Theoretical and Computational Acoustics 2001

The following sections are included:

• PREFACE

• CONTENTS

In the 20th Century, an important contribution to the modeling of wave propagation prediction is the Parabolic Equation (PE) approximation method. Five PE review literatures have been reported. One of them is a comprehensive review of the PE development in the twentieth century; this paper gives its summary.

Marine sediments support compressional and shear waves, both of which show weak dispersion and an attenuation that scales essentially as the first power of frequency. Such behavior emerges naturally from a new, linear theory of wave propagation in saturated unconsolidated granular media. The theory is based on the physics of inter-granular shearing, taking account of the microscopically rough surfaces of the mineral grains and the molecularly thin film of pore fluid that separates grains. A stochastic treatment of stress-relaxation occurring at points of contact between asperities, combined with a strain-hardening argument for the behavior of the thin fluid film as shearing progresses, leads to two wave equations, one for compressional and the other for shear waves. From these equations, expressions emerge for the dispersion and attenuation of both types of wave. These theoretical expressions for the phase speeds and attenuations are well behaved at all frequencies and are in agreement with the available experimental evidence.

The performance of sonar system strongly depends on the ocean environment. The real underwater acoustical channel is time/space variant. The system gain of sonar designed based on optimum detection theory often considerably degraded due to the model mismatch. Therefore the robust signal processing, which is less sensitive to the model mismatch, is necessary. The robust signal processor works well in real ocean environment. The model-based sonar system is a system which takes ocean environment as a part of integrate sonar. The design philosophy of this kind of sonar system is described in this paper. The basic idea of robust signal processing and model match concept are described and used to design model-based sonar system.

Today's minimum requirements for ocean acoustic models are to be able to simulate broadband signal transmissions in 2D varying environments with an acceptable computational effort. Standard approaches comprise ray, normal mode and parabolic equation techniques. In this paper we present a computationally efficient modal approach as implemented in the PROSIM model. There are three key elements to improved efficiency: (1) Local mode calculations for the depth-separated wave equation are done analytically within layers where the sound speed varies linearly in 1/c². This approach was adopted from the ORCA model [E.K. Westwood, C.T. Tindle and N.R. Chapman, JASA 100, 3631-3645 (1996)], which has been shown to be considerably faster than mode models using numerical integration in depth, particularly when the problem involves many hundreds of modes. (2) Range dependence is handled within the adiabatic approximation (no mode coupling), allowing a mode-by-mode spatial redistribution of energy in accordance with the change in local mode properties along the propagation range. (3) Frequency interpolation of local mode properties is performed as in ORCA, i.e. accurate determination of mode wavenumbers and depth functions are required only for a limited number of discrete frequencies (around one in twenty) within the band of interest. Numerical examples show that the PROSIM model in its current configuration is much faster than standard, less optimized models such as C-SNAP and RAM.

Mathematical modeling of long-range sound propagation in a 3-D inhomogeneous media with time-dependent parameters is a computationally-intensive problem. In underwater acoustics, the general problem can be significantly simplified by taking into account characteristic properties of the ocean as an acoustic medium. In this paper, we present a simple and efficient analytic technique to approximately reduce 3-D and 4-D problems of underwater sound propagation to better studied 2-D problems. The reduction is made possible by relative slowness of time dependence of environmental parameters and smallness of their horizontal gradients. The horizontal inhomogeneity and non-stationarity of the ocean, although weak, are often non-negligible and will be shown to have important observational consequences. Unlike brute-force numerical approaches, the perturbation theory provides an insight into physics of sound propagation in 3-D inhomogeneous, time-dependent ocean.

Recent investigations of propagation models suggest that models need to be "bench-marked" each time they are applied to a different type of scenario. It is important that the user has properly input the various model parameters and checked their validity. Some of this "benchmarking" can be done using the wedge test cases of Jensen and Ferla, '90. Additional testing can involve more complicated scenarios, e.g., multiple sediment layers with range-dependence. Thus, there is a need for a variety of "benchmark" results. These results must often go beyond analytic solutions (such as those for an ideal wedge with point or line sources and perfectly reflecting boundaries). There are very few analytic solutions. Consistency checks between different types of models plus initial comparisons with carefully generated and accepted benchmark results are highly desirable.

The following sections are included:

• Introduction

• Dr. Warren Denner

• ASIAEX – A brief overview of the 2000-2001 field programs

• Acoustic Modeling Results

Extracting the bottom back-scattering information from reverberation data in shallow-water waveguide is an attractive but difficult issue. In previous works, some a priori assumption (for instance, the Lambert's law) has been made in order to solve this problem. In this paper, new approaches are proposed. The modal back-scattering matrix can be extracted directly from reverberation data without any a priori assumption on scattering coefficient.

The Split-Step Fourier (SSF) algorithm is an explicit, unconditionally stable, and efficient method for solving a class of forward propagation wave equations in ocean acoustics that are obtained by making parabolic approximations. A general motivation of the SSF algorithm is given in terms of the concepts of semigroups, operator splitting, path integrals, Trotter's product formula, and the Baker-Campbell-Hausdorff expansion. Then an explicit numerical scheme for the SSF algorithm is introduced for the boundary conditions and input data appropriate to the underwater acoustics problem, and it is shown that this scheme is consistent, unconditionally stable, and convergent. A numerical example of high frequency (30 kHz) pulse propagation in shallow water is included.

