Index of content:
Volume 73, Issue 1, January 1983

RST analysis of monostatic and bistatic acoustic echoes from an elastic sphere
View Description Hide DescriptionWe readdress the basic problem of the scattering of acoustic waves by an elastic sphere, now under the dissecting knife of the ResonanceScattering Theory (RST), with the purpose of illustrating the power of the method, and the versatile options and information it offers. These options include ease of understanding, novel physical interpretation of the phenomenon, and striking calculational simplicity. The principal findings presented here include: (a) the actual modal resonances, quantitatively separated from the corresponding modal backgrounds in the frequency (the ‘‘acoustic spectrogram’’—already a target‐identification tool) domain, a n d in the combined frequency and mode‐order domain (the ‘‘response surface’’). (b) The bistatic plots of the scattering cross section, to illustrate the point that if the cross section is redetermined at certain selected observation angles, the resonance contributions from each individual mode can actually be isolated from those of all other modes. (c) A study of the nulls or dips present in the partial waves (i.e., modes), and in the summed cross section. We show the cause and physical meanings of these dips analytically and computationally, in both instances. (d) A derivation of the analytic conditions predicting the nulls and also the influence of the elasticresonance(SEM) poles in the scattered echoes. These conditions, which emerge from our scattering approach, are shown to be in agreement with early results of Love [A T r e a t i s e o n t h e M a t h e m a t i c a l T h e o r y o f E l a s t i c t y (Dover, New York, 1944)], obtained on a purely vibrational basis. A tungsten carbide sphere is used in all the examples, since this is a favorite target for experimental calibrations. Our future work will underline the intimate connection between his direct approach, and that (leading to the solution) of the inverse scattering problem for sonar target identification.

Scattering by spherical elastic layers: Exact solution and interpretation for a scalar field
View Description Hide DescriptionWe consider an infinite space with a given set of concentric elastic spheres which are solicited by SH waves emitted by an external point source. The exact solution for the wave propagation in every zone of this pattern is determined. It is shown that the ratios of determinants which define the solution of this problem can be expanded in series of which each term can be explicit in terms of reflection and refraction coefficients. This second expression of the exact solution allows one to give an accurate interpretation of all terms to describe the excitation of resonance of a multilayered spherical structure and to facilitate the study of miscellaneous physical phenomena (resonance, diffusion, and interference spectra).

A method for constructing wave velocity and density profiles from the angular dependence of the reflection coefficient
View Description Hide DescriptionIf the acoustic wave velocity and the density of an inhomogeneous medium are dependent on one of the coordinates only, then it is possible to express the logarithmic derivative of the pressure along this coordinate as a nonlinear first‐order differential equation. In the present work a particular finite difference approximation to this equation is found which allows one to obtain the reflection coefficient as a ratio of two polynomials involving the sine of the angle of incidence. In this way one can find, in a very simple way, the unknown coefficients of the two polynomials from the values of the reflection coefficient for different angles, and at the same time one can determine the wavevelocity and the density, from the unique continued fraction expansion of this rational representation for two different frequencies.

Estimation of the area function of human ear canals by sound pressure measurements
View Description Hide DescriptionA method to determine the ‘‘area function’’ of a duct, i.e., the variation of the cross section along the middle axis of the duct is described. It is shown that it is sufficient to measure the sound pressure at three locations in the duct to get knowledge of the area function. The calculated approximation fits the true area function the better the broader the bandwidth of the sound pressure signals is chosen. The bandwidth is bounded by the condition that no higher modes of sound propagation must be excited. In principle, the area function can be determined by solving a linear system of equations and executing a simple iterative algorithm. Some more work however is necessary to stabilize the computation against unavoidable small measuring errors. The calculated area function reproduces the acoustical transformation characteristics with good precision. In spite of this the calculated area function may disagree with the real area function in some regions, especially at the edges. Such errors occur, if the deviations of the area function from constant cross section correspond to frequencies outside the range of measurement. The method was applied to determine the area function of human ear canals and the results are presented.

Acoustic propagation in random layered media
View Description Hide DescriptionWe study how the presence of small scale random layering may affect the interpretation of acoustic scattering data. Our methods include theoretical analysis based on the idea of localization of states, in conjunction with Monte Carlo simulations of particular layered systems suggested by interesting geophysical examples. We show that the random layering can produce dramatic effects by leading to a cutoff frequency for transmission through the system.

