Index of content:
Volume 79, Issue 5, May 1986

Application of a Neumann‐series method to two problems in acoustic radiation theory that are formulated in terms of Green’s functions
View Description Hide DescriptionTwo problems in radiation theory are analyzed. In the first problem, two acoustic sources in a fluid interact via their radiated pressure field. Each source may have an arbitrary shape and an arbitrary nonuniform surface vibration. The two sources may be placed in the fluid in an arbitrary fashion, so that there is no preferred orientation of one with respect to the other. A Neumann‐series method is used to simplify that term in the integral equation expressing the pressure field which describes the interactions taking place between the sources. By use of this method, the pressure field can be expressed in terms of the Green’s functions of the individual sources, rather than in terms of an abstract a d h o cGreen’s function defined on the disjoint surface that comprises the surfaces of both sources. This field formulation is used to express the self and mutual radiation impedances of the sources. A physical description of the interactions taking place between the sources is given. This description is suggested by the form of the equation describing the pressure field that results when the Neumann‐series method is used. In the second problem, the exterior Neumann and Dirichlet Green’s functions are analytically determined for a radiating acoustic source with an arbitrary shape and with an arbitrary nonuniform surface vibration. These Green’s functions are constructed by using a Neumann‐series method to simplify that term in the integral equation expressing the pressure field which describes the interaction of the source with its own radiation. In principle, the exterior Green’s functions, which are expressed in terms of the geometry of the source and the free‐space Green’s function for the Helmholtz equation, are calculable for any source by performing a series of elementary mathematical operations.

The null‐field approach in series form—The direct and inverse problems
View Description Hide DescriptionThe direct and inverse scattering problems in two‐dimensional acoustics at a fixed frequency are considered. It is shown how the null‐field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three‐dimensional and vector problems.

Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients
View Description Hide DescriptionThe accuracy of a new method for measuringultrasonicbackscatter coefficients was tested, using narrow‐band pulses and well‐defined media having scatterers randomly distributed in space. Experimentally determined values agree very well with theoretical values for wide ranges of experimental parameters, these ranges being applicable in measurements made on human soft tissues. An important outcome is that the method yields accurate results for scattering media positioned anywhere from the nearfield through the farfield of the nonfocused transducers employed. In addition, backscatter coefficients can be determined for a broad range of gate durations.

Scattering of elastic waves by a smooth rigid movable inclusion
View Description Hide DescriptionThe null‐field approach is used for treating the problem of scattering by a smooth rigid movable inclusion of finite mass. The solution is obtained both directly and as a limit of the solution to the problem of scattering by a smooth elastic obstacle. The infinite mass limit for the smooth rigid movable inclusion is also investigated, and it is shown that while the immovable smooth spherical rigid body violates Rayleigh’s law, the movable inclusion does not. Numerical examples for spheres and spheroids are presented.

Spherical wave scattering by an elastic solid cylinder of infinite length
View Description Hide DescriptionThe problem of the scattering of a spherical acoustic wave by an elastic (lossless) solid cylinder of infinite length immersed in an infinite, inviscid fluid medium is investigated theoretically. The solution is obtained by imposing appropriate boundary conditions (involving stress and normal displacement) at the fluid–solid interface on the relevant differential equations. In order to be able to solve the differential equations, an approximation is made that is equivalent to assuming that the most significant additional contributions to the scatteredwave appearing in the fluid (compared with the contributions to the scatteredwave arising in the incident plane‐wave case) are those associated with the waves propagating along the z axis within the solid. Numerical results are presented for a 1000‐Hz wave incident on a 2‐cm‐diam metallic cylinder in water. This is a low k a calculation (where k is the wavenumber in the fluid and a is the radius of the scatterer). Several different metals are considered. The results are compared to those obtained for an incident plane wave. The scattered pressure wave resulting from an incident spherical wave is shown to differ in amplitude by as much as 20 dB from that resulting from an incident plane wave. (This difference is nontrivial, i.e., it does not result from a minor repositioning of nulls in the scattering pattern.)