In shallow water, the reverberation is often the most severe limiting factor in relation to the use of active sonar systems for target detection, and it is of great interest to underwater acousticians. In this paper, some recent progresses on the experiment and theory of shallow water reverberation are presented. The data were obtained from several experiments at different sites in different seasons. Some of the progresses include: (1) Ray-Mode reverberation model, which can be used to calculate the reverberation intensity; (2) Coherent reverberation model, which was developed to explain the observed oscillation phenomenon of the reverberation loss; (3) The bottom inversion from the reverberation; (4) Theoretical and experimental results of the spatial correlation of the reverberation, (5) Experimental results of the direction of the reverberation, etc.

The following sections are included:

• Introduction

• Theory

• Numerical modeling

• Conclusions

• References

In September-October of 1999 under the JESAEX (Japan/East Sea Acoustic Experiment) project, in the coastal zone of the Japan Sea experimental studies were performed on the acoustic monitoring of water temperature variations [1]. Using multiplex phase-manipulated signals comprising M-codes with the center frequencies 250, 366, 406 and 604 Hz, the water medium on stationary traces was insonified [2-4]. During May-November 2000, a series of experiments were conducted with the aim of improving the instrument support for monitoring acoustic propagation using a two-way propagation technique of measuring ocean flows. The second part of our paper is devoted to monitoring a field of temperature. Experimental and theoretical cross-correlation functions among transmitting and receiving signals are used in one reconstruction method for identification of oceanographic parameters.

This paper presents an investigation of the effect of bottom roughness on the sound propagation in shallow water with a thermocline. The data obtained from several shallow water transmission loss experiments in summer indicate that there is an anomalous transmission loss at 2 to 4 kHz when the source and the receiver are both above the thermocline. With an assumption of bottom roughness, a theory based on the coupled mode model is presented to explain this phenomenon. The relationships between the anomalous loss and the frequency, the rms interface height of the bottom, the interface correlation length are also given out. Those relations may be able to be used to estimate the seafloor roughness.

Acoustical method is the only method of ocean remote observation. Compared with other waves, the absorption less of acoustic wave by sea water is very lower. The IOC (Intergovernmental Oceanographic Commission) has carried out the GOOS(Global Ocean Observation System) program. Some parameters of ocean environment can be measured by means of the inversion of the acoustic field theory. In this paper the following applications are discussed.

a) Profiling of the current with Doppler or correlation technique.

b) Inversion of sea surface wave direction and/or wind field by means of bottom-mounted acoustical device.

c) Measurement the sound velocity, attenuation, temperature, salinity of sea water and the density and particle diameter spectrum of suspended sediment in sea water, by means of standing wave resonance of sweep-frequency ultrasonic wave as well as acousto-optical diffraction.

d) Detection the detail topography with SAS (Synthetic Aperture Sonar).

e) The progress of underwater information transmission using the state-of-the-art acoustic field theory.

Under the exact wave formulation in the spatial-time domain statistical boundary value problem of the scattering of sound pulses incident on the randomly fluctuating layered medium is considered. For the solving of this problem the analytical-numerical approach has been developed by us earlier and results of a statistical simulation for different durations of the original incident pulses and various thicknesses of a random medium layer were presented. Analysis was carried out for the statistical moments, correlation functions and power spectral densities of the backscattered wave field. Comparison with the results of an approximate asymptotical analysis, carried out earlier by another authors for this problem, finds out the number of differences. In this paper we examine both the results of the exact statistical simulation and the approximate analytical ones and as a generalization propose some rather simple approximations for the description of the statistical moments of the backscattered field in the region of nonstationarity. Studying of the considered problem has both the fundamental significance for theoretical acoustics and the practical application for the interpretation of data associated with the time pulse probing of the ocean water column and the sediments of a bottom.

On the basis of the exact wave approach and under the using of the imbedding method ideas the correspondent software was developed and calculations of acoustical fields in the shallow sea from monochromatic source, radiating in the middle frequencies 500 – 1000 Hz, were carried out. We investigate the influence of real factors presenting in the coastal ocean area on the possibilities of a reception of useful signal from the source. Namely these are the antiwaveguide sound speed depth stratification, having the features of the thermocline origin, the presence of a bottom, having some complex impedance structure, finally the presence of a complex field of the underwater acoustical noise. For the last factor the possibilities of imitative simulation are considered. Sound field simulation is carried out on an example of the scalar and vectorial power characteristic behaviour, namely these are the acoustical energy (scalar intensity) and the sound power flux density (vector). The considered models and calculation results can be useful for the prognosis of results of experimental measurements in a coastal ocean area.

The time evolution of tidal vortices generated in a small channel of the Seto Inland Sea, Japan was measured by the coastal acoustic tomography system (CATS) composed of five moored-type acoustic stations during March 2-3, 1999. The vortex fields were not well reconstructed by the conventional damped least squares method because of the failure of sound transmission for part of the station pairs. The reconstruction of vortex fields were significantly improved by assimilating the station-to-station differential travel time data into a barotropic ocean model, applying the ensemble Kalman filter technique.

Ocean cold water mass is an oceanographic phenomena that oceanographers concerned many years. Method of monitoring the cold water mass is a problem to be solved urgently. A new tomography approach, modal wave-number tomography (MWNT), is proposed for monitoring the cold water mass and numerical simulation is carried out for Yellow sea cold water mass. The perturbation theory is used to invert the coefficients of empirical orthogonal functions of sound speed profile (SSP) from the local modal wave number perturbation. The numerical simulation shows that the modal wave number tomography can invert average SSP, in particular, is of potential to monitoring range dependent SSP structure.