Visualization of acoustic scattering by elastic cylinders at low k a
View Description Hide DescriptionTheoretical studies have shown the great importance of circumferential waves in acoustic scattering by infinite elastic cylinders immersed in a fluid. The ‘‘Resonance Isolation and Identification Method’’ (RIIM) has allowed us to separate the resonances produced by circumferential standing waves by recording the free vibrations which are still present for some frequencies after the forced excitation has stopped. In this paper, the problem of acoustic scattering is studied by a stroboscopic Schlieren visualization method at low frequencies (0<k a<30). We attempt to establish that the acoustic beams generated in specific directions, when the cylinder is no longer excited, are actually issuing from the standing wave antinodes.

Precise model measurements versus theoretical prediction of barrier insertion loss in presence of the ground
View Description Hide DescriptionRecent work on barrier performance, aimed at quantifying effects due to interference and scattering by atmospheric turbulence, has also indicated systematic deviations between measured results of diffraction and those predicted by widely used theories embodying known approximations. Modelexperiments indoors (no atmospheric turbulence and a known, controllable ground impedance) show systematic deviations, independent of frequency, up to 12 dB when compared with either first‐order approximations of Macdonald’s diffraction theory or that of Kirchhoff. More exact evaluation of the diffraction integral [e.g., Hadden and Pierce, J. Acoust. Soc. Am. 6 9, 1266–1276 (1981)] or diffraction theory based on layer potential theory [e.g., Daumas, Acustica 4 0, 212–222 (1978)] provide agreement with measured results within ±1.5 dB under all experimental configurations investigated. An empirical correction based solely on such geometrical considerations as distance of source and receiver to the barrier and height of barrier, i.e., on the value of the angle of diffraction at the edge of the barrier, provides similarly good agreement. To obtain precise theoretical predictions of barrier performance under a wide range of configurations it is suggested that one use either calculations based on the recent formulation of Hadden and Pierce or some simple approximate theory plus this empirical correction based on diffraction angle.

Coupling of Lamb waves with the aperture between two elastic sheets
View Description Hide DescriptionThe interaction between ultrasonicLamb waves and the region of rigid contact between two elastic layers is explored analytically. This problem is a model of two sheets welded in this region, and is of great practical importance in a nondestructive evaluation of welds. To uniquely define the dimensions of the region on the basis of the transmitted ultrasonic signal, the experimental conditions are selected in such a manner as to avoid resonance in the region of contact. Our experiments show that there exists a one‐to‐one relationship in the form of T∼d ^{3} between the transmission coefficientsT of the wave through a region of contact having diameter d.

Properties of a periodically stratified acoustic half‐space and its relation to a Biot fluid
View Description Hide DescriptionThe problem of the reflection of acoustic waves from a periodically layered acoustic half‐space is solved exactly at all frequencies. Pass and stop bands and the associated complex slowness surfaces of propagation through the periodic medium are found. In the low‐frequency limit, when the wavelength normal to the layering is much greater than one period, it is shown that there always exists a homogeneous ideal fluid with the same reflection properties as the layered medium. However, the effective acoustic medium that models transmission as well as reflection at low frequencies has a bulk modulusK _{eff} given by 〈K ^{−} ^{1}〉^{−} ^{1} where the brackets denote a volume weighted average, and, for the inertial density appearing in the equations of motion, the effective medium has a t r a n s v e r s e i s o t r o p i c d e n s i t y t e n s o r with ρ_{∥}, the density component parallel to the layering, given by 〈1/ρ〉^{−} ^{1}, and ρ_{⊥}, the density component perpendicular to the layering given by 〈ρ〉. The range of wavespeeds speeds as a function of angle is shown to parallel the range of speeds in a Biot fluid as a function of the Biot coupling parameter. Normal propagation corresponds to full locking of the two phases in a Biot material and parallel propagation corresponds to the fully uncoupled case. The relation between the angle of propagation θ measured from the normal to the layering, and the corresponding Biot coupling parameter α is that α−1 is proportional to cot^{2} θ.