Numerical study of material properties of submerged elastic objects using resonance response
View Description Hide DescriptionScattering from many fluid‐loaded elastic solid objects yields echo responses that are characteristic of rigid scatterers, with the exception of narrow resonance regions. These resonances are related to Rayleigh‐type surface disturbances and thus are determined by the material properties of the scatterer. The purpose of this study is to predict resonances for several representative elasticmaterials to ascertain material properties that can be associated with resonance phenomena. This study was carried out using the T‐matrix technique to determine backscattered form functions for spheroids composed of brass,aluminum,nickel, steel, molybdenum, and tungsten carbide. Either low density or low shear velocities have been determined to yield results that do not correspond to simple rigid backgrounds, though for all cases shear velocity alone practically determines the resonance location. A Rayleigh surface standing wave argument was found to be useful in deriving simple expressions that determine resonance locations as a function of shear velocity alone. These expressions compare well with numerical calculations.

Directional dependence of ultrasonic propagation in textured polycrystals
View Description Hide DescriptionThe theory of ultrasonicwave propagation in textured polycrystals presented previously [S. Hirsekorn, J. Acoust. Soc. Am. 7 7, 832–843 (1985)] for propagation and polarization parallel to symmetry axes of texture is generalized to calculate the directional‐dependent scattering coefficients and phase and group velocities of plane waves with arbitrary propagation and polarization direction as a function of frequency. The analytical calculations are carried out for cubic polycrystals with orthorhombic texture symmetry. Numerical evaluation is done for an austenitic weld metal. The results are compared to those of the frequency‐independent description of ultrasonic propagation.

Inversion of the two‐dimensional S H elastic wave equation for the density and shear modulus
View Description Hide DescriptionIn this paper, a method for simultaneously reconstructing the two‐dimensional shear modulus and density configurations from S H data is presented. The method requires a complete suite of surface receivers for each of t w o source locations. This is in contrast to previous multiparameter, multidimensional inversion methods, which require complete source coverage for each one of a continuous set of surface receiver locations. The method presented here is a multiparameter extension of the recent single source, single parameter, multidimensional acoustic velocity inversion procedure of Esmersoy, Oristaglio, and Levy [J. Acoust. Soc. Am. 7 8, 1052–1057 (1985)]. A simple change of variables allows the multiparameter procedure to be directly applied to the acoustic inversion problem, i.e., to identify the multidimensional acoustic velocity and density configurations.

Stopbands in a corrugated parallel plate waveguide
View Description Hide DescriptionThe stopband structure of a two‐dimensional periodically corrugated waveguide is investigated by means of the null‐field method. In the numerical examples, the walls are assumed to be hardwalled with symmetric, antisymmetric, or one‐sided sinusoidal corrugations. The interaction pattern and the stopband widths are determined for the lowest modes. The resonances are found to be of the stopband type, and it appears that geometrical symmetries of the boundaries suppress well‐defined sets of resonances.

Three‐dimensional acoustic analysis of circular expansion chambers with a side inlet and a side outlet
View Description Hide DescriptionThe effects of higher‐order acoustic modes upon the performance of a circular expansion chamber with its inlet and outlet located on the cylinder wall are investigated. The sound distribution in the expansion chamber is obtained, theoretically, considering the influences of the diffractedsound pressuregenerated by the presence of end plates in the chamber. A theoretical method for the estimation of the transmission loss is suggested by using the derived four pole parameters, assuming that the chamber can be modeled as a piston‐driven circular rigid cylinder. A series of experimental observations are taken for verification, and it is found that they are in good agreement with the theoretical results. The influences of those modes are studied for a variety of chamber lengths and combined inlet/outlet locations. As a result of this study, it is concluded that the diffractedsound pressure affects the performance of the expansion chamber considerably in the range above the cutoff frequency of the n=1, m=0 asymmetric mode, and that the plane‐wave theory can hardly be adopted even in the low‐frequency range where the fundamental mode prevails.

Frequency division method for AE source characterization
View Description Hide DescriptionThis paper describes a new algorithm based on a frequency‐division method by which a solution to the inverse source problem of acoustic emission can be obtained. The algorithm is applicable to cases in which the acoustic emission signal can be expressed as a bilinear equation involving weighted Green’s functions or its spatial derivatives, convolved with a common source–time function. Through division of the detected signals in the frequency domain, the common source–time function is eliminated and the ratios of the weighting constants can be recovered by solving a set of linear equations. Using these recovered ratios, the source–time function can be found from the signal at any receiver point. Application of this algorithm is made to process the signals simulating the formation of a microcrack in an elastic plate modeled with a second‐order moment tensor. Dynamic information about the source is recovered by processing the signals at two receiver points. If four receivers are used, the second‐order tensor can be recovered at each frequency of the recorded signals. Additional information about the source can be extracted by diagonalizing and rotating the recovered moment tensor. The effect of noise on the inversion scheme is also considered.

