Index of content:
Volume 91, Issue 2, February 1992

Single‐scattering approximations for coefficients in Biot’s equations of poroelasticity
View Description Hide DescriptionThree single‐scattering approximations for coefficients in Biot’s equations of poroelasticity are considered: the average T‐matrix approximation (ATA), the coherent potential approximation (CPA), and the differential effective medium (DEM). The scattering coefficients used here are exact results obtained previously for scattering from a spherical inclusion of one Biot material imbedded in another otherwise homogeneous Biot material. The CPA has been shown previously to guarantee that, if the coefficients for the scattering materials satisfy Gassmann’s equation, then the effective coefficients for the composite medium satisfy Brown and Korringa’s generalization of Gassmann’s equation. A collection of similar results is obtained here showing that the coefficients derived from ATA, CPA, or DEM all satisfy the required conditions for consistency. It is also shown that Gassmann’s equation will result from any of these single‐scattering approximations if the collection of scatterers includes only spheres of fluid and of a single type of elastic solid.

Analysis of three‐dimensional acoustic scattering from doubly periodic structures using a source model
View Description Hide DescriptionA novel solution is presented for the problem of three‐dimensional acoustic scattering of a time‐harmonic plane wave from doubly periodic structures. The general problem is first reduced to a consideration of the fields over a suitably defined unit cell. Sets of fictitious doubly periodic and properly modulated patch sources are used to simulate the fields in the homogeneous regions crossed by the unit cell boundaries. Sets of fictitious point sources are used to simulate the fields inside the regions completely enclosed within the unit cell. The complex amplitudes of the fictitious sources are adjusted to satisfy the boundary conditions at a selected set of points on the boundaries between the regions. The suggested solution procedure is simple to implement and is applicable to doubly periodic structures composed of homogeneous regions of arbitrary shape. Structures comprising acoustically rigid and soft boundaries can also be handled by the procedure. The method has been tested for accuracy by studying the cases of scattering from arrays of spherical scatterers and from doubly periodic sinusoidal surfaces.

Acoustical resonances of solid elastic cylinders: Parametric study and introduction to the inverse problem
View Description Hide DescriptionIn underwater acoustics, target recognition is of major importance. In order to reach such a goal, one must solve the inverse problem to deduce from the received echoes the signature of the target. A rigorous method would require an inversion of the exact direct problem which is, at the present time, not feasible because of the complexity of the analytical expressions involved even for an isotropic, homogeneous, elastic cylinder. After trying different approximate alternative methods of resolution of such problems in this lab, a new method is presented in this paper. Its basis is a parametric study of resonance frequencies and widths, and some simple approximate expressions deduced from this work are used to solve the inverse problem. This study leads to a new classification of most resonances connected with the polarization of the corresponding waves. The frequencies of such resonances are given by very simple equations that can be very easily inverted. After carrying out this procedure, this model is applied to the characterization of the mechanical properties of a cylinder with a given radius a (longitudinal wave velocity c _{ L }, shear wave velocity c _{ T }, density ρ_{2} ). The first application concerns simulated resonance spectra and the second application concerns real experimental measurements. The results obtained are very satisfactory.

Inversion of acoustic material signature of layered solids
View Description Hide DescriptionThe simplex inversion technique is applied to the acoustic material signature (AMS) of layered solids to obtain thickness and material properties of layers. It is shown in this paper that by careful application of the simplex technique, it is possible to extract much information about a multilayered specimen from its AMS. A brief review of different inversion techniques is also given in the paper from which the relative advantages of using the simplex alogrithm over other techniques are derived.

Acoustic scattering from an immersed plane multilayer: Application to the inverse problem
View Description Hide DescriptionThe authors deal with acoustic scattering from a plane multilayered structure. This structure is composed of a first plastic elastic layer, a thin water layer, and a second aluminumelastic layer. A pulse excitation is used to obtain the scattered spectra and the resonance spectra at normal and oblique incidence. These spectra give information about the structure. The spectra of resonances due to the guided waves provide information about the resonant character of the two elastic solids. The farther the guided waves propagate in a layer, the more resonant the layer is. These guided waves are Lamb waves of the solid layers. The backscattered signal is formed by a series of echoes which arise from the different layers of the structure. With simple experiments, involving in particular a temporal filtering of the reflected signal, it is shown that it is possible to obtain three of the four parameters which characterize the solid layers: the phase velocities of the longitudinal waves, thicknesses, and densities. Measurements of absolute reflection coefficients are necessary for the determination of densities. The phase velocities of the shear waves in the two solid layers are obtained by studying the propagation of the guided waves.

