Volume 103, Issue 2, February 1998
 GENERAL LINEAR ACOUSTICS [20]


Acoustic scattering on an elastic plate described by the Timoshenko model: Contact conditions and uniqueness of the solution
View Description Hide DescriptionIt is shown that the known solution of the acoustic scattering problem on a supported elastic plate, described by the Timoshenko model, should be corrected. The general formulation of the contact conditions that imply the reciprocity principle is given. Sommerfeld’s formula and the “optical” theorem for the model are formulated. They lead to the uniqueness of the solution. The numerical comparison of the effective cross section of scattering from supported elastic plate described by the Kirchhoff and by the Timoshenko models is presented.

The Rayleigh equations for a multisinusoidal periodic surface
View Description Hide DescriptionIt was shown by Purcell [J. Acoust. Soc. Am. 100, 2919–2936 (1996)] that the Rayleigh equations in the Fourier domain for the reflection coefficients for scattering of a plane wave from a pressurerelease sinusoid are valid if the maximum slope of the sinusoid This current work finds the corresponding constraint sufficient for the validity of Rayleigh’s equations for a more general periodic surface consisting of a finite sum of sinusoids. The mathematical basis of the derivation of the Rayleigh equations from the Helmholtz integral formula is a Fourier series given by Oberhettinger. Unlike the single sinusoid case developed by Purcell (referenced above), the analysis of the general periodic surface given here requires an analytic continuation argument. In addition, a set of (infinite linear) equations of the “second kind” is derived for the reflection coefficients for the general periodic surface. This guarantees that the truncation solutions for the reflection coefficients converge and are unique. The matrix elements involved in these equations of the second kind require the numerical evaluation of a finite integral (a generalization of Bessel’s integral for the integer index Bessel functions) and all calculations required can be performed by desktop computing.

Propagation of Love waves in a transversely isotropic fluidsaturated porous layered halfspace
View Description Hide DescriptionThe dispersion equation for Love wave propagation in a layer lying over a halfspace is derived. Both media are assumed to be transversely isotropic fluidsaturated poroelastic solids with principal axes perpendicular to the surface. The analysis is based on the Biot’s theory. The dissipation due to fluid viscosity is considered and therefore the dispersion equation is complex and intractable analytically. An iterative procedure is developed to solve this equation. Two situations are discussed in detail: (i) an elastic layer overlying a poroelastic halfspace and (ii) a poroelastic layer lying over an elastic halfspace. Dispersion curves and attenuation curves of Love waves are plotted for these two cases. In addition, the upper and lower bounds of Love wave speeds are also explored.

A threedimensional finite difference code for the modeling of sonic logging tools
View Description Hide DescriptionThis paper presents a numerical program for the simulation of elastic wave propagation and scattering in threedimensional (3D) cylindrical coordinates based on the firstorder velocitystress finitedifference scheme on staggered grids. Both Liao’s and Lindman’s absorbing boundary conditions are implemented for the exterior boundaries to efficiently truncate the computation domain for elongated 3D well logging problems. Symmetric and antisymmetric boundaries in azimuthal and axial directions are also implemented in the code to further reduce the size of the problem. Included for the first time with this code are very large and complex geometrical structures such as the whole slotted sleeve housing of a sonic welllogging tool which typically involves hundreds of millions of unknowns. The calculation for such a large problem only takes a couple of days on a fourprocessor SGI Power Challenge machine. Different types of slotted sleeve models are studied for sonic logging tools. Simulation results show that different slotted sleeves vary widely in delaying and attenuating the pipe waves which travel along the tool housing. A new slotted sleeve structure with three horizontal slot sections for every vertical slot period is proposed for better performance. A dipole source is found to produce much cleaner waveforms than a monopole source.

A recursive Green’s function technique for acoustic scattering from heterogeneous objects
View Description Hide DescriptionA fast, efficient algorithm for computing acoustic fields scattered by inhomogeneous objects in an otherwise homogeneous space is presented. The algorithm, called the Recursive Green’s Function Method (RGFM), constructs the domain Green’s function by recursively combining known Green’s functions from smaller subdomains. The fields on the scatterer surface are then computed using a boundary integral formulation. Proper implementation of the RGFM results in a storage requirement of and computational costs of and for two and threedimensional problems, respectively, where is the total number of discrete points in the inhomogeneous region. Results are compared with those obtained from exact solutions to show the accuracy of the method.

Calculation of acoustical scattering from a cluster of scatterers
View Description Hide DescriptionThe problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed. The problem is formulated by using the matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm (MLFMA) and the fast multipole method–fast Fourier transform (FMMFFT) method, and the efficiency of the two methods is compared. It was observed that, in general, the MLFMA performs better than the FMMFFT algorithm. However, when the scatterers are distributed uniformly on a rectangular grid, the FMMFFT algorithm performs as good as the MLFMA. The accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres.