Numerical techniques for three‐dimensional steady‐state fluid–structure interaction
View Description Hide DescriptionThe application of a coupled finite element–boundary integral approach to the analysis of vibrating elastic, arbitrary‐shaped, three‐dimensional structures in an infinite media is presented. A feature of the methodologies implemented is the utilization of the isoparametric concept from finite elementtheory. The same shape functions are used both to model the surface of the structure and to interpolate the acoustic variables over the radiating surface. The boundary integral formulation used for the exterior acoustic field is a formulation that ensures uniqueness and existence of solutions for the entire frequency range. An assessment of the accuracy was made after comparing the solutions attained for a vibrating submerged sphere for various excitation loads and frequencies with the exact solution.

Diffraction of an explosive transient
View Description Hide DescriptionBlast walls and barriers are often proposed for reducing noise from explosives or artillery fire. The calculation of blast noise reduction by three‐dimensional finite difference flow field codes is expensive and computer time intensive. For many cases, the source strength at the barrier is neither very strong nor very weak (acoustical). In this paper, a combined theoretical and experimental model study to investigate the range of validity of linear transient diffraction theory is described. Also, this paper further investigates the use of a finite wave propagation program to extend the utility of the linear calculation. This study demonstrates that the linear theory, with an accurate input waveform, can predict the insertion loss to within 1.5 dB for peak sound‐pressure levels less than 162 dB at the barrier. For the source levels greater than 162 dB, the prediction significantly underestimates the insertion loss.

Equilibrium shapes of acoustically levitated drops
View Description Hide DescriptionThe quantitative determination of the shape of liquid drops levitated in an ultrasonic standing wave has provided experimental data on the radiation pressure‐induced deformations of freely suspended liquids. Within the limits of small deviations from the spherical shape and small drop diameter relative to the acoustic wavelength, an existing approximate theory yields a good agreement with experimental evidence. The data were obtained for millimeter and submillimeter drops levitated in air under 1 g, where g is the sea level gravitational acceleration.

Air absorption affected by Doppler shift
View Description Hide DescriptionIt is the purpose of this paper to present the effect of air absorption on the intensity of sound produced by a moving point source, particularly the role of the Doppler effect in the absorption. It has been assumed that a nondirectional point source is moving along a straight line, with subsonic speed, in a homogeneous atmosphere. The investigation is based on a geometrical approach.

The interaction of airborne sound with the porous ground: The theoretical formulation
View Description Hide DescriptionThe surface of the ground is modeled as that of an air‐filled poroelastic soil layer of known thickness overlying a semi‐infinite nonporous elastic substrate. Using a modified form of the Biot–Stoll differential equations for wave propagation in fluid‐saturated porous media, propagation constants for the two possible dilatational waves and the shear wave in the poroelastic layer are determined. The dilatational waves are identified as a fast wave, moving predominately in the solid frame, and a slow wave, moving predominately in the pore air. The elastic moduli in the substrate are assumed to be those of the solid grains of which the poroelastic soil layer is composed. Intergranular friction in the soil and substrate is assumed to be negligible. Boundary conditions at the air–soil interface and at the porous soil–substrate interface are applied to determine, numerically, the displacement amplitudes of the allowed wave motions. From the incident and reflected amplitudes at the air–soil interface, the normalized ground surface impedance is calculated as a function of angle of incidence and of frequency. In this paper, the response of the pore fluid and frame to airborne acoustic waves is considered and those ideas will be pursued in a later publication. The predicted impedance at normal incidence is compared with measurements of the impedance of a sandy soil for which measurements of the various parameters required by the theory are also available. The predictions of impedance are found to be in tolerable agreement both with measured data and with predictions of a simpler model of the surface as that of a rigid porous semi‐infinite homogeneous medium. Calculations of the surface impedance as a function of angle of incidence suggest that the porous medium is locally reacting.