Coupled-WKB perturbation and generalized perturbation relations for scattering matrices in arbitrary background are derived. The relations can be considered as a linear integral mapping between model perturbation and scattering data perturbation. Sensitivity kernel is introduced to investigate the functional dependence of the scattering data on the model parameters. Numerical simulation shows that for strong range dependent waveguide, the sensitivity kernels for the Coupled-WKB and the general perturbations behavior completely different. It means that the usual perturbation tomography inversion has its limit for the inversion of strong range dependent model since the Frechet derivative of the scattering matrix with respect to the model parameter has remarkable nonlinear property.

The uncertainties in the predicted acoustic wavefield associated with the transmission of low-frequency sound from the continental slope, through the shelfbreak front, onto the continental shelf are examined. The locale and sensor geometry being investigated is that of the New England continental shelfbreak with a moored low-frequency sound source on the slope. Our method of investigation employs computational fluid mechanics coupled with computational acoustics. The coupled methodology for uncertainty estimation is that of Error Subspace Statistical Estimation. Specifically, based on observed oceanographic data during the 1996 Shelfbreak Primer Experiment, the Harvard University primitive-equation ocean model is initialized with many realizations of physical fields and then integrated to produce many realizations of a five-day regional forecast of the sound speed field. In doing so, the initial physical realizations are obtained by perturbing the physical initial conditions in statistical accord with a realistic error subspace. The different forecast realizations of the sound speed field are then fed into a Naval Postgraduate School coupled-mode sound propagation model to produce realizations of the predicted acoustic wavefield in a vertical plane across the shelfbreak frontal zone. The combined ocean and acoustic results from this Monte Carlo simulation study provide insights into the relations between the uncertainties in the ocean and acoustic estimates. The modeled uncertainties in the transmission loss estimate and their relations to the error statistics in the ocean estimate are discussed.

Internal waves are the primary source of ocean variations in shallow waters. The ability of passive source localization may be degraded by mismatch between forward modals and data because of the activities of internal waves. In this paper, the effects of Garrett-Munk and solitary internal waves on broadband coherent matched-field processing (MFP) and incoherent MFP are investigated. The true data fields are simulated with the presence of internal waves. The replica fields are calculated using the average sound profile (without internal waves). It is shown that both coherent MFP and incoherent MFP give the correct source location if there are Garrett-Munk internal waves only. For the case that there are solitons or both Garrett-Munk internal waves and solitons, incoherent MFP shows a complex ambiguity surface structure and fails to localize the source correctly. However, coherent MFP demonstrates consistent true source localization and small sidelobes.

On the base of the generalized phase integral (WKBZ) theory, the coupled-mode parabolic-equation (CMPE) method is studied in the range-dependent waveguides. The CMPE solution is a hybrid model expressed in terms of the normal modes and mode coefficients, which uses a PE approach in a radial direction and normal modes in depth direction. Considering the contribution of the imaginary part of the eigenvalues, a fast and accurate approach for eigenvalues-finding of local normal modes is presented. The complex eigenvalues are solved from a transcendental equation in real plane. This approach is conveniently applied to other normal mode model. Combining the improved WKBZ theory and CMPE theory, a new range-dependent propagation model is studied. Examples are presented to illustrate the accuracy and efficiency of CMPE.

Usually, in the conventional processing of marine data, the velocity variations of the sea water are not considered because their effects have no influence on the imaging reconstruction, especially at large depths; but in the time-lapse monitoring of hydrocarbon reservoirs, the velocity time-variations in the seawater should be measured and compensated.

We show here the application of a 3-D tomographic tool on a 4-D real data set and the results of the depth imaging by considering or not the velocity variations in the sea water in the time-lapse analysis. The results here presented evidence also how the huge number of time-lapse surveys in various parts of the world may be of great importance for oceanography, since can be a way of monitoring temperature time and space variations within the oceans.

The gain limitation of dense spacing linear array in high or low wave-number noise field is discussed in this paper. The gain expression is derived, and the computer simulation is carried out, which compares the processing results obtained from the towing trial. The theoretical model is consistent with the experimental results, so it can be used to predict the actual array gain. When high wave-number noise predominates, about 30dB gain limitation can be achieved with the aperture of about two wavelengths, which far exceeds the conventional theoretic gain; while low wave-number noise will lead to gain degradation.

Ocean ambient noise is highly correlated with surface wind speed. The relationship between ambient noise and wind speed is given based on the measurements made in recent years. It is shown that ocean ambient noise levels (NSL) are linear with the logarithm of surface wind speeds at usual wind speeds: SPL = a + b*log₁₀V. The coefficients, a and b, depend not only on measurement positions and frequencies, but also on wind speeds. Their values determined at a frequency usually adapt only in the wind speed range from 1.5 to 18m/s (corresponding to the Pushurl wind level of 1 to 6). Our study shows that the coefficients of the previous expression must be re-determined in the cases when the wind level is higher than 15m/s and when the wind speed is lower than 1.1 m/s (corresponding to 0-1 sea state). The expression provides the basis for estimation of wind speed by ambient noise within ±10% of the measured value in wind speed range of 0.1-22.6m/s when the coefficients are determined piecewise. Results of surface wind speed estimation by ocean ambient noise are given. The best estimation frequency band is 2kHz-5kHz.

