Normal Mode Theory Research Articles

Sequential Inversion for Geoacoustic Parameters in the South China Sea Using Modal Dispersion Curves

This paper presents an inversion scheme that uses two wideband explosive sources based on modal dispersion characteristics to synchronously obtain sound speed profile and bottom parameters. The signal received on a single hydrophone in a vertical line array is decomposed into a series of propagation modes within the framework of normal mode theory, and the dispersive characteristics of the modes are analyzed using time–frequency analysis. Time-warping transform is applied to resolve the propagation modes of explosive sources from the South China Sea in 2007. The relative travel time differences of the propagation modes are used to invert the environmental parameters, including empirical orthogonal function coefficients, sediment thickness, sound speed, density, basement sound speed and density. Sediment density and basement density are constrained by Hamilton’s empirical relationship, and relative mode energy ratio is used to estimate the bottom attenuation coefficient. An adaptive simplex simulated annealing algorithm is used for the inversion. The reliability of the inversion results is verified via uncertainty analysis. The transmission loss calculated using the deduced parameters matches the experimental data well.

Nonlinear dynamics induced by linear wave interactions in multilayered flows

Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\unicode[STIX]{x1D6FE}$ are derived, and their stability evaluated. Depending upon the initial condition and $\unicode[STIX]{x1D6FE}$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different nonlinear phenomena such as the formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\unicode[STIX]{x1D6FE}$ where normal-mode theory predicts neutral stability. Such rich nonlinear phenomena are not observed in a 2D dynamical system resulting from the two-wave problem, which reveals only stable and unstable nodes.

Modeling of acoustic pulse signal propagation in deep sea using the Maslov canonical operator

An asymptotic solution to the problem of sound pulse propagation in deep sea is derived using the Maslov canonical operator. An example of a waveguide with the Munk sound speed profile and a point source is considered, and an asymptotic expression for the pulse signal time series at the receiver is obtained. The asymptotic solution is compared with the solution computed using the normal mode theory.

Vertical discretizations giving optimal representation of normal modes: general equations of state

Previous work has identified a number of vertical discretizations of the non‐hydrostatic compressible Euler equations that optimally capture the propagation of acoustic, inertio‐gravity, and Rossby waves. Here, that previous work is extend to apply to a general equation of state, making it applicable to a wider range of geophysical fluid systems. It is also shown that several choices of prognostic thermodynamic variables and vertical staggering which were previously thought to be suboptimal can, in fact, give optimal wave propagation when discretized in an appropriate way. The key idea behind constructing these new optimal discretizations is to ensure that their corresponding linear system is equivalent to that of a certain, most fundamental, optimal configuration.

Ocean ambient noise model considering depth distribution of source and geo-acoustic inversion

An ocean ambient noise model is established considering source depth distribution. The model is used to analyze the effect of source depth on the vertical characteristics of ambient noise field. The analyses are explained and validated by normal mode theory. The energy of normal mode excited changes with source depth. Effects of different order normal modes are different. The high order modes raise up the equivalent seabed reflection loss, whereas the low order modes depress it. It is found that the seabed sound speed, density and source depth all have significant influences on equivalent seabed reflection loss at large grazing angles. So the source depth should be taken into account and the model is used in geo-acoustic inversion. Two sets of experimental data in a bandwidth of 200-525 Hz are used to obtain geo-acoustic parameters. The results show that the geo-acoustic parameters inverted from ocean ambient noise and from sound propagation data are similar. The mean value of inverted source depth tends to be smaller as frequency increases, which demonstrates that wind waves become dominant over ship noise. The average of inverted source depth values in the band(>400 Hz) is very small when sea state is higher than grade 3, which is consistent with the result from the Monahan's bubble theory.