Multiple scattering of sound by correlated monolayers
View Description Hide DescriptionEarlier results for scattering by random uncorrelated planar distributions, and by doubly periodic planar configurations of relatively arbitrary obstacles, are generalized to pair‐correlated nonsymmetrical monolayers. The existing development for parallel cylinders in terms of the Zernike–Prins one‐dimensional pair function p(x), is extended to analogous two‐dimensional distributions specified by p(R) for aligned impenetrable disks. We obtain the average multiple scattered transmitted and reflected waves, and an energy conserving approximation of the differential scattering cross section per unit area. Simplified forms are developed to facilitate determing p by inverting measured data. Closed form low‐frequency results are derived for identical ellipsoids aligned nonsymmetrically to the plane of centers, and the array multipole‐coupling processes are discussed in terms of functions of p and their approximations.

Reflection and scattering of sound by correlated rough surfaces
View Description Hide DescriptionPair correlations and the effects of dense packing are included to extend existing results on reflection and scattering by distributions of protuberances (bosses) on rigid or free base planes. The earlier energy conserving forms for the specularly reflected wave, the surface impedance, and the differential scattering cross section per unit area, are obtained in terms of a transform of the scattering amplitude of an isolated boss. Low‐frequency approximations are developed with emphasis on the roles of the packing density and multipole coupling effects, and explicit results are given for semi‐elliptic cylinders and hemi‐ellipsoids. For lossless bosses, the reflection coefficient has a minimum, and the incoherent scattering a maximum, at the packing density corresponding to maximum fluctuations in the number of bosses per unit area. Multipole coupling effects may be misinterpreted in data inversion programs as changes in boss shape; if such effects are not included, then, e.g., hemispheres may be mistaken for hemiellipsoids broadened along the base plane and shortened along the normal.

Sound pressure distribution about the human head and torso
View Description Hide DescriptionPublished and unpublished data on sound pressure distributions about the human head and torso were collected and analyzed. The results of the analysis are presented graphically. The data include measurements on both humans and manikins in both progressive wave and diffuse fields. The results can be used to aid selection of locations for the microphones of noise dosimeters and hearing aids and design of spectrum shaping circuits for these microphones. An appendix discusses procedures for determining whether it is appropriate to assume a progressive wave or diffuse field when these data are applied to actual situations in the field.

Nonlinearity acoustical parameter and its relation with Rao’s acoustical parameter of liquid state
View Description Hide DescriptionSound‐speedproperties of liquids are calculated in terms of the easily measurable parameters on the assumption that the isochoric temperature derivative of the sound speed is the dominant factor with significant contribution to influencing the thermo‐acoustic properties. Using this simple model it is shown that Beyer’s parameter of nonlinearity, Rao’s exponent, Carnevale and Litovitz’s exponent, molecular constant relating internal pressure and cohesive energy density, the microscopic Grüneisen parameter, and the thermodynamic or pseudo‐Grüneisen parameter are all simply related to each other. The parameter of nonlinearity has been shown to depend on both Rao’s exponent and that of Carnevale and Litovitz. The calculated results are in reasonably good agreement with both the empirical and experimental results for various liquids.

The modulated ultrasonic whistle as an acoustic source for modeling
View Description Hide DescriptionA compound modulated whistle source has been developed which produces broadband sound of high intensity in air over the 16 to 160‐kHz frequency range. The development and characteristics of a prototype whistle source are described, and the details given of the final whistle design. A compound source of three such whistles is shown to have an adjustable frequency spectrum, of particular application to modelingexperiments.

The polytropic exponent of gas contained within air bubbles pulsating in a liquid
View Description Hide DescriptionGas bubbles that are driven into oscillation within a liquid pulsate with amplitudes that are determined largely by the thermodynamic behavior of the gas contained within the bubble. Theoretical approaches to the solution of the motion of the radius of the bubble normally involve the polytropic exponent of the gas. Measurements are presented of this exponent for three different gases and for a wide range of values of the exponent. Comparison with available theories indicates good agreeement between measurements and predictions provided the gas bubble is not driven near one of its harmonic resonances.

Nonlinear oscillations of gas bubbles in liquids: An interpretation of some experimental results
View Description Hide DescriptionMeasurements are presented in this paper of the pulsation amplitude of an individual air bubble that is levitated in a glycerine–water mixture by a stationary acoustic wave operating at a frequency of 22.2 kHz. Observations of the bubble pulsation for a wide range of bubble sizes demonstrate the existence of the n=2 harmonic resonance (ω≊ω_{0}/2). The available theoreticalinformation on linear and nonlinear bubble oscillations is adapted to apply to the specific experimental conditions. Comparisons made with the theory show excellent agreement between measurements and predictions.