Method of statistical energy analysis is used to analyze self-noise of underwater fluid-coming structure excited by turbulent boundary layer pressure fluctuation. The estimation formula for the self-noise is obtained.

Many methods have been offered in the treatment of the interface conditions of the three-dimensional parabolic wave equation. There are mainly two improved methods in handling irregular boundary interface: the stair approximation and the slope approximation. The accuracy of each method has been shown in this paper by numerical results.

The aim of this paper is to present a non-standard equation for studying the acoustic propagation in multidirectional non-uniform mean flows. This equation is obtained using a mixed Eulerian-Lagrangian description. Some elementary concepts are recalled, as well as their physical significance and the specificity of the Eulerian-Lagrangian description is outlined. Then, equation that governs sound propagation is derived from Euler equations. When the associate variational formulation is directly used, solutions are corrupted by spurious rotational modes. In this paper, a mixed variational formulation is proposed to avoid this problem. Finally, results obtained by finite element discretization are compared with semi-analytical solutions obtained from the Pridmore-Brown equation. The ability of the method to take into account convection and refraction phenomena is shown. Furthermore, wave propagation in a complex geometry is investigated.

Experiments were performed to verify the validity of our strategy to improve numerical stability on impedance inversion and investigate the influence of different extracting impulse response on the inversion. The acoustic impedance profiles are reconstructed from impulse responses, the results identify to sample data or handbook data with high precision. It is shown that our algorithm is effective in practice and the inverted impedance is less dependent on the methods of extracting the impulse responses.

The 1^st class μ(D) and 2^nd class μ(D^-1) rhythm estimators of time sequence signals are given in this paper, on the basis of transforming signal properly. It is pointed out that the product μ(D) μ(D^-1) = 1 for the periodic signals with rhythms, and μ(D) μ(D^-1) >1 for no-periodic signals. Accordingly the ship noise rhythms are analyzed, and the results are obtained with this method to processing noise of a certain type ship.

The paper presents a high-resolution beamforming method for non-uniform line array. This method first predicts the signal on the uniform position from the actual receive data, and then use Levinson algorithm to estimate the degree of arrival (DOA). Using multi-frequency to accumulate the output, we can get robust high-resolution beamforming. Experiment is also conducted to show that the method can play a robust role in real passive DOA detection, even in cases that the line array are not uniform.

A method previously developed for asymptotic Green's functions is applied to the determination of the time-domain asymptotic Green's function of Biot's poro-elasticity.

A method based on fractional calculus is applied to the generalized finite-difference implementation of equations describing pore gas motion in a rigid porous matrix.

A simple model of seismic wave attenuation combining anisotropy with anelastic effects is constructed. The anelastic response is based on the Cole-Cole relaxation function. Time-stepping finite-difference and ray-asymptotic methods of numerical solution are discussed.

Recent studies reveal that the effect of pore structure on acoustic wave velocity at low effective pressure can be much greater than that of porosity in fractured and/or porous rocks. For two limestones of a given porosity, the difference in compressional velocity can be as large as 2.5 km/s or even larger. In such cases, seismic interpretation for lithological analysis and fluid detection are much complicated. In this paper, we propose a theoretical model for the velocity-porosity relationship including the effect of pore structure on acoustic wave propagation. A formation geometrical factor is introduced to characterize the influence of pore structure on wave velocity, which is effectively quantified as a polynomial function of porosity. We show that pore structure plays an important role in controlling sonic and seismic velocities, as observed in sonic and acoustic measurements. Using numerical modeling, we conclude that the pore structure effects on wave velocity profoundly affect acoustic signals in both amplitude and phase. These results can help to explain many pitfalls of amplitude versus offset (AVO) analysis.

Time-lapse seismic monitoring is a technique to monitor reservoir with repeat seismic prospecting in different time during the period of reservoir production. In this paper, the method of normalization of time-lapse data is presented. And normalization for amplitude, frequency and phase of time-lapse seismic data in non-reservoir is done to obtain the equalized data. The normalization processing was applied in time-lapse seismic data of TPT block in Daqing Oilfield. The results indicate that the differences of time-lapse seismic data in non-reservoir can be eliminated with the normalization processing. While, the differences of time-lapse seismic data occur, which are related with the distribution of production wells. These reasonable identity and difference of these seismic data can analyze and interpret the dynamic variation in reservoir.

Deterministic multiple scattering formulations based on Twersky's approach and the polymerisation technique are presented. These lead to full-scale simulations of elastic-wave multiple scattering in fibre-reinforced composites. It is shown that the convergence and accuracy of the composite models depend on the geometry of the composite structure and the strategy used for the polymerisation. The paper explains why the multiple scattering formulation may not converge if precautions are not taken during polymerisation. A strategy is proposed to improve convergence. Results are presented for the case of an incident harmonic SH wave impinging on a bounded Ti/SiC composite region, the plane of propagation being orthogonal to the fibres' axis.

The angular spectrum theory is currently used to analyse the diffraction fields generated by a finite aperture source on surface of the anisotropic medium. Only is the anisotropy of the propagation velocity regarded in such a sclar theory. So this theory is insufficient and exists many shortcomings. Since the elastic wave in solid is the vector field, the diffraction patterns for various wave modes, various components of each mode and various excitated sources are different generally. The angular spectrum theory neglected actual physical field can only give a relative distribution of phenomenological field.

According to the generalized excitation theory in our previous works, the diffraction elastic wave fields excitated by the surface finite-aperture source are obtained in this paper. As an example, the calculation results of the diffraction fields for YZ quartz are given and compared with that of the angular spectrum theory.

