two-layered anelastic Earth

The Earth's rotational normal modes depend on Earth model used, including the layer structures,principal inertia moments of different layers and the compliances. This study focuses on providing numerical solution of the rotational normal modes of a triaxial two-layered anelastic Earth model without external forces but with considering the complex forms of compliances and the electromagnetic coupling between the core and mantle. Based on the present knowledge of the Chandler wobble(CW) and Free Core Nutation(FCN), we provide a set of complete compliances which could be used for reference in further investigations. There are eight rotational normal mode solutions, four of which might exist in nature. However, in reality only two of these four solutions correspond to the present motion status of the prograde CW and the retrograde FCN. On one hand, our numerical calculations show that the periods and quality factors(Qs) of CW and FCN are respectively 434.90 and 429.86 mean solar days(d) and 76.56 and 23988.47 under frequency-dependent assumption, and the triaxiality prolongs CW about 0.01 d and has hardly effect on FCN. On the other hand, we analyze the sensibility of compliances and electromagnetic coupling parameter on the periods and Qs of CW and FCN and find the sensitive parameters with respect to them.

Thermodynamics under the impact of thermal fluctuations and quasi-normal modes of Euler-Heisenberg AdS BH in the framework of NLED

We study the impact of thermal fluctuations on the thermodynamics,quasi-normal modes,and phase transitions of an anti-de Sitter Euler-Heisenberg black hole(BH)with a nonlinear electrodynamic field.An anti-de Sitter Euler-Heisenberg BH with a nonlinear electrodynamic field is composed of four parameters:the mass,electric charge,cosmological constant,and Euler-Heisenberg parameter.We calculate thermodynamic variables such as Hawking temperature,entropy,volume,and specific heat,which comply with the first law of thermodynamics.First,we use this BH to determine the thermodynamics and thermal fluctuations with the Euler-Heisenberg parameter to distinguish their effect on uncorrected and corrected thermodynamical quantities.We derive the expression for corrected entropy to study the impact of thermal fluctuation with simple logarithmic corrections on unmodified thermodynamical potentials,including Helmholtz energy,pressure,Gibbs free energy,and enthalpy.The Euler-Heisenberg parameter improves BH stability at large radii.Second,we analyze the local stability of the proposed BH,and the phase shifts of the BH are also investigated using temperature and specific heat.When there is a decrease in charge and an increase in r_(+)andα,the temperature shifts from an unstable region to a stable one.Similarly,increases in local stability are observed with each of these parameters.Third,we use null geodesics to deal with the effects of nonlinear electrodynamics on the quasi-normal modes of the Euler-Heisenberg anti-de Sitter BH.The null geodesics provide the angular velocity and Lyapunov exponent of the photon sphere,which are the same as the real and imaginary parts of the quasi-normal modes in the eikonal limit.

SINGULAR CHARACTERISTICS OF NONLINEAR NORMAL MODES IN A TWO DEGREES OF FREEDOM ASYMMETRIC SYSTEMS WITH CUBIC NONLINEARITIES

Nonlinear normal modes in a two degrees of freedom asymmetric system with cubic nonlinearities as singularity occurs in the system are studied, based on the invariant space in nonlinear normal modes and perturbation technique. Emphasis is placed on singular characteristics as the linear coupling between subsystems degenerated. For nonresonances, it is analytically presented that a single-mode motion and localization of vibrations occur in the system, and the degree of localization relates not only to the coupling stiffness between oscillators, but also to the asymmetric parameter. The parametric threshold value of localization is analytically given. For 1 : 1 resonance, there exist bifurcations of normal modes with nonlinearly coupling stiffness and asymmetric parameter varying. The bifurcating set on the parameter and bifurcating curves of normal modes are obtained.

Arrival time fluctuation of higher order normal modes caused by solitary internal waves

The sensitivities of the normal modes arrival time to solitary internal waves （IWs） are analyzed by using the SW06 environments. Simulation results show that the arrival time of mode 1 is relatively stable. But, there are some higher-order normal modes which arrive earlier than mode 1, and fluctuate with the appearance of solitary IWs. Explanation of the phenomenon is given based on ray theory. It is shown that, when thermocline falls down to some depths, those higher-order modes with a group of definite grazing angles mainly propagate above the thermocline and arrive earlier.

