Journal of Computational Acoustics

As a first step towards investigating multiple scattering of acoustic waves by a few scatterers, this paper presents a detailed study of the case that sound is scattered by a pair of arbitrary targets. An intuitive scheme is developed for calculating the effective scattering function. An approximate formula is derived and presented in a simple form. To verify the approach, it is applied to the special case of two rigid spheres, as the exact solution exists for this special situation. It is shown that the approximate results compare favorably with that from the exact solution for a wide range of situations. Through the comparison, the validity range of the approximate approach is inferred. Some ambiguities in the previously published results are also pointed out.

The ambient noise model CANARY calculates noise coherence and array noise response by treating the noise sources as surface distributions rather than points. This assumption leads to simplification of the propagation, even in range-dependent environments, and by allowing variations in the source density one can represent distant storms or groups of ships. Included is a description of the numerical algorithms used to calculate coherence. Some recently developed analytical solutions for uniform source distributions and uniformly sloping seabed³ are used as test cases for CANARY. Some additional examples demonstrate CANARY's performance in more realistic environments and conditions including wind and shipping sources, and comparisons are made with the noise model RANDI.

The role of 3-D effects has long been debated in the ocean acoustics community. Typically, 3-D problems are handled in an approximation that uses a fan of bearing lines to build up a 3-D image of the acoustic field with many 2-D slices. This is the so-called N by 2-D approach and it is probably safe to say it is adequate for simple calculations of the SONAR range-of-the-day. On the other hand, acoustic models are now finding an important role in matched-field processing (MFP) where they are integrated with the signal processing to correct for the channel effects. These applications demand much more from the models but, somewhat surprisingly, recent broadband experiments have revealed that the models are up to the task. The main features of the echo patterns are well-predicted by such models and can be reliably exploited, at least in areas where the ocean bottom is fairly flat. The challenge now is to push MFP to more realistic scenarios with complicated topography and potential 3-D effects. One phase of the December 1997 Key West LWAD (Littoral Warfare Advance Development) experiment was designed to address exactly this issue. We use this Key West scenario as a backdrop to discuss modeling approaches combining beams and modes and the impact of the resulting 3-D variation on MFP for towed arrays.

To assess the adiabaticity of sound propagation in the ocean is very important for acoustic field calculating (forward problem) and tomographic retrieving(inverse problem). Most of the criterion in the literature is too restrictive, specially for the nongradual ocean structures. A new criterion of adiabaticity is suggested in this paper. It works for nongradual ocean structures such as front and internal solitary waves in shallow-water.

This work focuses on the way the pressure field generated by a dilatational load changes inside a channel waveguide filled with a homogeneous fluid, in the presence of an irregular floor. The problem is solved in the frequency domain using the boundary element method. The Green's functions used are obtained by superposing virtual acoustic sources defined so as to generate the free or rigid surfaces of the fluid channel. The responses in the time domain are obtained by means of Fourier transforms, using complex frequencies. Irregular floors of varying sizes are modeled. The main features and spectral representation of the signals scattered by these inclusions are described. They are used to elucidate the most important aspect of wave acoustics, which can be used as the basis for the development of nondestructive testing and imaging methods.

Three-dimensional infinite elements for exterior problems of time-harmonic acoustics are developed. The infinite elements mesh only the outer boundary of the finite element domain and need not match the finite elements on the interface. A four-noded infinite element, based on separation of variables in spherical coordinates, is presented. Singular behavior of associated Legendre functions at the poles is circumvented. Numerical results validate the good performance of this approach.

Acoustical tomographical reconstruction of the location and sizes of spatially localized, random perturbation in a shallow-water oceanic waveguides is developed for mid-range distances. As examples of such perturbations, clouds of bubbles arising due to breaking wind waves and shoals of fishes are discussed. The ray approximation is used for the acoustical field description. The oceanic environment is presented as a medium with a background of uniformly distributed volume and surface random inhomogeneous fluctuations. Complex acoustical repetitive pulses are used as the probing signal. The receiving system consists of an array of spatially distributed hydrophones. The signals received from all hydrophones are processed on the basis of a tomographic algorithm including matched-filter processing in the spatial and frequency domains for reconstruction of the random time-delay statistical moments (average value and dispersion), which are descriptors of the observed random spatially localized inhomogeneity. As a specific model of random perturbation, a three-dimensional Gaussian cloud of fluctuations associated with parameters of a fish shoal is investigated. A vertically distributed array of receivers and a directed source of LFM-pulses are tested as the prototype of the tomographical system in a shallow waveguide with typical hydrology. Computer simulations of the tomographical reconstructions are conducted taking into account investigation of the influence of the level of additive noise on the accuracy of the reconstructed parameters.