The zigzag dispersion curves of Rayleigh wave are usually obtained in practical exploration especially in multi-layered media contained the low-velocity zones. The mechanism of zigzag dispersion curves of Rayleigh wave in multi-layered media is studied in this paper, and the characteristics of zigzag shapes in different models are also discussed. It is found that the number of zigzag shapes decreases with the decrease of the number of the low-velocity zones in multi-layered media. The shapes of the zigzag dispersion curves can reflect the position of the low-velocity zones. In addition, the effects of parameters of medium models on the zigzag dispersion curves are investigated in detail.

In this paper, the time reversal method in anisotropic elastic solid is theoretically studied for the first time. The transversely-isotropic anisotropic medium (6mm) is modeled as the anisotropic elastic solid. And unidirecional glass-reinforced epoxy-fiber is chosen as the material of the 6mm anisotropic medium. Time reversal acoustic field is numerically investigated by ray approximation method. The focused acoustic field has different characteristics in different direction. The focused field is also symmetric about the principal axes. It is found that the width of the principal lobe of the focused acoustic field reaches the minimum in the maximum group velocity direction and reaches the maximum in the minimum group velocity direction. The relation of time reversal acoustic field to the parameters of anisotropic medium is also studied in detail.

The importance of anisotropic phenomena in wave propagation and processing of seismic data is now widely recognized by the exploration community. Transverse isotropy with a horizontal symmetry axis (HTI media) is the simplest azimuthally anisotropic model used to describe fractured reservoirs that contain parallel vertical cracks. This paper puts forward the double profile for the horizontal reflectors in HTI media with any strength of anisotropy. The double profile is obtained through doubling the offset and traveltime of seismic data. Moreover, the equations of P-S wave normal-moveout (NMO) velocity with arbitrary symmetry and any strength of anisotropy are obtained. The converted-wave NMO velocity is controlled by the azimuthal angle, the vertical velocity of incident wave, the vertical velocity of reflected wave and anisotropic parameters (Thomsen parameters). Using azimuthal NMO velocities of P-P wave, P-SV wave and P-SH wave in multicomponent seismic data, we can estimate the velocities of SV-, SH-wave and anisotropy parameters, including Thomsen parameter γ, which has close relationship with crack density.

For multilayered media, the recursive equations of NMO velocity of each layer are developed, which can be applied to HTI media with arbitrary symmetry and any strength of anisotropy. The formula developed here is the same as Dix equation. The numerical results indicate that the S-wave velocity can be exactly computed by using P-P wave and P-S wave NMO velocities in various HTI media, which provides reliable basis for the inversion of Thomsen parameters in 2D multicomponent seismic data.

Active Noise and Vibration Control (ANVC) means generation of intentional sound and/or vibration fields in order to change those caused by existing sources. It has applications in the industry, including the car and aircraft industries, structural design and audio engineering among others. Active noise and vibration control involves several disciplines that link mechanics with Hi-Tech. It necessitates knowledge of physical and theoretical acoustics, electro acoustics, control engineering and electrical engineering.

Combining the multiresolution analysis (MRA) with tomography, this paper proposes a new method of crosshole seismic tomography named the wavelet multiscale seismic traveltime tomography. The method overcomes the drawbacks of linearized inversion depending on initial model and easily trapped by local minimum, and greatly improves the performance of linearized inversion. The results of numerical modeling and field application show that the main advantages of the new algorithm are: reaching the global minimum; inversion results less depending on the initial model; good stability; high resolution and high quality of tomography, and is suitable for non-uniform media with high contrast velocity, and also provides different resolution inverse results benefiting geological interpretation of tomographic images.

Scattering field by penny-shaped crack gains much concern in NDT. In this paper, the surface of the penny-shaped crack is regarded as pressure-released. Because it coincides with a degenerate coordinate surface of oblate spheroidal coordinate system, the separation of variable method is applied to solve the scalar wave equation to calculate scattering field. One kind of oblate angle functions, which can be expanded as the series of associated Legendre functions, is chosen as the solution of one separated differential equation. It is a solution around the ordinary point and convergent within the neighborhood area of origin. The oblate radial functions are asymptotic expansions to the solutions around the irregular singularity. They can be expressed as the series of spherical functions. The coefficients of these series are calculated by continued fraction method and normalized by properly chosen terms respectively. For the desired accuracy in numerical computation, the series are truncated according to the characteristics of asymptotic series. For normal and oblique incidence, under various kinds of conditions, the scattering field is calculated and analyzed. The directivity and low-frequency characteristics of the scattering far field on different conditions are discussed and compared with each other.

A fast adaptive wavelet-based method, named Multiresolution Finite Difference (MRFD) is first proposed to simulate the wave propagation in multi-layered media with general boundary. It is a promising method for complex media where large gradient and dramatic variety occur because of little computational burden and robustness. Numerical results derived from the gepphysics exploration, proves the efficacy and potential of new scheme.

Traditional techniques of multiscale inversion pay a lot attention to the truth that the local minima at coarser scale are far apart from each other, thus provide opportunities for finding the global minimum efficiently at the coarser scale and then switch to the finer scales. But they have not taken full advantage of the coherent relation between scales. In this paper, a method of multiscale-combined inversion (MCI) is first proposed to exploit the coherent relation between scales, compared to the traditional techniques, which we prefer to call multiscale-independent inversion (MII). Since the model (or signal) at coarser scale is a "smoother" version of the model (or signal) at finer scale, inversion result derived from the coarser scale may be considered as certain kind of constraints in the following scales, not merely as a good initial guess for the next scale. MCI is a promising method because of some advantages such as little dependence on initial model, efficiency of convergence and robustness. As one realization of MCI, a new formulation of scale-constrained Least Mean Square (LMS) method has been constructed, which we point out is a certain kind of generalizations of the traditional finite power constrained LMS. Several numerical results show the effectiveness and potential of the method.