ParametricVibration Analysis of Pipes Conveying Fluid by Nonlinear Normal Modes and a Numerical Iterative Approach

Nonlinear normal modes and a numerical iterative approach are applied to study the parametric vibrations of pipes conveying pulsating fluid as an example of gyroscopic continua.The nonlinear non-autonomous governing equations are transformed into a set of pseudo-autonomous ones by employing the harmonic balance method.The nonlinear normal modes are constructed by the invariant manifold method on the state space and a numerical iterative approach is adopted to obtain numerical solutions,in which two types of initial conditions for the modal coefficients are employed.The results show that both initial conditions can lead to fast convergence.The frequency-amplitude responses with some modal motions in phase space are obtained by the present iterative method.Quadrature phase difference and traveling waves are found in the time-domain complex modal analysis.

IMPLICIT NONLINEAR NORMAL MODE INITIALIZATION:A MULTIGRID APPROACH

The implicit nonlinear normal mode initialization (INMI) is applied to a tropical limited area shallow water model in spherical coordinates.The boundary condition for the INMI scheme is based on the boundary formulation of the model.The INMI scheme is found to be very efficient in suppressing spurious gravity wave oscillation and providing a well balanced initial data set for the model.The INMI scheme involves solving a number of elliptic type equations with varying complexity.and hence an efficient numerical technique is required for solving such equations.In order to make INMI computationally more attractive,we are employing the multigrid method for solving all the elliptic type equations in the INMI scheme.The numerical procedures for the development of such multigrid solvers are briefly described.

A Normal Mode Stability Analysis of Numerical Interface Conditions for Fluid/Structure Interaction

In multi physics computations where a compressible fluid is coupled with a linearly elastic solid,it is standard to enforce continuity of the normal velocities and of the normal stresses at the interface between the fluid and the solid.In a numerical scheme,there are many ways that velocity-and stress-continuity can be enforced in the discrete approximation.This paper performs a normal mode stability analysis of the linearized problem to investigate the stability of different numerical interface conditions for a model problem approximated by upwind type finite difference schemes.The analysis shows that depending on the ratio of densities between the solid and the fluid,some numerical interface conditions are stable up to the maximal CFL-limit,while other numerical interface conditions suffer from a severe reduction of the stable CFL-limit.The paper also presents a new interface condition,obtained as a simplified characteristic boundary condition,that is proved to not suffer from any reduction of the stable CFL-limit.Numerical experiments in one space dimension show that the new interface condition is stable also for computations with the non-linear Euler equations of compressible fluid flow coupled with a linearly elastic solid.

Extracting the interference components of normal modes in shallow water waveguide using singular value decomposition method

The normal mode interference characteristic in shallow water waveguide is a valu- able topic in the fields of underwater acoustic. A method for extracting the interference components of normal modes from broadband acoustic propagation data recorded by a single hy- drophone without any prior information is present in this paper. First, a Hermitian matrix is formed by the power spectral density. Second, a singular value decomposition （SVD） is performed on the Hermitian matrix to obtain the orthonormal eigenvectors, which are proportional to the interference components of normal modes. The fundamental equations of the new extracting method are derived based on normal mode and waveguide invariant theory. And the validity of the present method is verified by the numerical simulation and experimental results. In addition, the extracted results of normal-mode interference components are intended to be used for passive ranging of broadband sources.

Complex eigenvalues and group velocities of normal modes in shallow water with a lossy bottom

For the stratified shallow water with a lossy bottom, the distribution and asymptotic behavior of mode eigenvalues in the complex plane are discussed on the basis of the Pekeris cut. The analysis shows that even in the shallow water with a low-speed lossy bottom there may be the proper modes which satisfy the radiation condition at infinite depth. It is also shown that when the ratio between the densities of the seawater and seabottom is close to one, there exist only a finite number of improper modes . An iterative method for evaluating the complex eigenvalues and group velocities of normal modes is presented and some numerical results are given.