The horizontal refraction method (HRM) of transverse current acoustics monitoring in the Fram Strait environment is modeled numerically. The current profile was constructed on the basis of experimental data on the West Spitsbergen Current (WSC) structure and the complex environment of the region presented by the sound speed profiles, which were derived from the experiments. The HRM is based on the measurements of the deflection of the horizontal modal rays due to the gradient of the refraction index produced by the transverse current. The signals received by the antennas in a receiving plane perpendicular to the propagation axis have a modal phase difference in proportion to the average transverse velocity, which is due to the different orientation of the acoustical path of the current. The interesting result is that even in a strong mode coupling environment the numerically obtained phase difference is quite close to its value as calculated by adiabatic approximation. The physical reason behind this is most likely that the mode coupling impact can be canceled at the sound signal propagation along the two very close paths.

The nonlinear constitutive equations of Kelvin's medium (a nonlinear elastic polar medium consisting of rotating particles) are obtained. We establish the analogy between basic equations of saturated elastic ferromagnetic insulators and basic equations of Kelvin's medium. The most general way of taking the couplings of magnetic and elastic subsystems into account is suggested. Wave processes are investigated from this point of view. All results are interpreted both in terms of mechanical medium and ferromagnets.

This study is a part of a project to develop a borehole sonar for acoustic imaging of the geological structure of the rock formation near to a deviated or horizontal borehole performed while drilling. The purpose of the sonar is to provide a direct measure of the distance and the direction to bed boundaries parallel to the borehole. This paper gives a theoretical analysis of the effect of the borehole and the drillstring on the received response on sensors positioned on the drillstring and proposes a method for processing the received signal to enable the determination of the direction of an incoming plane wave and time of arrival. In the first part of this paper, the response at any position in a fluid filled borehole is determined for an incoming plane P-wave. It is shown that the response is quite complicated and consists of several vibrating modes with resonance structure. Secondly, the paper presents how the received signals can be decomposed and the individual modes can be resolved by utilizing the orthogonal properties of the mode functions. The individual mode functions are resolved by performing a spatial Fourier transform of the sensor signals. A requirement is that the sensors are uniformly distributed around the circumference of the drillstring and that the number of sensors is at least two times the number of significant modes. It is demonstrated that the spatial and temporal characteristics of the resolved modes can then be exploited to determine the time of arrival and the angle of direction of the incoming reflected waves.

In this work, the analytical and experimental examination of the problem of diffraction effect is treated. In the laboratory, the diffraction phenomena have been mainly due to the beam spread of the ultrasonic plane wave propagating through a viscoelastic material. In fact, this effect has been found to be related, essentially to the attenuation and dispersion losses on a viscoelastic material. In this work, a frequency domain system identification approach is applied to determine an optimal function correcting the beam spread effect in both the normal and oblique incidences for a large frequency band (300 kHz–3 MHz). The Maximum Likelihood Estimator is applied to the magnitude and phase of the measured beam patterns of the used transducers in order to determine the model parameters. The calibration procedure is also discussed. Once the proposed model is established, the propagation through a viscoelastic plate is described and a comparison with measurements is done to validate the investigated model. The obtained longitudinal and shear attenuation and dispersion of the ultrasound in the viscoelastic plate are compared with those obtained by applying the complex harmonic plane waves combination model.

Two new derivations of "vector" parabolic equations (PE) for use in acoustic propagation have recently been published. In these cases, PEs have been derived from first principles and incorporate velocity fluctuations of the medium as two additional vector terms. In the simpler case, large spatial-scale velocity fluctuations can be accommodated. In the more general case, multi-scale velocity fluctuations can be accommodated.