A modal domain beamforming approach for depth estimation by a horizontal array

Distinguishing and recognizing water targets and underwater targets has been the focus of passive sonar detection. The depth of the target is closely related to the physical characteristics of the signal. In the shallow water waveguide, the normal mode theory can be used to give a good explanation to the acoustic signal physical properties. In this paper, a new method of beam forming in horizontal array modal domain is proposed. Under the condition of predicting target azimuth, the difference in acoustic path between the horizontal array elements corresponding to the direction of the target signal can be calculated according to the azimuthal information, and the phase delay of each normal mode component of the acoustic signal can be obtained. The horizontal wave number varies with order of normal mode, so each order of the normal mode has a specific phase delay. By using the beam forming principle, when the phase of a certain order of normal mode is compensated for, the output of the superposition of the signal on each element is the modal intensity of the normal mode. After obtaining the target signal modal intensity of each order, based on the shallow water condition, the modal intensities of sound source excitation at different depths are obtained as the reference mode intensities of the sound source at corresponding depths in the shallow water waveguide by simulating on Kracken software. Then, calculating the correlation coefficient between the target signal modal intensity of each order and the reference modal intensity of the sound source at each depth, we search for the maximum value of the correlation coefficient. The reference depth corresponding to the maximum value of the correlation peak is the estimated value of the target depth calculated by the method. Based on physical causes and characteristics of the normal modes, in this paper, the influences of the parameters such as the element number of horizontal array, depth of receiving array, signal-to-noise ratio, velocity profile, waveguide depth, azimuthal estimation accuracy, effective array length and application frequency band on the performance of this method are analyzed. The simulation results show that the algorithm can estimate the depth of the sound source effectively by using the signal sample with a bandwidth of 300 Hz when the signal-to-noise ratio is -10 dB. The wider the frequency band, the longer the effective array length, and the more the array element number, the higher the accuracy of azimuth estimation will be, which will bring beneficial effects to the depth estimation with the method. In addition, the depth estimation performance of the proposed method is still robust when the waveguide conditions such as the velocity profile and the seafloor parameters are disturbed.

Inversion of seabed attenuation by using single mode extracted by warping transform

Seabed is an important part of the marine environment and it has a significant influence on sound propagation. Considering the fact that geoacoustic parameters are directly acquired with difficulty and complexity, a lot of researchers have focused on the inversion of them. The seabed attenuation coefficient is insensitivity to the matching field. However it has great effects on the transmission loss, mode amplitude ratios, etc. It can be inverted from measurements of these quantities. In this paper, we present an inversion scheme based on warping transform technique for estimating the seabed attenuation coefficient. It utilizes an equivalent seabed model which is constructed by using a prior and posterior knowledge. The dispersion characteristics of normal modes can be observed using the time-frequency analysis of the explosive signal recorded. The dispersion curve can be used to invert the seabed sound speed and density. The results presented by other scholars in the same circle are cited in this paper that focuses on how to obtain the seabed attenuation. Warping transform technique is used to separate and extract the normal modes. The main advantage of warping transform is that it can transform the time-frequency spectrogram into linear relationship which makes it easier to extract the normal modes. The feature of this paper lies in determining the distance normalized normal mode transmission loss. If the depths of receiving hydrophone and the explosion source are constant, the plot of normalized normal mode transmission loss versus distance is a straight line from the normal modes theory, which can be used to obtain the attenuation factor of real part of pressure. Then the seabed attenuation coefficient of the shallow water acoustic model can be calculated. In order to verify the effectiveness of this method, the warping transformation technology is used to separate and extract the first three modes from the simulated Gaussian pulse signal which is obtained in a simulated environment which is similar to the real marine environment. The extracted results are completely consistent with the numerical results. After that, the impulsive signal data collected in the Yellow Sea are analyzed according to the scheme process, and the relationship between the seabed attenuation and frequency is =0.581fk1.86(dB/m) in a range from 150 Hz to 550 Hz. The results are in good agreement with those obtained by other scholars in the same circle. On the other hand, the inversion results of seabed attenuation from different modes can be used for comparison at the same frequency, which can be a good support for the result.

Infrared laser spectroscopy of the n-propyl and i-propyl radicals: Stretch-bend Fermi coupling in the alkyl CH stretch region.