Target scattering calculations with the parabolic equation method
View Description Hide DescriptionThe parabolic equation technique is used to solve the Helmholtz equation in the presence of scatterers of arbitrary shape, in two and three dimensions. The scattered field is computed directly, using nonhomogeneous boundary conditions on the scattering object to represent the incident field. Effectively this decouples the PE paraxial direction from the direction of incidence. For convex objects the whole range of scattering angles can be covered with a small number of narrowangle calculations. Finitedifference implementations involve tridiagonal matrices in two dimensions and more general sparse matrices in three dimensions. The resulting codes can be used to solve scattering problems for objects ranging in size from a few wavelengths to hundreds of wavelengths. The method has been tested against analytical solutions for soft and rigid circular cylinders in 2D and soft and rigid spheres in 3D, showing good agreement at all scattering angles.

An iterative solver of the Helmholtz integral equation for highfrequency acoustic scattering
View Description Hide DescriptionHighfrequency scattering from convex and nonconvex bodies is studied using an iterative algorithm. The key point of the method is a selfadjoint formulation of the Helmholtz integral equation, which ensures the convergence of the iteration process toward the true solution. For all investigated structures with different surface impedances fast convergence could be observed. The number of surface elements of the scatterer varies from about 6000 to 60 000 and the calculations are performed in the highfrequency range with Helmholtz numbers between 20 and 63. Even for a scatteringstructure with nearly 60 000 boundary elements, all computations could be carried out on a regular personal computer.

A stochastic model for wave localization in onedimensional disordered structures
View Description Hide DescriptionThis paper presents a probabilistic study of the effects of structural irregularity on wave propagation along an infinite 1D chain. A general integral equation method based on Markov chain theory is used to determine the phase probability density function (pdf) at the scatterers distributed irregularly along the chain. The scatterers could be atoms in a onedimensional crystal, or ribs on a flat plate or membrane. The integral equation derived for the phase pdf is simplified considerably when the scatterers are distributed completely randomly or quasiperiodically. In these cases, the integral equations may be asymptotically solved for the phase density functions in the limit of weak or strong scattering; the localization factors are then obtained. The present approach is quite general and is directly applicable to any disordered onedimensional system consisting of identical scatterers that are arranged according to a probability distribution function. The validity of the present asymptotic solutions is examined and verified by comparing against the existing analytical solutions for simple atomic or mechanical disordered systems.

Torsional waves in lossy cylinders
View Description Hide DescriptionThe pure shear problem is one of relative mathematical simplicity and includes the essential physics common to more complicated cases, where multiple and coupled deformations occur. In this sense, the analysis of torsional waves serves as a pilot problem for investigating the influence of anisotropy and/or anelasticity on solution behavior. We obtain the kinematic and dynamic properties of torsional axially symmetric harmonic waves propagating in an infinitely long circular cylinder. The medium is transversely isotropic and dissipative, with its symmetry axis coincident with the axial axis of the cylinder. For an elastic cylinder each mode has a cutoff frequency and below that frequency there is no propagation. For tubes made of quartz and aluminum Lucite, we found that the existence of the cutoff frequencies depend on the degree of anisotropicattenuation, i.e., if the axial quality factor is greater than the transverse quality factor, the modes propagate at all frequencies. In contrast to the elastic case, the Poynting vector and the energy velocity have a component along the radial direction, whose values depend on the transverse attenuation. The presence of intrinsic attenuation confines the energy near the (elastic) cutoff frequencies while the radial distribution of the energy is governed by the geometrical features of the cylinder.

On the velocities of localized vibration modes in immersed solid wedges
View Description Hide DescriptionThe approximate theory of localized elastic waves in immersed solid wedges earlier developed for wedges with small values of the apex angle [V. V. Krylov, Proc. IEEE Ultrason. Symp., Cat. #94, CHO 793–796 (1994)] has predicted that the effect of water loading results in velocity decrease for wedge modes travelling in the subsonic regime of wave propagation. The results of this theory, in particular the absolute values of wedge wave velocity calculated for slender Plexiglas wedges, agree well with the corresponding experiments. The present study demonstrates that for relative values of wedge wave velocity, as compared with those for wedges in vacuum, this theory provides good quantitative agreement with the experiments on Plexiglas samples also for large values of the apex angle. In addition to this, a generalization of the theory is undertaken to describe the effect of heavier wedge material and a supersonic regime of wave propagation. The corresponding results show good agreement with the existing velocity measurements in immersed brass wedges.

Temporal deconvolution of lasergenerated longitudinal acoustic waves for optical characterization and precise longitudinal acoustic velocity evaluation
View Description Hide DescriptionThe laser thermoelastic generation of ultrasound is a promising technique with many potential applications, but it is also a complicated process with many physical phenomena involved. Contrary to a conventional piezoelectric transducer generation, which is a surface phenomenon, a laser generation can activate acoustic sources within the material by optical penetration of the excitation wavelength, resulting in asynchronous wave arrivals at a given point. More generally, in the ideal case of a nondispersive isotropic material, the laserultrasonics displacement signals result from temporal convolutions between optical penetration, laser pulse duration, and laser spot extension effects. In this paper, a deconvolution technique is presented that extracts the laser pulse duration contribution from the experimental displacement signals. This deconvolution scheme applied to onedimensional experiments, in which the laser excitation is spread over a sufficiently large area on the front side of the sample, allows the measurement of the optical absorption coefficient of the material at the excitation wavelength and also a precise evaluation of its longitudinal acoustic velocity.