In this paper we report on a series of two-dimensional numerical experiments which compares sound propagation predicted from traditional PEs with sound propagation predicted from these two "vector" PEs. Two types of velocity fields are simulated. One, suitable for approximating an atmospheric boundary layer, is a field in which velocity has only a horizontal component, but whose magnitude can depend on height, i.e., v=v_x(z). The other is a field having random spatial fluctuations over a range of length scales and could be suggestive of atmospheric turbulence. In both cases celerity inhomogeneities are also included.

Results suggest that at least, in two dimension, the standard PE using an effective index of refraction is not accurate to describe the effects of the mean and turbulent velocity on sound propagation near the ground. We suspect that in three-dimensional problems, the added terms in the "vector" PEs will significantly increase in importance.

Uniformly asymptotic frequency-domain solutions for a class of hyperbolic equations with singular convolution operators are derived. Asymptotic solutions for this class of equations involve additional parameters — called attenuation parameters — which control the smoothing of the wavefield at the wavefront. At caustics the ray amplitudes have a singularity associated with vanishing of ray spreading and with divergence of an integral controlling the rate of exponential amplitude decay. Both problems are resolved by applying a generalized Kravtsov–Ludwig formula derived in this paper. A different asymptotic solution is constructed in the case of separation of dispersion and focusing effects.

The parameters of diffraction tomography methods for inhomogeneity reconstruction is investigated. For a two-dimensional case with linear multi-element transceiver one single frequency and two multi-frequency measurement schemes are introduced. For these schemes the data regions in the spatial frequency domain of inhomogeneities function are presented. The new two-stage approach is developed for resolution investigation and image quality estimation. In the first stage, it is proposed to find the resolution for ideal measurement conditions (infinite measurement aperture, ideal transducers). Then in the second stage for real measurement conditions, it is enough to determine only the degradation factor which is calculated for all measurement schemes dependent on measurement parameters. For examination of this approach also the images for complicated inhomogeneity model are calculated. It is shown that resolution and point spread functions cannot fully describe image quality, especially for complicated data regions in the spatial frequency domain of the inhomogeneities function. Many numerical examples are presented.

Boundary integral equation approaches and their discretization into boundary element methods (BEM) have been useful to obtain solutions for numerous problems in dynamic elasticity. Well documented advantages over domain approaches are dimension reduction, relatively easy fulfillment of radiation conditions at infinity, and high accuracy of results. In spite of dimension reduction, the computational cost at high frequencies may easily exceed the capacity of computing facilities. To overcome this problem, Galerkin's ideas may be used. The Indirect Boundary Element Method (IBEM) equations are the starting point of the proposed methodology. The boundary force density is expanded in terms of a complete set of functions. Weighting functions from the same complete set are used to minimize the error of this approximation. Once a significant subset is selected, the size of the resulting linear system is much smaller than that of the IBEM method as currently applied. Moreover, with appropriate trial functions, some matrix operations can be reduced to Fourier transformations.

In what follows, the formulation and some examples for scalar problems are presented. Simple 2-D topographies are studied, but the extension to 3-D realistic configurations may well be treated on the same basis.

A theoretical study is made of the flow induced by a piston, performing rotational sinusoidal vibrations in a uniform two-dimensional planar duct. A frequency of the forced vibrations is close to a cut-off frequency. It is shown that exactly at a cut-off frequency the steady-state response represents a transverse periodic shock wave, propagating between duct walls and decaying algebraically in the far field.

Due to the complexity of the seafloor as an acoustical system, it is very important to validate wave propagation models and test inversion methods first in the laboratory. A known sediment is placed in a water-filled tank, in such a way that a smooth water–sediment interface is obtained with minimal air bubbles retained in the sediment. Two broadband piston transducers are used to carry out reflection experiments at oblique incidence. The placement of the emitter and receiver is symmetrical with respect to a vertical plane perpendicular to the sediment surface. The aim of this work is to find physical parameters of the sediment from the reflected signals. A general viscoelastic model that considers losses due to absorption and dispersion along the propagation of the wave through the sediments is proposed. A rational transfer function is used to model the viscoelastic modulus of the bulk sediments and a comparison with constant Q viscoelastic modelling is performed. The estimation of the model parameters is elaborated using a Maximum Likelihood Estimator in the frequency domain.

