Index of content:
Volume 71, Issue 5, May 1982

Wave propagation in nonhomogeneous thin elastic rods subjected to time‐dependent velocity impact
View Description Hide DescriptionResponse of a thin long nonhomogeneous elastic rod subjected to a time‐dependent velocity impact is studied by the use of similarity transformations. Similarity representation of the original equation of motion and the associated conditions has been obtained in the form of an ordinary differential equation with variable coefficients and the corresponding boundary conditions. A general solution of the similarity representation is obtained in the form of series. Relations between the parameters of the problem for which the results hold have been obtained in the form of inequalities. The displacement and stress distribution in the rod is obtained for the admissible values of the parameters of the problem and the results are plotted for some of their discrete values. It is found that the stress is discontinuous at the wave front only in the special case of a homogeneous rod subjected to constant velocity impact; in all other cases the stress is found to be continuous at the wave front.

Analysis and computation of the acoustic scattering by an elastic prolate spheroid obtained from the T‐matrix formulation
View Description Hide DescriptionThe T‐matrix formulation is used to compute the form function of an elastic prolate spheroid. The method allows acoustic scattering computations to be made for finite bodies at frequencies into the resonance region, and the lowest order resonance observed is, as expected, due to the excitation of a Rayleigh surface wave.

Scattering from inhomogeneities in layered structures
View Description Hide DescriptionIn the present paper we give a general two‐dimensional formalism for scattering in a multilayered medium with a finite inhomogeneity present. The formalism is based on the Tmatrix method, introduced by Waterman [J. Acoust. Soc. Am. 45, 1417 (1969)]. The formalism is developed in detail for the geometries consisting of one layer with an obstacle situated either inside or below the layer. More general geometries with an arbitrary number of layers are also discussed. Numerical results for the one‐layer case are given. Extensions in several directions are indicated.

Scattering at a rough boundary—Extensions of the Kirchhoff approximation
View Description Hide DescriptionA Helmholtz integral formula gives the acoustic field reflected from a rough surface, in terms of the values of the acoustic field and of its normal derivative at all points on the surface. For surface reflection problems in which one, or the other, of these surface field values are specified, we can make use of this formula to derive a boundary surfaceintegral equation on the unspecified field values. Further, a series solution of this equation can be constructed by iteration, the zeroth‐order term of the series being the result predicted by a Kirchhoff approximation. The nth‐order iterate of this series requires an n‐fold integration over the region of the rough surface, an integration that does not in general converge absolutely for unboundedly large surfaces. For a pressure release surface, we consider the first two iterates in detail and using stationary phase approximations replace the required integration by summations. In this way the source of the convergence difficulty is made clear and is demonstrated to result from the iteration procedure. We also demonstrate that the Kirchhoff approximation is not a complete high‐frequency approximation and that multiple reflections and shadowings need be incorporated. These reflections and shadowings are associated with the stationary points of the higher order iterates in the series solution. Finally, we consider a proper manner for renormalizing the series solution, thereby removing the convergence difficulties.

On extending Biot’s theory of low‐frequency acoustic scatter about a rough fluid–rigid interface to more general acoustic media
View Description Hide DescriptionWe extend Biot’s [J. Acoust. Soc. Am. 44, 1616–1622 (1968)] boundary conditions for multiple scatter at low frequencies to a medium consisting of a fluid wholespace containing a layer of spherical fluid inclusions. At low frequencies the layer of inclusions has a scatteringeffect equivalent to that of a set of linear boundary conditions acting on the center of the layer. These boundary conditions are of order kτ, where k is the wavenumber and τ is a measure of the size of the inclusions. The boundary conditions predict that a normally dispersed acoustic boundary wave can propagate along the plane of inclusions for specific acoustic contrasts between inclusions and wholespace. When the acoustic contrast is small the boundary wave exists only when the compressional velocity of the inclusions is slower than that of the wholespace. When the contrast is large, boundary waves exist on some layers of high‐velocity inclusions as well.

Studies in reverberation. II. Scattering of sound by a cylindrical vortex embedded in a fluid at rest
View Description Hide DescriptionA solution is obtained for the scattering of a plane sound wave incident normally on a cylinder of fluid rotating uniformly in a fluid at rest. In the long wave approximation, the amplitude of the scatteredwave is proportional to the peripheral velocity of the cylinder and vanishes in the forward and backward directions as well as in directions normal to the direction of propagation of the incident wave.

Acoustic resonance frequencies of deformed spherical resonators
View Description Hide DescriptionSpherical acoustic resonators are powerful tools for precision acoustic measurements. The frequencies of the radially symmetric modes of such resonators are determined mainly by the volume of the resonator: the first nonvanishing corrections to the resonance frequencies owing to imperfections in the boundary shape are of second order. In this work, arbitrary axisymmetric deformations are considered. Boundary shape perturbation theory is used to show that the first‐order correction vanishes for constant‐volume deformations. A general expression for the second‐order frequency shift is derived. The results are applied to estimate the effect of machining errors in the construction of spherical resonators. With standard high‐quality machining, it is possible to match the mean radii of two hemispheres used to fabricate a spherical resonator to within three parts in 10^{4}. Calculations have been carried out for the first seven radial modes for the case of a resonator assembled from hemispheres with mean diameters differing by this amount. The resonance frequencies do not differ from those of a perfect sphere with the same volume by more than 1.3 parts in 10^{6}. Other probable shape imperfections are likely to introduce even smaller errors.