Numerical modelling of elastic wave propagation in heterogeneous viscoelastic media with free-surfaces has been studied by using various schemes in the past. The principal methods are finite-element, finite-difference, pseudo-spectral and spectral-element methods. Finite-element methods are very flexible in handling perfectly elastic models with free surfaces. Spectral-element methods are special higher-order finite-element formulations. The latter is one of the best methods to tackle free surface problems as they balance the requirements of complexity, accuracy, and computation time. However the implementation is complicated compared with finite difference methods. Pseudo-spectral methods are accurate but time-consuming, and have difficulty in handling strongly curved or rugged stress-release surfaces although they handle smooth free surfaces reasonably well. Finite-difference methods use fine grids to treat irregular stress-release surfaces and are easily implemented. This work focuses attention on using a staggered finite-difference method. The finite-difference method was chosen because it is simple and the codes are portable. Moreover, this method provides a convenient environment to implement complicated boundary conditions. Based on the variable-order algorithms in finite-difference methods, we extended the existing imaging and variable-order finite-difference algorithms to heterogeneous viscoelastic media with rugged free-surfaces. The implementation of finite-difference algorithms incorporated with curved stress-release boundaries are studied. Under stable conditions with constraints, the proposed method works effectively for modelling wave propagation for 2-D viscoelastic models. With the use of gradually variable orders in space, the unstable problem associated with spatial derivative calculations has not been seen. Results obtained by our algorithms are compared with those using vacuum schemes for representing free surfaces. We also test the viscoelastic model against a perfectly elastic model for high values of quality factor. Our numerical investigations demonstrate that the algorithm is efficient and effective. It can also be extended to 3-D viscoelastic heterogeneous media without problems.

A general three-dimensional (3D) theoretical model for simulating the behaviour of micromachined electrostatic ultrasonic transducers has been developed. The model takes into account the transducer's geometry, the properties of material used, tension in diaphragm, and bias voltage, and then gives a complete mathematical formulation. Some special cases of it are briefly discussed also. The paper further applies the general model and its simplified varieties to predict natural frequencies of a surface micromachined ultrasonic transducer. The comparison between these theoretical values and measured results indicates that the 3D model gives accurate prediction and is a generalized model for describing the behaviour of various electrostatic audio and ultrasonic transducers.

For anisotropic media the knowledge of group velocities and phase velocities of ultrasonic waves is very desirable. Knowing one set of these velocities, the elastic constants can be obtained. In this paper, the magnitude of the group velocity as a function of its direction is established. The magnitude of the group velocity and as well as the phase velocity for any given direction can be numerically calculated by this set of equations.

Ultrasonic transmitting transducers produce harmonic components that are normally lost due to the limited bandwidth of the receiving transducer. In pulse-echo mode it is natural since the same transducer is used both as transmitter and receiver. However, when using a separate receiver the higher harmonic components emitted by the transmitter can be used for improving the temporal resolution in ultrasonic imaging. This paper presents an application of this technique to the non-destructive evaluation (NDE) for improving quality of ultrasound B-scans acquired in the immersion inspection of solids.

Firstly, the presence of higher components in the signal received from copper specimen is proven in a set-up consisting of two ultrasonic transducers with different center frequencies. A narrowband transmitter has lower center frequency while the receiver has a broader band and a higher center frequency. The received harmonic components, although much weaker than the echo in the fundamental frequency band, can be combined to enhance the signal to noise ratio and the resolution in B-scan images. Secondly, the broadband signals acquired in this set up are processed using algorithms enhancing the temporal resolution. It is shown that the presence of higher harmonics results in an improved temporal resolution. Practical application to NDE of electron beam weld in copper is used to illustrate the proposed technique.

In this paper a novel algorithm is proposed for efficiently and accurately implementing the angular spectrum approach (ASA) to calculating curved as well as planar transducers with axisymmetry. In contrast to the existing algorithms, called k-space algorithms, in which the angular spectrum approach (ASA) is implemented in the k-space (the wave vector space), the proposed algorithm is called angular space algorithm because the ASA is implemented in the angular space in which the polar angle is the only variable for an axisymmetric transducer. In the angular space algorithm the problem of undersampling the Green function can be overcome so that the aliasing error in the algorithm can be eliminated. The angular space algorithm is formulated from a general case where the transducer is arbitrarily curved, and a double integral needs to be calculated. In the case of axisymmetric transducers, the double integral reduces to the single integral with a single variable of polar angle. The k- and angular space algorithms have been applied to calculating the field from a spherically curved transducer, and the results from both algorithms have been compared with the analytical results. The comparison has shown that the efficiency and accuracy of the angular space algorithm are much better than those of the k-space algorithm.

In this paper, the sound directionality patterns produced by a nonseparable source located on a rigid cylinder of infinite length have been investigated for the case in which the source strength may be represented as a sum of separable sources by use of addition theorems. The result shows that the patterns of amplitude and phase distribution over the whole farfield space for the source are dependent of the frequencies of sound waves, circumference of the cylinder, and the strength of the source piston. It reveals that the observation points and the frequencies are important to the techniques of the boiler tube leak trace. This result can be applied to the technique of acoustic leak detection.