The problem of determining the shape of an object from far-field data is considered. We present a method, originally formulated in Ref. 1 and furtherly modified in Ref. 3, for the solution of this ill-posed nonlinear inverse problem whose main features are:

• the method is exact, that is no low- or high-frequency approximation is considered;

• it is not necessary to know the number of scatterers and whether or not the scatterers are penetrable by the waves;

• if the medium is not penetrable, it is not necessary to know whether the obstacle is sound-hard or sound-soft;

• in the case of an inhomogeneous scatterer, the method provides the shape of the inhomogeneity.

The method is particularly simple since it requires only the solution of a linear Fredholm integral equation of the first kind whose integral kernel is the far-field pattern. The numerical instability due to ill-conditioning can be reduced by using regularization algorithms such as Tikhonov method where the regularization parameter is chosen by using Morozov's discrepancy principle generalized to the case where the noise affects the kernel of the integral operator.

We present an adaptive finite element method for solving the acoustic wave equation. Using a global duality argument and Galerkin orthogonality, we derive an identity for the error with respect to an arbitrary functional output of the solution. The error identity is evaluated by solving the dual problem numerically. The resulting local cell-wise error indicators are used in the grid adaptation process. In this way, the space-time mesh can be tailored for the efficient computation of the quantity of interest. We give an overview of the implementation of the proposed method and illustrate its performance by several numerical examples.

We present a tutorial introduction to the 3-D finite-difference modeling of seismic ground motion in elastic and viscoelastic media with special emphasis on its computational efficiency. We consider four basic types of the finite-difference schemes — the displacement-stress, displacement-velocity-stress and velocity-stress schemes on a staggered grid, and displacement scheme on a conventional grid. Their memory requirements in the case of perfectly elastic medium, elastic medium with aposteriori approximate attenuation correction, and realistic viscoelastic medium are reviewed. We also present application of the powerful optimization techniques to the 3-D fourth-order displacement-stress and displacement-velocity-stress modeling in the case of viscoelastic medium whose rheology is based on the generalized Maxwell body. Description of a medium using material cell types and use of a discontinuous grid with combined memory optimization makes it possible to simulate earthquake ground motion in realistic large-scale models.

Complete families in a given function space are sets of linearly independent functions, a linear combination of which can approximate any other function with arbitrarily high accuracy. Outgoing cylindrical wave functions are one such family, used to represent the scattered wave in exterior boundary value problems for the scalar Helmholtz equation in two spatial dimensions. When the incident wave is plane and the scattered wave is represented by a series of said functions, which converges up to the boundary of the obstacle, the obstacle is said to be in the Rayleigh class. One shall further distinguish between Dirichlet–Rayleigh and Neumann–Rayleigh obstacles, according to the applicable boundary condition. Discs are trivial obstacles of these classes. Ellipses of eccentricity η such that η²<1/2 were shown to be in the Dirichlet–Rayleigh class by Barantsev et al. in 1971, who used the saddle point method to asymptotically estimate the Fourier scattering coefficients. Herewith, another one parameter family of obstacles is constructed by the same method. It is also shown that the same obstacles are in the Neumann–Rayleigh class. The relevance of these results to the numerical treatment of scattering problems is briefly discussed.

The problem of reflection/transmission of plane waves in homogeneous isotropic dissipative media is investigated. In dissipative media, the elastic moduli, seismic velocities and slowness vectors are complex-valued and frequency-dependent. Results of extensive computations of R/T coefficients for plane waves incident at a plane interface are presented and discussed. In all cases, the "reference R/T coefficients" are considered, computed for a fixed frequency equal to the reference frequency f_r, for which the elastic moduli are specified. The influence of angle of incidence i and attenuation angle γ of the incident wave on the coefficients is studied in detail. In general, the differences between the moduli of the R/T coefficients for dissipative and nondissipative models are usually small for realistic Q's and γ's. These differences are more distinct in critical and post-critical regions for the reflection coefficients, particularly for small Q's, large contrast of Q's across the interface, and for large |γ|'s (say, |γ|>50°). The differences between the phases of R/T coefficients for dissipative and nondissipative media may be larger. For some γ's, the signs of the phases for dissipative and nondissipative media may be roughly opposite, even for very high Q's.