A formulation of multiple scattering by many bounded obstacles using a multicentered, T supermatrix
View Description Hide DescriptionThe acoustic scattering from many interacting, bounded, three‐dimensional obstacles has been treated by several authors [see, for example, V. Twersky, J. Math. Phys. 8, 589 (1967) or B. Peterson and S. Ström, J. Acoust. Soc. Am. 5 6, 771 (1974)]. In particular, Peterson and Ström extended the single‐obstacle, transition (T) matrix formalism to several obstacles (including all orders of multiple scattering) by using the translation properties of the spherical basis functions to translate the multiply scattered fields of each obstacle to a common origin. Later, their formalism was extended to treat elastic wave scattering by using spherical vector basis functions [A. Boström, J. Acoust. Soc. Am. 6 7, 399 (1980)]. However, for numerical results, the present state of development is cumbersome to apply to more than two obstacles and convergence of the rescattering matrices is sensitive to obstacle separation. In this paper, a multicentered, T‐matrix formalism for acoustic and elastic wave scattering is given, based on field expansions centered on each obstacle. Like previous approaches, it also incorporates the single‐obstacle T matrix and sums the multiple scattering series exactly. The result is simpler and less numerically sensitive to the separation of the obstacles. The extension of the formalism to treat the scattering from infinite systems that periodically repeat an arbitrary group of obstacles is also given. Numerical calculations based on the exact formalism for scattering from a linear array of as many as ten spherical obstacles are presented. The validity of approximating the scattered field from larger finite arrays by the field scattered from finite sections of an infinite array is considered.

Numerical solution of the direct scattering problem through the transformed acoustical wave equation
View Description Hide DescriptionA transformation is applied to the acoustical wave equation to obtain a new equivalent form that does not contain gradients of the pressure. A new technique, based on the spectral method, is developed for the numerical solution of the direct time domain scattering problem. Modeling techniques for obtaining accurate solutions are discussed and numerical examples are presented.

Computation of the far‐field time domain solution of the scattering problem
View Description Hide DescriptionThis paper presents a new technique for calculating the time domain (transient) far‐field scattered pressure. The scattering problem is divided in two steps; the first step evaluates the field distribution inside the scatterer, and the second step generates the far‐field scattered pressure by 3‐D Radon transform of these data for each time step and summing over time. The algorithm results in considerable saving in CPU time and memory by simplifying the calculation along the path from scatterer to receiver. This technique can also be used in two dimensions.

Acoustic scattering from arbitrarily shaped multiple bodies in half‐space: Method of moments solution
View Description Hide DescriptionIn this work, the method of moments solution is applied to calculate the scattered or radiated fields from arbitrarily shaped, multiple rigid bodies located in an infinite half‐space. Both cases, that is, when the bodies are away from the infinite plane and when the bodies are situated in contact with the infinite plane, have been treated. It is noted that no special treatment was required when the bodies are in actual contact with the infinite plane. The numerical technique presented in this work is simple, efficient, and accurate. Numerical results are presented for certain canonical shapes and compared with alternate formulations.

Acoustic wave propagation in a cylindrical borehole with fractures
View Description Hide DescriptionThe problem of acoustic wave propagation in a cylindrical borehole possessing a finite number of transverse discontinuities is studied. The field behavior is modeled through Green’s function techniques and an integral equation is formulated to solve for the acoustic field everywhere within the structure. Asymptotic forms are investigated to speed the numerical convergence of our solution. To solve the integral equation both the method of moments and a low‐frequency approximation are employed. The reflection coefficient in the time and frequency domains is studied. After presenting solutions for the one and two fracture case, the analysis is generalized for many fractures.

Scattering from a finite cylindrical shell
View Description Hide DescriptionAn asymptotic solution of scattering from a finite slender cylindrical shell is derived from the two‐variable expansions method by modifying the usual concept of secular terms. An analytical expression for the scattered far field is obtained from a superposition of sectorial spherical harmonics that have been used to describe scattering from slender bodies. The contribution of the scattered resonant pressure due to the elasticity of the material is analyzed and some numerical computations are made in monostatic and bistatic cases for rigid and elastic shells to evaluate the importance of the radiation of the resonant modes in the far field.