The n-propyl and i-propyl radicals were generated in the gas phase via pyrolysis of n-butyl nitrite [CH3(CH2)3ONO] and i-butyl nitrite [(CH3)2CHCH2ONO], respectively. Nascent radicals were promptly solvated by a beam of He nanodroplets, and the infrared spectra of the radicals were recorded in the CH stretching region. Several previously unreported bands are observed between 2800 and 3150 cm-1. The CH stretching modes observed above 3000 cm-1 are in excellent agreement with CCSD(T) anharmonic frequencies computed using second-order vibrational perturbation theory. However, between 2800 and 3000 cm-1, the spectra of n- and i-propyl radicals become congested and difficult to assign due to the presence of multiple anharmonic resonance polyads. To model the spectrally congested region, Fermi and Darling-Dennison resonances are treated explicitly using "dressed" Hamiltonians and CCSD(T) quartic force fields in the normal mode representation, and the agreement with experiment is less than satisfactory. Computations employing local mode effective Hamiltonians reveal the origin of the spectral congestion to be strong coupling between the high frequency CH stretching modes and the lower frequency CHn bending/scissoring motions. The most significant coupling is between stretches and bends localized on the same CH2/CH3 group. Spectral simulations using the local mode approach are in excellent agreement with experiment.

A review of research progress in air-to-water sound transmission**Project supported by the National Natural Science Foundation of China (Grant Nos. 11434012 and 11674349).

International and domestic research progress in theory and experiment and applications of the air-to-water sound transmission are presented in this paper. Four classical numerical methods of calculating the underwater sound field generated by an airborne source, i.e., the ray theory, the wave solution, the normal-mode theory and the wavenumber integration approach, are introduced. Effects of two special conditions, i.e., the moving airborne source or medium and the rough air-water interface, on the air-to-water sound transmission are reviewed. In experimental studies, the depth and range distributions of the underwater sound field created by different kinds of airborne sources in near-field and far-field, the longitudinal horizontal correlation of underwater sound field and application methods for inverse problems are reviewed.

Spatial correlation of the high intensity zone in deep-water acoustic field**Project supported by the National Natural Science Foundation of China (Grant Nos. 11434012 and 41561144006).

The spatial correlations of acoustic field have important implications for underwater target detection and other applications in deep water. In this paper, the spatial correlations of the high intensity zone in the deep-water acoustic field are investigated by using the experimental data obtained in the South China Sea. The experimental results show that the structures of the spatial correlation coefficient at different ranges and depths are similar to the transmission loss structure in deep water. The main reason for this phenomenon is analyzed by combining the normal mode theory with the ray theory. It is shown that the received signals in the high intensity zone mainly include one or two main pulses which are contributed by the interference of a group of waterborne modes with similar phases. The horizontal-longitudinal correlations at the same receiver depth but in different high intensity zones are analyzed. At some positions, more pulses are received in the arrival structure of the signal due to bottom reflection and the horizontal-longitudinal correlation coefficient decreases accordingly. The multi-path arrival structure of receiving signal becomes more complex with increasing receiver depth.

Error Analysis on Bearing Estimation of a Towed Array to a Far-Field Source in Deep Water

Multipath arrivals received by a towed array with different arrival angles result in a bearing estimation error (BEE) when conventional beamforming is used in the real ocean waveguide. This paper investigates the errors based on simulated data, focusing on their dependence on the range and bearing of a far-field broadband source. The BEE varies periodically with range, reaching local minima and local maxima at the edges of convergence zones (CZs). The estimated source bearing always lies between the true source bearing and the broadside direction of the towed array. The physical mechanism is analyzed by using normal-mode theory for theoretical interpretation and the ray method for perceptual analysis. Based on conventional beamforming, the approximate analytical solution of the beam-forming output is derived in terms of normal modes and can predict the BEE quantitatively. The ray trace is used to illustrate the range dependence. The BEEs for different ocean depths are also discussed and the results can be used as auxiliary information for source localization.

Low frequency beamforming in shallow water environments

Ray theory is a high frequency approximation of the wave equation solution. Beamforming is a spatial filtering method that uses the ray theory to estimate the arrival angle from the array recordings. However, the ray theory is restricted to situations where water column is much deeper than the propagated wavelength and fails when water depth is smaller compared to the acoustic wavelength (i.e., in shallow-water low-frequency propagation). In this situation, the underwater sound propagation is modeled by normal-modes theory, the exact solution of the wave equation. In the theory of normal modes, each mode propagates at a different incident angle and the total field is composed of a discrete sum of the propagation modes excited at the broadcast frequency. In this situation, the beamforming performance degrades because of the interference between individual mode incident angles. In this presentation, the performance of the conventional beamforming technique is investigated when low frequency signals are prop...