At present a lot of piezoelectric broad band ultrasonic transducers for both, medical and NDE purposes, use as active material piezoelectric ceramic composites. A lot of scientific and technological research has been devoted to the optimisation of piezocomposite materials, in order to increase the transducer band and efficiency. In a typical transducer based on piezocomposites the active material is mounted on a soft lossy backing and one matching layer is placed on the front, radiating face of the transducer with the aim to match the acoustic impedance of the medium and to enlarge the bandwidth. In this paper an optimisation work is shown to demonstrate that a composite configuration can be used in the matching layer, in order to improve the efficiency and the band of the transducer. An approximated two–dimensional analytical model has been used to optimise the design of a composite–structured matching layer in the case of 2–2 composites, obtaining different results for the polymer and piezoceramic composite phases; a design technique is suggested in order to improve the transducer performance. With the aim to verify the proposed design criterion, a transducer prototype, based on a 2–2 piezocomposite, with a composite matching layer was realised. On this sample we measured the electrical input impedance and the insertion loss and we compared the obtained results with those of a transducer classically matched to the load. The obtained results confirm the computed improvements in the transducer performance and justify the proposed design approach.

Spatial resolution in modern ultrasound imaging systems is limited by the high cost of large aperture transducer arrays, which means large number of transducer elements and electronic channels. A new technique to enhance the lateral resolution of pulse-echo imaging system is presented. The method attempts to build an image which could be obtained with a transducer array aperture larger than that physically available. We consider two images obtained imaging the same object with two different apertures, the full aperture and a sub aperture, of the same transducer. A suitable artificial neural network (ANN) is trained to reproduce the relationship between the image obtained with the transducer full aperture and the image obtained with a sub aperture. After a suitable training, the network is able to produce images with almost the same resolution of the full aperture transducer, but using a reduced number of real transducer elements. All the computations are carried out on envelope-detected decimated images: the overall computational cost is low and the method is suitable for real time applications. The proposed method was applied on experimental data obtained with the ultrasound synthetic aperture focusing technique (SAFT), giving quite promising results. Real-time implementation on a modern full digital echographic system is currently being developed.

The following sections are included:

• Introduction

• Regular mesh of long duct

• Conclusions and future investigations

• References

The following sections are included:

• Introduction

• The modal basis approach

• Hierarchical error estimation

• Multigrid preconditioned subspace iteration

• Numerical results

• Conclusions

• References

Our work is devoted to the solution of large scale (kl = 10…100π) three dimensional radiation and scattering problems covered by the time harmonic Helmholtz equation. We present an application of the Regular Grid Method and Multilevel Fast Multipole Method to acoustic scattering problems. These methods lead to a memory requirement of what enables us to solve exterior scattering or radiation problems with several ten-thousands of unknowns. In two computational examples we show the efficiency of these methods.

Time dependent analysis of the dynamic damped behaviour of continua are mathematically modelled by partial differential equations. One gets uniqueness, existence and stability (well posed problems) by the implementation of the correct initial boundary conditions. However, by taking memory effects under consideration, any change in the past of the system changes the future dynamic behaviour. Classical damping descriptions failed when describing the behaviour of many materials, like teflon. This is because in classical theory the operators are local ones. The implementation of fractional time derivatives into the partial differential equations is an alternative technique to overcome these problems. Now the time derivative operator is a global one, memory effects in structure borne sound are calculable.

In this presented paper the theory of fractional time derivative operators and their application in continuum mechanics is briefly sketched. The main result when using this methods for damping behaviour is that a global operator is needed which takes the whole history into account. We call this theory the functional calculus method instead of the well known fractional calculus with the use of initial conditions.

To show the effectivity of this method the impulse response of a viscoelastic rod is compared with measurement. It is shown that the damping behaviour is described very much better than by other models with comparable few parameters. Moreover it is the only one that works in a wide frequency range as well as it can describe the dispersion of the resonance frequencies. The implementation of this damping description in a Boundary Element Code is an application in dynamics of 3D continua in frequency domain.

This paper deals with a 2d scattering problem of elasto-acoustic plane waves in elastic isotropic and homogenous media. An indirect BEM formulation is applied to obtain the solution of a problem with mixed boundary condition in both bounded and unbounded domains. A system of integral equations with singular kernels is derived. The variational formulation allows direct integration of singular terms and Hypersingular integral, coming from the traction representation, is treated based on analytical transformation. An associated algorithm is implemented. Several examples and discussions, showing stability and accuracy of this method are presented.