Radiation from a double layer jet
View Description Hide DescriptionThe paper deals with the radiation of aerodynamic sound due to two symmetrically placed point sources on either side of a double layer jet flow. These sources are regarded as the driving forces in the acoustic medium. In consequence, the illumination due to a point source on one side is supplemented by the transmission due to the point source on the other side, which results in an enhanced radiation on each side. The analysis provides an expression for the reflexion coefficient which is valid at an interface that separates a fluid at rest from a fluid in motion [which is in perfect agreement with the one obtained by Miles, J. Acoust. Soc. Am. 2 9, 226–228 (1957), Ffowcs Williams, J. Fluid Mech. 6 6, 791–816 (1974), and Dash, J. Sound Vib. 4 9, 365–377; 379–392 (1976)] and predicts an aerodynamic zone of silence, also known as the valley of relative silence, which prevails parallel to the vortex interface off the source locations and appears in the form of sharp clefts around the jet axis. This deep valley in the directivity pattern deepens as the jet velocity or the ambient‐to‐jet density ratio ( ρ_{0}/ρ_{1}) increases. Marked by the conspicuous absence of highly directional beams, the formation of symmetric directivity patterns around the jet axis resembling a kidney‐shaped, lung‐shaped, or heart‐shaped structure has been vividly illustrated in Figs. 5–8. The emergence of these structures depends on the jet speed and the ambient‐to‐jet density ratio ( ρ_{0}/ρ_{1}). This investigation also suggests that the sharp, steep, dual dimples formed in the directivity pattern are characteristics to the transonic and supersonic flows at ρ_{0}/ρ_{1}=1 whereas the balloon‐type, puffed‐up, dual loops formed symmetrically at close angles to the jet axis in the forward flow direction are solely characteristics to the supersonic flows at all values of the ambient‐to‐jet density ratio, ρ_{0}/ρ_{1}≥1.

A propagator matrix method for periodically stratified media
View Description Hide DescriptionThe propagator matrix formalism of F. Gilbert and G. E. Backus is a convenient general method for treating stratified media. In this paper, the Gilbert–Backus formalism is used in conjunction with Sylvester’s matrix theorem to obtain an exact solution for wave propagation in a periodically stratified medium of arbitrary thickness. Explicit expressions are derived for the simple case of a periodically layered fluid. These expressions are used to investigate several interesting limiting cases. Application of the method to the more complicated case of a periodically layered solid is briefly discussed.

Sound propagation over Dickins Seamount in the Northeast Pacific Ocean
View Description Hide DescriptionThe acoustic propagation losses between a 230‐Hz cw source and a multi‐hydrophone receiving system were measured over Dickins Seamount in the Northeast Pacific. The source was towed at depths of 18 and 184 m. The receiving system had hydrophones spaced in depth from 323 to 633 m. The measurements were made to a maximum range of 130 km with the receiver located at a range of 60 km from the seamount peak. The results show that the seamount cast an acoustic shadow over the receiver, increasing the propagation loss by up to 15 dB, when the source was shallow and in a position which enabled the seamount to intercept all of the deep refracted source energy. Back reflections from the seamount with levels 6 to 13 dB below the direct signal level were present when the shallow source was 3 to 5 km from the seamount peak. Downslope reflections enhanced the direct signal by up to 10 dB when the shallow source was within 3 km of the peak. Acoustic shadowing and reflection effects were minimal in the results for the deep source because most of the source energy propagated along the sound‐channel axis above the seamount peak. The analysis indicates that ray theory is adequate for describing the reflection effects of the acoustic propagation but does not account for all of the acoustic energy in the shadow zone.

Improved methods for determining eigenfunctions in multi‐layered normal‐mode problems
View Description Hide DescriptionThis paper derives and illustrates the utility of the derivative of a determinant that arises from the solution of the wave equation in a multi‐layered medium. The horizontal wavenumbers at which the determinant is zero are the eigenvalues of the solution. The derivative of the determinant with respect to the horizontal wavenumber therefore can be an aid in finding the roots (zeros). More importantly, the eigenfunction normalization constant is shown to be a function of this derivative. Two types of boundary are discussed, an impedance boundary and an infinite half‐space. The normalization using the derivative of the determinant is shown to apply to both types. The derivative of the determinant is also useful in evaluating mode group velocities. An algorithm for the efficient determination of the derivative of a determinant is given. This algorithm includes a rapid computational technique that applies special properties of matrix operations and utilizes the sparseness of the determinant.