The investigation of the method for measuring the low-frequency radiated sound power in a reverberation tank

In order to measure the low-frequency radiated sound power of the underwater sound source in a reverberation tank, the measuring method based on the characteristic acquisition of the testing sound field is proposed. Based on the normal-mode theory, the spatial averaging properties of the reverberation sound field with the ideal boundaries is firstly derived, of which the sound field transfer relationship from the reverberation field to the free field is obtained. In addition, for the purpose of obtaining the characteristic of the testing sound field with the general boundaries, the numerical method based on the ACTRAN software is utilized. According to obtaining the characteristic of the testing sound field in advance, the low-frequency radiated sound power can be got by measuring the square pressure in a reverberation tank. The proposed method is checked by the experiment measurement. The test results show that by utilizing the proposed method, the low-frequency radiated sound power of the underwater sound source has been accurately measured in the glass tank below the Schroeder cut-off frequency, of which the radiated sound power level deviation is less than 2 dB compared with the free field.

Interference pattern of the sound field in the presence of an internal Kelvin wave in a stratified lake.

Internal Kelvin waves (IKWs) initiated by rotation of the Earth are one of the main hydrodynamic phenomena in large stratified lakes where baroclinic Rossby radius of deformation is smaller than the horizontal scale of the lake. IKWs can be identified using the spectra of internal waves, where in the presence of IKWs, the inertial frequency is at maximum. IKWs play a rather important role in the lake's dynamics for different processes, both in the water layer and sediment, especially at the periphery of lake. Due to influence of internal waves on the sound propagation, acoustical methods can be used for estimation of behaviour of IKWs. In this paper, the spatiotemporal variability of the mid-frequency (∼1 kHz) sound field in the presence of IKWs in a deep stratified Lake Kinneret is studied using numerical simulations based on normal-mode theory. Due to the specific character of perturbation of the water layer, IKWs can cause specific variations of interference pattern, in particular, a significant shift of the sound interference pattern both in spatial and frequency domain. These shifts can be easily measured and used for reconstruction of IKW parameters.

Exact vs. asymptotic spectral densities in the Garg-Onuchic-Ambegaokar charge transfer model and its effect on Fermi's golden rule rate constants.

The Garg-Onuchic-Ambegaokar model [J. Chem. Phys. 83, 4491 (1985)] has been used extensively for benchmarking methods aimed at calculating charge transfer rates. Within this model, the donor and acceptor diabats are described as shifted parabolas along a single primary mode, which is bilinearly coupled to a harmonic bath consisting of secondary modes, characterized by an Ohmic spectral density with exponential cutoff. Rate calculations for this model are often performed in the normal mode representation, with the corresponding effective spectral density given by an asymptotic expression derived at the limit where the Ohmic bath cutoff frequency is much larger than the primary mode frequency. We compare Fermi's golden rule rate constants obtained with the asymptotic and exact effective spectral densities. We find significant deviations between rate constants obtained from the asymptotic spectral density and those obtained from the exact one in the deep inverted region. Within the range of primary mode frequencies commonly employed, we find that the discrepancies increase with decreasing temperature and with decreasing primary mode frequency.

Inversion of the Sound Speed With Radiated Noise of an Autonomous Underwater Vehicle in Shallow Water Waveguides

In December 2013, an experiment using an autonomous underwater vehicle (AUV) as a moving source for water column sound speed profile (SSP) inversion was conducted in Mogan Lake, China. With in-situ conductivity, temperature, and depth (CTD) measurements, the empirical orthogonal functions were first constructed to represent the SSP. Radiated noises of the AUV were received on a vertical line array and processed via acoustic field matching to obtain estimates of the range-independent SSP, along with the position and velocity of the AUV and the water column depth. The frequency of the radiation noise of the AUV was approximately 14 kHz. At such high frequency, even though the source motion was slow, a significant Doppler shift/broadening was observed. To incorporate the Doppler effects to better match the measured data, a forward acoustic model is derived based on the waveguide Doppler and normal mode theory. An analytical solution of the forward model is obtained for arbitrary signal integration intervals with a monochromatic source, moving radially relative to the receiver. Through simulations and experimental data processing, the feasibility of using an AUV source for water column SSP inversion has been demonstrated; it is also shown that the waveguide Doppler model is more effective compared with the model that does not consider the waveguide Doppler effect.

