Time-domain Green Function Research Articles

Improvement of a Green's Function Estimation for a Moving Source Using the Waveguide Invariant Theory.

Understanding the characteristics of underwater sound channels is essential for various remote sensing applications. Typically, the time-domain Green's function or channel impulse response (CIR) is obtained using computationally intensive acoustic propagation models that rely on accurate environmental data, such as sound speed profiles and bathymetry. Ray-based blind deconvolution (RBD) offers a less computationally demanding alternative using plane-wave beamforming to estimate the Green's function. However, the presence of noise can obscure low coherence ray arrivals, making accurate estimation challenging. This paper introduces a method using the waveguide invariant to improve the signal-to-noise ratio (SNR) of broadband Green's functions for a moving source without prior knowledge of range. By utilizing RBD and the frequency shifts from the striation slope, we coherently combine individual Green's functions at adjacent ranges, significantly improving the SNR. In this study, we demonstrated the proposed method via simulation and broadband noise data (200-900 Hz) collected from a moving ship in 100 m deep shallow water.

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

The effect of fastener clip fatigue for high-speed railway on vehicle-track dynamic interaction: Numerical analysis and probabilistic evaluation

To investigate the vehicle-induced fatigue damage evolution of fastener clips and their impact on the vehicle-track (VT) system, a numerical dynamic model is established to achieve probabilistic evaluation. The “rain flow” counting method is adopted to describe the stress cycles of the hazard area of the clips, and the S-N curve is used to predict the fatigue life of the clips. The solution of the dynamic equation of motion is obtained by the time-domain Green's function method, which is an effective method to improve computational efficiency. A versatile iterative algorithm is adopted to solve the nonlinear system considering the damaged clips. In the probabilistic evaluation framework, a novel random field method is employed to depict the spatial variability of clips’ strength. The applicability of the model established in this work in aspects of expressing random damage of fasteners and evaluating the dynamic behavior of the system is demonstrated by different numerical examples. The results show that the dynamic responses of the train and track system significantly exacerbate as the number of train cycles increases, which is closely related to the variability and non-uniform damage of the fasteners. The probability of cumulative damage to the fasteners induced by the train follows a three-stage rule, and the fatigue damage of the non-uniform fastener system caused by the train is greater than that of the uniform situation. In addition, the short wave components of track irregularity will significantly exacerbate the damage to fasteners and therefore degrade the status of the railway line. As the driving speed increases, the damage to the fasteners also increases, but the impact of vehicle speed on the fastener damage is smaller than that of track irregularity, especially in the range of fatigue damage greater than 0.9. Therefore, maintaining the geometric smoothness of the track, controlling the uniformity of the mechanical properties of the fasteners, and reducing the driving speed are feasible measures to extend the service life of the fastener system, thereby ensuring the operation of high-speed trains.

Three-dimensional time-domain Green's functions and spatial impulse responses for the van Wijngaarden wave equation

An exact analytical three-dimensional time-domain Green's function is introduced for the van Wijngaarden wave equation when the coefficients of the two loss terms satisfy a specific relationship. This analytical Green's function, which describes frequency-squared attenuation in acoustic media such as water, enables the subsequent derivation of new expressions that describe the lossy spatial impulse response for a circular piston. Initial time-domain assessments, which compare the Green's functions for the van Wijngaarden, Stokes, and power law wave equations using the attenuation and sound speed for water, indicate that these three lossy wave equations yield nearly identical results at distances greater than or equal to 10 μm. Lossy spatial impulse responses are also evaluated with increasing distance in and near the paraxial region of a circular piston radiating in water to reveal some interesting time-domain interactions between frequency-squared attenuation and diffraction. Similar behaviors are also demonstrated for the lossy far-field spatial impulse. In addition, the convergence is demonstrated for two analytically equivalent expressions applied to numerical computations of the lossy spatial impulse response. The results show that these new expressions are ideal for describing and explaining fundamental interactions between frequency-squared attenuation and diffraction in the time-domain.

An efficient and highly accurate singularity extraction method for the evaluation of transient potentials of stratified media

AbstractAn efficient and highly accurate method for extracting the singularity of scalar potential time‐domain Green's function (TDGF) of layered media is presented. The proposed method is based on the Taylor series expansion and is exploited in the finite‐difference (FD) technique. The presented method is accurate for the singularity behavior of the potential near the source. Also, the effect of singularity extraction box size on the calculation error of Green's functions is explained. Numerical results demonstrating the accuracy of the proposed singularity extraction method for calculating the TDGF of a PEC‐backed dielectric slab are presented. It is shown that the accuracy of the evaluated TDGF is improved by an order of magnitude using the proposed singularity extraction technique. Also, the effect of changing the spatial and temporal steps of the FD scheme on the calculation error is studied.