The nearfield acoustical holography (NAH) based on the boundary element method (BEM) is one of the indirect identification methods for vibro-acoustic source properties. In this method, the sound radiation and transmission between a vibrating source and a hologram plane is modeled by the vibro-acoustic transfer matrix using the boundary integral equation (BIE). Distribution of surface velocities of the source can be reconstructed by multiplying the inverse of the calculated vibro-acoustic transfer matrix and the measured field pressure vector. In principle, the pressure data can be measured at a surface with any shape of the nearfield plane including the conformal one. In this technique, the field pressure should be measured as close to the source as possible in order to acquire precise field information including the non-propagating evanescent waves for an accurate source reconstruction. In three-dimensional problems, a propagating wave in the free field can be a Green function satisfying the Helmholtz equation. In the acoustical BIE, i.e. Kirchhoff-Helmholtz integral equation, monopole and dipole Green functions are proportional to 1/R or 1/R² when R denotes the distance between source and field point. This means that there is a serious singularity problem in the nearfield. In order to overcome this dilemma, the non-singular boundary integral formulation utilizing propagating plane waves is employed in the NAH. With this kind of BIE, all singularities included in the conventional acoustical BIE can be removed. Consequently, the field pressure can be determined precisely in the very closed field and the acoustical parameters can be continuously calculated on the surface. This can result in the improvement of the BEM-based NAH. In this study, the nonsingular BIE is reformulated to deal with the foregoing matter and the holography equation is derived by using this formulation. Simulations and experiments are performed for an exterior acoustic model in order to demonstrate the advantageous characters of the present method. These include the prediction accuracy of nonsingular BEM (NBEM), and the improvement of reconstruction result in NAH. When the NBEM is used for NAH, it is observed that the reconstruction error is improved more and more by making the hologram plane to nearing the source surface. In particular, the improvement of reconstructed result is distinguishable when the distance becomes less than about 20 % of the characteristic length.

Based upon a stochastic and asymptotic formulation of the acoustic model, the diffusion effects of a random multilayer structure on wave propagation are analyzed for a totally refracting random medium on which waves are incident obliquely. Using a two step decomposition method, the random phase processes of the reflection coefficients are asymptotically represented in terms of the infinitesimal generators of the Kolmogorov backward equations obtained in the two divided regions of propagating and evanescent waves. The marching limit process of the phase processes throughout the whole region is expressed by the product of two evolution operators and a transformation given at the turning point. Finally, the transition probability density of the random phase process solving the Komogorov-Fokker-Planck equation is explicitly approximated by an infinite dimensional functional construction.

The application of finite element method (FEM) in analyzing the waves propagation in cylinders is presented in this paper. The finite element equation is obtained by the virtual displacement theorem and solved by the central difference method. The spatial and temporal steps of discretization of the finite element computation are discussed. In order to compute high-frequency problems, a new sub-matrix method was set up and applied in the computations. The displacement fields, excited by a pulsed line-source with 5 MHz central frequency, are computed and displayed by gray plots. The waveforms of receiving points on the surface of the cylinders, identified as A-scan results, are also shown. The comparison between the FEM and experimental results shows that the FEM is a good way for analyzing the waves propagation in cylinders.

Aimed to time relay partial differential equation (PDE), 2-D wave equation, a method of joint using Finite Element Method (FEM) and Finite Difference (FD) Method in spatial domain, which is named as Finite Element – Finite Difference Method (FE–FDM), has been proposed in this paper. By using the semi-discretization technique of FEM in spatial domain, the origin problem can be written as a coupled system of lower dimensions PDEs that continuously depend upon time and part of space. FDM is employed to solve these lower dimensions PDEs. The concept and theory of this method have been discussed in this paper. A numerical example of 2-D wave reverse-time propagation shows the excellent performance and potential of it.

The following sections are included:

• Problems and strategy

• Imaging by Common Reflection Point Set(CRPS) based on models

• Example

• Discussions

• References

The following sections are included:

• Introduction

• Basic Theory

• Results

The wake/stator interaction has been predicted using an N-S solver, in order to obtain unsteady pressure fluctuations on blade surface and the variety of unsteady vortex. So the noise can be computed. The equations are solved using a finite volume time-integration Ni scheme. For improving the ability to simulate the turbulent flow , K-ε turbulence models are utilized for the turbulent closure. Then the numerical calculations on a turbine and an axial compressor cascades are carried out in this paper.

The following sections are included:

• Introduction

• Representation by potentials

• Uniqueness and stability for the inverse source problem

• Approximation and completeness results

• References

The basic MVU (minimum variance unbiased) beamforming designs are defined to satisfy the general design constraints of (1) no signal distortion (boresight constraint) and (2) minimizing output noise power. The aim of applying MVU beamforming in seismic data processing is to extract seismic reflection signals. For this purpose, additional criteria in MVU beamforming for coherent noise, such as multiples, rejection are (1) zero or minimize the response to coherent (multiple) and (2) constrain random noise amplification. The solution of this problem can be obtained by using the method of Lagrange multiplier. The main challenger for the implementation of beamforming is the way to calculate the inversion of an ill conditional matrix to obtain the solution. This paper will show the stability, resolution and error study of different inverse operators for the ill conditional matrix of implementation of beamforming in seismic data processing.

In this paper, the finite element method (FEM) is applied to analyze the sound characteristic of the tiles with the periodic arraying spherical and cylindrical cavities. The reflection and transmission coefficients are obtained. Moreover, numerical results are obtained for tiles with different periodic structures and the effect of the cavities' size and shape is analyzed.

Beamlet propagation and imaging using Gabor-Daubechies frame (G-D frame) decomposition with local perturbation theory is developed and tested. The method is formulated with a local background velocity and local perturbations for each window of the wave field decomposition. The propagators and phase-correction operators are obtained analytically or semi-analytically by one-way operator decomposition and screen approximation in beamlet domain. The numerical tests on point-spreading responses in homogeneous and laterally varying media and the imaging example for the SEG-EAGE salt model demonstrate the validity and the great potential of this approach applied to seismic wave propagation and imaging.

Mode-coupling induced by internal solitary waves in shallow-water is investigated by numerical simulation. It has been found that mode-coupling is very sensitive to frequency, mode number as well as water depth.