A source range and depth estimation method based on modal dedispersion transform

The wideband source localization is analysed widely in shallow water. It is pointed out that its performance is poor when the number of array elements is few or the ocean environment is uncertain. A method of estimating the range and depth is studied by using a single hydrophone based on the relationship of the horizontal wavenumber difference between two modes with the waveguide invariant for low frequency underwater acoustic pulse signals in a range-independent shallow water waveguide. This localization method estimates the source range by using the rangedispersion two-dimensional（2D) plane focus phenomenon and also the source depth by matching the modal energy. So it can separately estimate the source range and source depth by single hydrophone. First, the signal received on a single hydrophone can be decomposed into a series of modes within the framework of normal mode theory. In order to obtain a better localization performance, the first few order modal dispersion parameters and waveguide invariant are regarded as the unknown parameters. And then the first few order modal dispersion parameters and waveguide invariant can be estimated by comparing the differences between the modal phase velocity calculated by Eq.(8) and that calculated by the Kraken model. Second, using the estimated dispersion parameters and waveguide invariant for dedispersion transform, the amplitudes of each normal mode can achieve maximum values but only when the range of the received signal after dedispersion transform is equal to the range of source. On range-dispersion 2D plane, there appears the sound pressure focus phenomenon, and this phenomenon can be used to estimate the source range. Simulation results from a shallow water Pekeris waveguide show that the time-frequency distribution represents well the dispersion characteristics of the underwater acoustic pulse signal and the dedisperision transform can eliminate this dispersion at the range of source, so that the source range can be estimated. Besides, the first few order modal signals received are clearly separated in time domain after dedispersion transform, and the first few order modal energy can be calculated accurately. So the source depth can be estimated by matching the modal energy. The errors in range estimation and depth estimation are little in simulation. Finally, the data collected from airgun sources during an experiment in the shallow water are used to verify the presented method, and the experimental results obtained using airgun sources on a straight line are shown. The presented method is very significant for estimating the range and depth in shallow water.

The use of normal forms for analysing nonlinear mechanical vibrations.

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

Nonlinear vibration modes of an offshore articulated tower

Compliant structures, due to their low stiffness and hostile environmental loads, may present large displacements and rotations, leading to significant nonlinear effects. This work applies the theory of nonlinear normal modes to investigate the nonlinear vibrations of a discrete two-degree-of-freedom conceptual model of an offshore compliant articulated tower. The effects of buoyancy, added mass, ocean currents and waves are considered in the analysis. The elastic restoring forces are modeled, based on Augusti's model, using two orthogonal rotational springs. The invariant-manifold approach is then applied to the equations of motion, and the resulting equations are solved through an asymptotic expansion. The derived nonlinear normal modes are then used to reduce the problem to a single degree-of-freedom nonlinear oscillator in each mode. The stability of the solution is investigated through Floquet theory and Mathieu charts. Multiplicity of stable and unstable modes is detected using Poincaré sections. Similar and non-similar modes are also identified. The results of the reduced order model are compared to the numerical solutions of the original equations of motion. The favorable comparisons between both solutions confirm that nonlinear normal modes are a good alternative for the nonlinear analysis of an articulated tower and similar offshore structures.

Design of broadband linear micromachined ultrasonic transducer arrays by means of boundary element method coupled with normal mode theory

In view of the maturity of fabrication processes for capacitive micromachined ultrasonic transducers (cMUTs), engineers and researchers now need efficient and accurate modeling tools to design linear arrays according to a set of technological specifications, such as sensitivity, bandwidth, and directivity pattern. A simplified modeling tool was developed to meet this requirement. It consists of modeling one element as a set of cMUT columns, each being a 1-D periodic array of cMUTs. Model description and assessment of simulation results are given in the first part of the paper. The approach is based on the theory of linear systems so the output data are linked to input data through a large matrix, known as an admittance matrix. In the second part of the paper, we propose reorganization of matrix equations by applying the normal mode theory. From the modal decomposition, two categories of eigenmodes are highlighted, one for which all cMUTs vibrate in phase (the fundamental mode) and the others, which correspond to localized subwavelength resonances, known as baffle modes. The last part of the paper focuses mainly on the fundamental mode and gives several design strategies to optimize the frequency response of an element.