Elastic wave scattering by an isotropic noncentrosymmetric sphere
View Description Hide DescriptionIn an isotropic, noncentrosymmetric elastic solid, the displacement field u has to be supplemented by the independent microrotation field φ. In this mirror‐asymmetric (or chiral) medium, six different wave numbers are possible: Of these, two represent longitudinal fields, and the remaining four are (circularly) handed. This paper is concerned with the scattering of elastic waves by a chiral sphere, and the effect of the chiral parameter on its scattering response.

Propagation of sound and the assignment of shape factors in model porous materials having simple pore geometries
View Description Hide DescriptionThe acoustical properties of a class of simple porous materials have been studied experimentally and theoretically. Rigid‐frame materials containing air‐filled pores of uniform cross section were investigated. Two modelporous materials were constructed, one with pores of rectangular cross section (0.0146×0.0172 cm) and one with pores of triangular cross section (0.037‐cm sides). The characteristic impedance and propagation constant were measured for frequencies between 50 and 4500 Hz and good agreement with exact theoretical predictions was obtained. The exact theoretical expressions for specific pore shapes (e.g., slitlike, square, and triangular) can be used to investigate the assignment of shape factors by various general but approximate theoretical models. It is demonstrated that the model introduced by Attenborough requires a shape factor that is frequency dependent. A theoretical model, appropriate for materials containing pores of uniform cross section, that correctly treats the low‐ and high‐frequency behavior, is presented. Given subsidiary measurements of flow resistivity, porosity, and tortuosity, a s i n g l e shape factor leads to very good agreement with the exact solutions for all frequencies.

Evaluating acoustic absorption coefficients by comparative analysis—Theory part
View Description Hide DescriptionAn analytical model has been developed for use in evaluating normal incidence acoustic absorption coefficients of materials. The model describes an acoustic impedance tube having a closed and partially absorptive boundary, corresponding to an absorber specimen mounted within an endcap affixed to the tube. The other boundary contains an acoustic excitation. Additionally, the model incorporates internal acoustic dissipation effects of the tube, which can be significant. Acoustical frequency response functions are computed based on the amounts of the two dissipation mechanisms. This composite analytical model is to be used for comparison to experimentally measuredacoustic pressure magnitudes so as to evaluate the acoustic absorption coefficient.

Evaluating acoustic absorption coefficients by comparative analysis—Experimental part
View Description Hide DescriptionA new method has been developed for evaluating normal incidence acoustic absorption coefficients of a material by making direct comparisons between experimental and analytical acoustic pressure magnitudes. The method uses an analytical model for an acoustic impedance tube having a closed and partially absorptive boundary, corresponding to an absorber specimen mounted within an endcap affixed to the tube. Additionally, the model incorporates internal acoustic dissipation effects associated with the apparatus. From the composite model, acoustical frequency response functions are computed based on the amounts of the two dissipation mechanisms. An algorithm is implemented whereby the response magnitudes measured at tube resonance frequencies are compared to corresponding analytical values so as to evaluate the absorption levels needed for a match. Since the absorption coefficients are evaluated for resonant modes only, a B‐Spline curve is used to interpolate the results. The normal incidence acoustic absorption coefficient is then specified at all frequencies within a measurement spectrum. Experimental results are presented for a foam acoustic material and a comparison is made to the results from the standing wave method (ASTM C 384‐90). The comparison shows a good correlation between the two.

Nonlinear pressure fields due to focused circular apertures
View Description Hide DescriptionThe use of high amplitude focused ultrasound fields is widespread in medical diagnosis and therapy but there has been relatively little work published that compares experimental measurements with appropriate theory for such systems. In this paper, comparisons are made between the measured continuous‐wave pressure field of a focused circular aperture operating at 2.25 MHz and a numerical solution of the nonlinear parabolic wave equation. The measurements were made in water using a 38‐mm diam plane circular transducer as the acoustic source with perspex lenses providing focusing, focal lengths of 440, 216, and 142 mm were examined. Results are presented for the amplitudes of the fundamental, second, and third harmonics along and across the acoustic axis of the source. In general, the agreement between experiment and theory is good.