Enhancing cross correlations of ocean ambient noise in the time domain based on random matrix theory.

Cross correlations of diffuse noise can be utilized to recover the time domain Green's function (TDGF) between two points. This principle allows for a variety of practical applications, such as seismic tomography, ocean thermometry, passive localization, etc. However, in the real ocean environment, loud interference noise sources usually bias the travel time estimates of the TDGF or result in poor recovery quality. To deal with this issue, a diffuse noise reconstruction approach is proposed to eliminate the influence of the strong interference noise by utilizing the time domain statistical property of ocean ambient noise recorded on single hydrophones with the help of random matrix theory. Simulation and experimental data analysis indicate that this algorithm can effectively extract the diffuse noise component from the ocean ambient noise field and retrieve the TDGF with a higher signal-to-noise ratio when coherent accumulation of cross correlations of the reconstructed diffuse noise is performed.

Numerical spatial impulse response calculations for a circular piston radiating in a lossy medium.

Exact analytical expressions for the spatial impulse response are available for certain transducer geometries. These exact expressions for the spatial impulse response, which are only available for lossless media, analytically evaluate the Rayleigh integral to describe the effect of diffraction in the time domain. To extend the concept of the spatial impulse response by including the effect of power law attenuation in a lossy medium, time-domain Green's functions for the Power Law Wave Equation, which are expressed in terms of stable probability density functions, are computed numerically and superposed. Numerical validations demonstrate that the lossy spatial impulse for a circular piston converges to the analytical lossless spatial impulse response as the value of the attenuation constant grows small. The lossy spatial impulse response is then evaluated in different spatial locations for four specific values of the power law exponent using several different values for the attenuation constant. As the attenuation constant or the distance from the source increases, the amplitude decreases while an increase in temporal broadening is observed. The sharp edges that appear in the time-limited lossless impulse response are replaced by increasingly smooth curves in the lossy impulse response, which decays slowly as a function of time.

Peak-time sensitivity kernels for noise cross-correlation envelopes.

The envelope of the time-lagged cross-correlation of an underwater noise field between two hydrophones can under certain conditions be used as a proxy for active acoustic receptions between the two locations enabling the study of ocean variability. Previous work looked at the sensitivity of cross-correlation peak amplitudes with respect to the distribution of the noise sources. The present study examines the sensitivity of the cross-correlation envelope peak times with respect to changes in the sound-speed distribution. A wave-theoretic scheme allowing for finite-frequency calculations in two and three dimensions, combined with the Born approximation for perturbations of the Green's function and the peak arrival approach, is used to obtain sensitivity kernels with respect to environmental (sound-speed) changes. These kernels provide a way to infer ocean structure from the cross-correlation peaks, considered as observables on their own and valid even in cases where the cross-correlation function does not approximate the time-domain Green's function between the two receivers. The sensitivity behavior is studied for different propagation conditions and noise-source distributions, ranging from spatially distributed uncorrelated noise sources to point sources, such as individual ships. Deviations from linearity are addressed and discussed.

A study on hydrodynamic analysis of ship with forward-speed based on a time domain boundary element method

The hydrodynamic analysis for ships is studied based on the three-dimensional (3D) time domain Green function (TDGF) method. The precise integration method (PIM) with variable dimension matrix is developed to evaluate the TDGF and its derivatives. The numerical convergence analysis for TDGF evaluation is carried out. Based on the TDGF method, the radiation problem and diffraction problem are solved by using impulsive response function. The hydrodynamic coefficients, wave exciting forces and motions for Wigley Ⅰ hull form ship are computed to verify the reliability and accuracy of the three-dimensional time domain program. The numerical results obtained by the proposed method show a good agreement with the previous experimental data and other published results.

Extrapolation of Green's functions using the waveguide invariant theory

The broadband interference structure of sound propagation in a waveguide can be described by the waveguide invariant, β, that manifests itself as striations in the frequency-range plane. At any given range (r) there is a striation pattern in frequency (ω), which is the Fourier transform of multipath impulse response (or Green's function). Moving to a different range (r+Δr), the same pattern is retained but either stretched or shrunken in ω in proportion to Δr, according to Δω/ω=β(Δr/r). The waveguide invariant property allows a time-domain Green's function observed at one location, g(r,t), to be extrapolated into adjacent ranges with a simple analytic relation: g(r+Δr, t) ≈ g(r,α(t-Δr/c)) where α = 1+β(Δr/r) and c is the nominal sound speed of 1500 m/s. The relationship is verified in terms of range variation of the eigenray arrival times via simulations and using real data from a ship of opportunity radiating broadband noise (200–900 Hz) in a shallow-water environment, where the steep-angle arrivals contributing to the acoustic field have β ≈ 0.92.