The acoustic transfer function at the surface of a layered poroelastic soil
View Description Hide DescriptionA model for the response of a poroelastic layered soil to an incident plane wave developed in a previous paper [Sabatier e t a l., J. Acoust. Soc. Am. 7 9, 1345–1352 (1986)] is used to predict the complex sound pressure within the upper poroelastic layer. The predictions both of phase velocity and attenuation of the slow wave associated primarily with propagation in the pore fluid are compared with measurements made with a specially constructed probe microphone. The agreement between theory and experiment is good. The predictions for a layered poroelastic soilmodel are compared numerically with those of a semi‐infinite rigid porous soilmodel and are found to differ only at frequencies higher than 1000 Hz. Analysis of the sensitivity to the theoretically predicted propagation constants in the poroelastic soil to the assumed value for the bulk rigidity modulus of the soil predicts that over the known range of rigidity moduli for soils it is possible to obtain a switchover between fast and slow propagation modes. This switchover occurs at the lower end of the possible range of values of the shear modulus. It is suggested that probe microphone measurements in air‐filled soils offer a way of measuring flow resistivity and of deducing the structural parameters required for application of the Biot–Stoll model to water‐saturated sediments.

Speed of sound in standard air
View Description Hide DescriptionThis paper describes the calculation of a new value for sound speed (c _{0}=331.29 m s^{−} ^{1}) in standard dry air at 0 °C and at a barometric pressure of 101.325 kPa. The maximum uncertainty is estimated to be approximately 200 ppm. The theory of the calculation is based on the equation of state, and includes the knowledge of γ/M which is derived from published theoretical and experimental thermodynamic data on the constituents of the standard atmosphere. Investigations which led to the general acceptance of the previous sound speed are examined, and there is good evidence to conclude that, in previous sound speed assessments, the maximum possible uncertainties were sufficient to encompass the above new sound speed. The variation of sound speed with carbon dioxide concentration and temperature is also discussed.

The transient wave fields in the vicinity of the cuspoid caustics
View Description Hide DescriptionA general formalism is presented for the study of the transient wave fields in the vicinity of caustics of arbitrary complexity. The caustics are classified using catastrophe theory. In the present study, a general introduction to the subject is presented and explicit results (numerical and analytical) for the cuspoid family of catastrophes are given. Numerical results are presented for the fold, cusp, and swallowtail catastrophes which show the smooth variation of the acoustic field as the control parameters are independently varied. These results vividly demonstrate the phenomenon of ray focusing, showing the number of rays which converge at each catastrophe, the resulting phase shifts they undergo, as well as their decay in shadow regions. Analytical results are given which describe the time‐dependent acoustic field on and at large distances away from each of the cuspoid catastrophes. Four sets of numbers are introduced which describe the behavior of the wave field in the vicinity of each catastrophe. These are: (1) an exponent which describes the temporal decay of the acoustic field on the most singular point of each catastrophe; (2) a number which describes the asymmetry of the time‐dependent acoustic field on the most singular point of each catastrophe; (3) an exponent which describes the time separation as a function of control parameter between individual ray arrivals as each of the control parameters are independently varied; and (4) an exponent which describes the asymptotic decay as a function of control parameter of each constituent ray arrival. Sets (1) and (3) are shown to be simply related to the s i n g u l a r i t y and f r i n g e indices previously introduced by Arnold [V. I. Arnold, Usp. Mat. Nauk. 3 0(5), 1–75 (1975)] and Berry [M. V. Berry, J. Phys. A 1 0, 2061–2081 (1977)], respectively. It is argued that many features of the frequency domain representation of the acoustic field in the vicinity of the catastrophes are most easily understood by examining the time domain representation.

The transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic caustics
View Description Hide DescriptionThe caustics of high‐frequency wave propagation may be classified using catastrophe theory. In this paper, we examine the transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic catastrophes. Analytical results are presented which describe the time‐dependent wave field due to an impulsive source on and at large distances in each control direction away from the most singular point of each of these three catastrophes. Numerical results are presented which show the smooth variation of these transient wave fields as each control parameter is independently varied. These results are compared to previously derived results on the transient wave fields in the vicinity of the cuspoid catastrophes (the cuspoid and umbilic catastrophes have coranks 1 and 2, respectively). It is found that the transient wave fields in the vicinity of the cuspoid and umbilic catastrophes differ with regard to the temporal structure of the wave field on the most singular point of each catastrophe, the manner in which these temporal structures unfold, and the phase shifts which individual rays undergo as a result of touching one of the catastrophes.