Weak shock interaction with a free‐slip interface at low grazing angles
View Description Hide DescriptionTheoretical/numerical investigations with the nonlinear progressive wave equation (NPE) model predict qualitative differences between behavior of a planar weak shock and a linear acousticwave of similar time signature incident downward at small grazing angle onto a free‐slip interface between two fluids. For the linear case, the model agrees well with an appropriate analytic benchmark solution. For the nonlinear cases considered, initial conditions evolve toward (a) supercritical reflection (steady wave with no Mach stem development) or (b) subcritical reflection (self‐similar behavior with Mach stem development). When the horizontal phase speed c _{0} of the incident wave barely exceeds the linear sound speed c _{ L } of the lower medium, an intermediate state can be obtained in which the incident shock discontinuity terminates on the interface;wave energy below the interface runs ahead and reforms a deep penetrating shock ahead of and below the interface disturbance. When c _{0} is below c _{ L }, the shock discontinuity terminates at the interface and does not reform in the lower medium.

Study of a thermoacoustic prime mover below onset of self‐oscillation
View Description Hide DescriptionThe frequency response of a thermoacoustic prime mover has been measured as a function of the mean gas pressure and temperature gradient across the prime mover stack. The quality factor Q and resonance frequency can be determined from the response. As the temperature gradient is increased, the Q increases, indicating a decrease in attenuation across the stack. At sufficiently large temperature differences (∼300 K), the resonator goes into self‐oscillation, indicating negative attenuation. Measurements are reported for helium and argon at pressures ranging from 170–500 kPa and temperature gradients ranging from zero to that required for onset of self‐oscillation. The results are explained in terms of a counterpropagating, plane‐wave analysis, based on techniques commonly used in porous media investigations. In general, the predictions of the analysis are in good agreement with experiment. The predictions of Q and the change in resonance frequency with mean gas pressure are within approximately 5% and 0.4% of measured values for the no temperature gradient cases. In the cases where temperature gradients are present, the agreement is quite good for the two highest mean pressures reported (370 and 500 kPa). There are some noticeable discrepancies at the lowest pressure (170 kPa). The reasons for these discrepancies are unknown.

A scattering model for nondifferentiable periodic surface roughness
View Description Hide DescriptionIce dendrites that form when seawater freezes produce a surface with periodic roughness. This surface may be modeled by joining a concave upward semiellipse to a concave downward semiellipse and replicating this fundamental surface element to form a surface that is everywhere continuous, but is not differentiable at the points where the two ellipses are joined (i.e., the derivative is infinite). This surface is regarded as realization of the class of periodic rough surfaces that are not differentiable at a finite number of points within a period and includes surfaces that are piecewise and nonpiecewise smooth. A model is described to calculate plane‐wave scattering from one‐dimensional realizations of these types of surfaces. To calculate scattering from these types of surfaces for the physical parameters of interest, the integral equation formalism of Holford [J. Acoust. Soc. Am. 7 0, 1116–1128 (1980)] is used and the surface roughness profile is approximated by truncating its Fourier series representation. This approximation provides a continuous and differentiable representation of the surface roughness profile and is necessary to guarantee a convergent solution for the surface field that is obtained as a solution to a Fredholm integral equation of the second kind. The scattering theory of DeSanto [Radio Sci. 1 6, 1315–1326 (1981)] that uses Green’s theorem and a modified physical optics representation of the surface field is used to determine the effect of the approximation on the surface field and scattering amplitudes in the domain in which the theory is numerically stable (This theory is not numerically stable for the physical parameters of interest.)
Since this technique may be applied to any nondifferentiable surface that may be represented by a Fourier series, several nondifferentiable periodic surfaces are constructed. Numerical calculations show that the amplitudes of the plane‐wave modes scattered from a continuous and nondifferentiable surface and from its truncated Fourier series representation are essentially identical over a wide range of incident grazing angles, acoustic frequencies, surface roughness, and surface types. Additionally, it is shown that when an integral equation is used to calculate scattering from a truncated Fourier series representation of a continuous and nondifferentiable surface,scattering may be calculated from surfaces considerably rougher than those considered previously.

Anderson localization of one‐dimensional wave propagation on a fluid‐loaded plate
View Description Hide DescriptionThe propagation of flexural waves along a fluid‐loaded plate with an irregular array of line attachments is examined. The system is shown to be equivalent to a one‐dimensional chain with a nearest neighbor coupling arising from the fluid‐loaded wave, and a long‐range coupling due to bulk waves in the fluid. The magnitude of the long‐range coupling is shown to be small compared to the nearest neighbor coupling for many applications. The irregularity will thus produce Anderson localization, at least for moderate distances. The equations of motion are numerically solved for two cases: (1) a sparse array for which the long‐range coupling is expected to be negligible, and (2) a dense array for which the long‐range coupling is fairly small, but not negligible. Unambiguous Anderson localizationeffects were observed in both cases.