Localization of a distant ship using a guide ship and a vertical array.

A method is presented for estimating the range of a distant ship in shallow water using a vertical array and a guide ship at a known range close to the array. The method involves a combination of four different approaches: blind deconvolution, waveguide invariant, virtual receiver (VR), and array invariant. (1) Blind deconvolution extracts a time-domain Green's function from the broadband acoustic source (guide ship). (2) The Green's function is extrapolated into adjacent ranges using the waveguide invariant, generating a horizontal array of synthetic guide sources. (3) Each guide source then turns into a VR where the output approximates the signal that the distant (objective) ship will produce at the location of the guide source. (4) The horizontal virtual array around the guide ship applies the blind deconvolution again to estimate the Green's function for the objective source, followed by the array invariant to estimate the distance between the two ships. The proposed method is demonstrated using a ship of opportunity radiating broadband noise (100-500 Hz) and a 56.25-m long vertical array in approximately 100-m water.

Accuracy evaluation of 3D time-domain Green function in infinite depth

An accurate evaluation of three-dimensional (3D) Time-Domain Green Function (TDGF) in infinite water depth is essential for ship’s hydrodynamic analysis. Various numerical algorithms based on the TDGF properties are considered, including the ascending series expansion at small time parameter, the asymptotic expansion at large time parameter and the Taylor series expansion combines with ordinary differential equation for the time domain analysis. An efficient method (referred as “Present Method”) for a better accuracy evaluation of TDGF has been proposed. The numerical results generated from precise integration method and analytical solution of Shan et al. (2019) revealed that the “Present method” provides a better solution in the computational domain. The comparison of the heave hydrodynamic coefficients in solving the radiation problem of a hemisphere at zero speed between the “Present method” and the analytical solutions proposed by Hulme (1982) showed that the difference of result is small, less than 3%.

A method based on LS-SVM to estimate time-domain Green function

How to calculate the 3D time-domain Green function and its gradients accurately and efficiently is the key to solving the hydrodynamic problem of ships. In this paper, the least square support vector machine (LS-SVM) theory was proposed for the first time to solve the ordinary differential equations (ODEs) satisfied by the time-domain Green function and the spatial derivatives in time domain. Furthermore, the computational advantages of the theory were discussed from three perspectives which were the accuracy, stability and rapidity of the calculation of the time-domain Green function. This approach was validated by comparing the obtained results with those of alternative methods which was found to effectively improve the accuracy and stability of calculation of the time-domain Green function and the spatial derivative.

Numerical evaluation of a fractional calculus model for the spatial impulse response of a circular piston

In lossless media, exact analytical expressions for the spatial impulse response describe the effect of diffraction in the time domain for several specific transducer shapes. At present, no exact analytical expressions are available for lossy media, so numerical evaluation of the spatial impulse response is required. For attenuation that follows a power law, the spatial impulse response is numerically evaluated by superposing contributions from time-domain Green's functions for the Power Law Wave Equation, which is a fractional calculus model for power law attenuation. To demonstrate examples of lossy spatial impulse responses obtained with these time-domain Green's functions, which are expressed in terms of maximally skewed stable densities, numerical results are computed for a circular transducer using different attenuation values and compared with the analytical result for a lossless medium. The results show that the numerically computed spatial impulse response for a lossy medium converges to the analytical result evaluated in a lossless medium as the value of the attenuation constant decreases. As the attenuation constant grows larger, the temporal extent increases, and the sharp edges are replaced by increasingly smooth curves, as observed in numerical evaluations of the lossy spatial impulse response evaluated in multiple locations.

Normal modes separation by use of ambient noise cross-correlation function in shallow sea

The possibility to extract the time-domain Green's function from ambient noise cross-correlation function was demonstrated by a number of authors. This approach has a weak point: the accumulation time of the noise signal. The accumulation time is significantly reduced when using mode signals and separating normal modes. Both numerical study of the spectrogram of the noise cross-correlation function and experiment in Arctic shelf demonstrated that dispersion curves observed in the spectrogram in the case of large distances between hydrophones, with a decrease in this distance are transformed into separate maxima localized both in frequency and in time delay. These maxima arise due to the presence of a stationary phase point near the minimum of the group velocities of the modes, and also due to modes interference, which is more efficient at small distances between hydrophones. At small distances, working with a signal in the frequency band where only the lowest hydroacoustic mode is excited has a number of advantages, the main of which is a decrease in the accumulation time. We demonstrate the change in the regime of modes separation with a change in the distance between hydrophones and discuss the optimal conditions for the applicability of the method.

A parametric evaluation of shear wave speeds estimated with time-of-flight calculations in viscoelastic media.