A practical definition of site effect, that accounts for both the seismological and engineering point of view, is the one given by Field⁷: "the unique behavior of a site, relative to other sites, that persists given all (or most) of the potential sources of earthquake ground motion in the region". This definition implicitly indicates the difficulties connected with a correct estimate (or prediction) of the site effect. We estimate theoretical site responses adopting several techniques and using different seismic motion types. Our results confirm that the identification of the site behavior with a set of resonance frequencies can be a very difficult task, especially when the amplification levels seem to be "azimuthally" dependent. We show that the adoption of simplified models can lead to misleading conclusions concerning the seismic response of sedimentary basins. The results suggest that, in order to perform an accurate estimate of the site effects in complicated geometries, it is necessary to make a parametric study that takes into account the complex combination of the source and propagation factors.

We consider the problem of the reconstruction of the shape of an obstacle from some knowledge of the scattered waves generated from the interaction of the obstacle with known incident waves. More precisely we study this inverse scattering problem considering acoustic waves or electromagnetic waves. In both cases the waves are assumed harmonic in time. The obstacle is assumed cylindrically symmetric and some special incident waves are considered. This allows us to formulate the two scattering problems, i.e. the acoustic scattering problem and the electromagnetic scattering problem, as a boundary value problem for the scalar Helmholtz equation in two independent variables. The numerical algorithms proposed are based on the Herglotz Function Method, which has been introduced by Colton and Monk.¹ We report the results obtained with these algorithms in the reconstruction of simple obstacles with Lipschitz boundary using experimental electromagnetic scattering data, that is the Ipswich Data^2,3 and in the reconstruction of "multiscale obstacles" using synthetic acoustic scattering data.

The finite-element method (FEM) with mass lumping is an efficient scheme for modeling seismic wave propagation in the subsurface, especially in the presence of sharp velocity contrasts and rough topography. A number of numerical simulations for triangles are presented to illustrate the strength of the method. A comparison to the finite-difference method shows that the added complexity of the FEM is amply compensated by its superior accuracy, making the FEM the more efficient approach.

A new numerical technique is developed for wave propagation in heterogeneous poroelastic media and mixed poroelastic, elastic and acoustic media. The scheme, based on a first-order hyperbolic Biot's system and a discretization mesh of triangles and quadrangles, solves the problem using integral equilibrium equations around each node, instead of satisfying Biot's differential equations at each node as in the finite-difference method. The surface topography and complex geometrical interfaces can be accurately modeled with the proposed algorithm by making the nodes of triangles and quadrangles follow the curved interfaces. The elastic (acoustic)/poroelastic interface conditions of complex geometry are introduced using the integral equilibrium equations around nodes at the interface based on the continuities of total stresses and velocities between the interface. The free-surface conditions of complex geometrical boundaries are satisfied naturally for the scheme. This work is an extension of the grid method for the heterogeneous elastic media to the heterogeneous poroelastic one. The proposed algorithm is successfully tested against an analytical solution for Lamb's problem when the algorithm is reduced to handle the elastic limit of the Biot's equations. Examples of wave propagation in a poroelastic half-space with a semi-cylindrical pit on the surface and mixed acoustic-poroelastic and elastic-poroelastic models with inclined interfaces are presented.

We apply a spectral element method based upon a conforming mesh of quadrangles and triangles to the problem of 2-D elastic wave propagation. The method retains the advantages of classical spectral element methods based upon quadrangles only. It makes use of the classical Gauss–Lobatto–Legendre formulation on the quadrangles, while discretization on the triangles is based upon interpolation at the Fekete points. We obtain a global diagonal mass matrix which allows us to keep the explicit structure of classical spectral element solvers. We demonstrate the accuracy and efficiency of the method by comparing results obtained for pure quadrangle meshes with those obtained using mixed quadrangle-triangle and triangle-only meshes.