Shear wave elasticity imaging (SWEI) uses an acoustic radiation force to generate shear waves, and then soft tissue mechanical properties are obtained by analyzing the shear wave data. In SWEI, the shear wave speed is often estimated with time-of-flight (TOF) calculations. To characterize the errors produced by TOF calculations, three-dimensional (3D) simulated shear waves are described by time-domain Green's functions for a Kelvin-Voigt model evaluated for multiple combinations of the shear elasticity and the shear viscosity. Estimated shear wave speeds are obtained from cross correlations and time-to-peak (TTP) calculations applied to shear wave particle velocities and shear wave particle displacements. The results obtained from these 3D shear wave simulations indicate that TTP calculations applied to shear wave particle displacements yield effective estimates of the shear wave speed if noise is absent, but cross correlations applied to shear wave particle displacements are more robust when the effects of noise and shear viscosity are included. The results also show that shear wave speeds estimated with TTP methods and cross correlations using shear wave particle velocities are more sensitive to increases in shear viscosity and noise, which suggests that superior estimates of the shear wave speed are obtained from noiseless or noisy shear wave particle displacements.

Extrapolating Green's functions using the waveguide invariant theory.

The broadband interference structure of sound propagation in a waveguide can be described by the waveguide invariant, β, that manifests itself as striations in the frequency-range plane. At any given range (r), there is a striation pattern in frequency (ω), which is the Fourier transform of the multipath impulse response (or Green's function). Moving to a different range (r+Δr), the same pattern is retained but is either stretched or shrunken in ω in proportion to Δr, according to Δω/ω=β(Δr/r). The waveguide invariant property allows a time-domain Green's function observed at one location, g(r,t), to be extrapolated to adjacent ranges with a simple analytic relation: g(r+Δr,t)≃g(r,α(t-Δr/c)), where α=1+β(Δr/r) and c is the nominal sound speed of 1500 m/s. The relationship is verified in terms of range variation of the eigenray arrival times via simulations and by using real data from a ship of opportunity radiating broadband noise (200-900 Hz) in a shallow-water environment, where the steep-angle arrivals contributing to the acoustic field have β≈0.92.

Time-domain analysis of side-by-side floating bodies by the three-dimensional hybrid Green function method

ABSTRACT In the present research, the three-dimensional (3D) time-domain hybrid Green function method is developed for simulating the coupled motions of two floating bodies with the side-by-side arrangement in waves. The fluid domain is decomposed into an inner and an outer domain by an imaginary control surface. Then, the Rankine source is adopted in the inner domain, while the transient time domain Green function is adopted in the outer domain. The side-by-side problem is numerically solved by means of the continuity condition adopted on the control surface. A Wigley hull ship and a rectangular barge with the side-by-side arrangement in beam wave are investigated, while the hydrodynamic interactions between them are taken into consideration. Based on that, the coupled motions in waves between a supply ship and frigate both with large flare, as well as with same forward speed and side-by-side arrangement are also investigated. A good agreement is obtained between numerical and experimental results for all case studies, which indicates that the present method and developed time-domain numerical solver is accurate enough and available to handle the side-by-side problem in ship and ocean engineering.

Cross-correlation sensitivity kernels with respect to noise source distribution.

The cross correlation of the underwater noise field recorded at two receivers conveys information about the time-domain Green's function between the two locations, provided that sufficient energy is channeled into the acoustic paths connecting these. The efficiency of this procedure depends on the locations and characteristics of the receivers and noise sources, as well as on the refraction properties of the ocean sound channel. The sensitivity of the finite-frequency noise cross-correlation function with respect to the location and amplitude of the noise sources is studied here, taking into account the refractive features of the ocean environment. The sensitivity kernel describing changes in the cross-correlation envelope due to changes in the noise source distribution is used to highlight noise-source locations with maximum potential impact on the cross-correlation output.

Application of the 3D time-domain Green's function for finite water depth in hydroelastic mechanics

Based on the boundary element method and the superposition principle of structural elastic modes, the three-dimensional time-domain Green's function in finite water depth is applied to hydroelasticity, and the theoretical basis of three-dimensional time-domain hydroelasticity is established. To address the difficulty and divergence of the three-dimensional time-domain Green's function in finite water depth, a numerical solution method that has high accuracy and good stability is developed using series expansion, asymptotic methods and fourth-order differential equations. Taking a large bulk carrier as an example, the corresponding predictions of resonance frequency, motion, wave load and hydroelastic response with speed in finite water depth are carried out. The numerical results are compared with the results of the three-dimensional frequency domain method and the time-domain method of inner and outer region matching and are verified by the towing model test. The three-dimensional time-domain hydroelastic theory and numerical method established in this paper are of great significance for fluid coupling analysis, structural dynamic response research and wave load prediction of complex floating bodies with speed.