Scattering of a cw plane wave by a pulse
View Description Hide DescriptionAn examination of the scattering of sound by sound indicates that the boundary of the interaction region is very important in determining not only the level of the scattered sound but also the frequency of the scattered sound. In the case of a pulse propagating through a cw plane wave (where the boundary of the interaction region is propagating at the sound speed) the frequency of the farfield scattered sound at a fixed angle is the cw frequency, Doppler shifted i.e., f _{ s }(ϑ, φ) = f _{ l }(1−cos ϑ)/(1−cos φ), where f _{ l } is the frequency of the plane wave, ϑ is the angle between the cw wave vector and the pulse wave vector, and φ is the observation angle with respect to the pulse wave vector in the interaction plane. The scattered signal has a strong maximum at the Doppler angle φ_{ d }, where the frequency of the scattered signal is equal to the sum frequency. The scattered signal at φ_{ d } observed at a finite distance from the interaction region thus appears as a pulse whose frequency sweeps from a value somewhat above the sum frequency to a value somewhat below the sum frequency. Previous detection of sum frequency in a cw crossed‐beam experiment has been attributed to nonlinear processes other than scattering from the interaction region. Positive results from this type of experiment would unambiguously demonstrate the existence of scattered sound by the angular dependence of the scattered signal.

Acoustic streaming reversal by a finite amplitude sound wave in a non‐Newtonian fluid
View Description Hide DescriptionTheoretical predictions are presented for the steady secondary flows (acoustic streaming) produced by the propagation of a finite amplitude planar acoustic wave in a non‐Newtonian fluid. The constitutive equation governing finite amplitude acoustic wave propagation in quiescent non‐Newtonian fluids is derived and consists of an integral expansion of the constitutive functional. Each successive term of the expansion is higher order in a small parameter related to the amplitude of the propagated wave. The linear term of the expansion governs infinitesimal wave propagation and the quadratic term governs nonlinear effects. These equations are used to study the acoustic streaming caused by the propagation of a finite amplitude planar acoustic wave in a channel which has a width greater than that of the acoustic wave. The results show that because of nonlinear viscoelasticeffects, the direction of the streaming can be opposite to that found for Newtonian fluids.

Scattering calculations using the characteristic rays of the coherence function
View Description Hide DescriptionWe use the two‐scale expansion developed previously to determine the intensity variation in the neighborhood of caustics, to show how we may calculate the scattering of acoustic radiation in a random medium with a variable speed of sound profile. Equations are derived which show how the coherence function varies along the characteristic rays (identical to the rays of geometrical acoustics in the parabolic approximation). Conditions are given for simplifying the equations for high‐frequency quasi‐isotropic scattering. The problems that arise when the medium is highly anisotropic are discussed and an approximate solution is given. It is also shown how to calculate the intensity variation due to scattering in the neighborhood of a caustic.

Short‐range acoustic transmissions through cyclonic eddies between a submerged source and receiver
View Description Hide DescriptionConsequences of eddy‐induced sound‐speed and current variations on acoustic propagation between a submerged source and receiver are considered using ray theory. For ranges of tens of km, those rays which exist for particular source and receiver depth and range values are determined and studied. Sound‐speed and current effects on per‐ray travel time and spreading loss are investigated. Changes in the former of 15 ms or more are demonstrated, depending on source and receiver locations and ray type. Then, eddyeffects on total‐field amplitude and phase are examined. For a cw sound signal of 400 Hz, variations of about 25 dB in amplitude and several cycles of phase are observed as source and receiver positions vary within a typical eddy.Eddy currents alone are shown to have a relatively significant effect away from the eddy center, when certain rays are present or absent, and when source and receiver change their orientations. Indeed, current effects by themselves can lead to variations of up to about 12 dB in amplitude and 0.9 cycles in phase.

Attenuation parameters from normal mode measurements
View Description Hide DescriptionA new method of determining the bottom attenuation coefficient in shallow water is described. Normal mode amplitudes are measured relative to the lowest mode at a single location. These relative mode amplitudes are compared with theoretical results to obtain the bottom attenuation coefficient. The method does not require source level calibration nor measurements at different ranges. Increased attenuation as modes approach cutoff is in good agreement with theory.

Observation of Raman–Nath optical diffraction in the phase grating plane
View Description Hide DescriptionPrevious studies of Raman–Nath optical diffraction have emphasized the characteristics of the light distribution in either the nearfield or the farfield of the optical phase grating produced by an acoustic beam. The plane in which this grating is located can be imaged outside of the acoustic medium and experimentally studied with a photodetector. This paper describes measurements of the light intensity variations which occur in the image plane containing the optical phase grating after spatial filtering is performed. Theoretical results are presented for values of the Raman–Nath phase parameter v?6 when the negative optical diffraction orders are removed by a spatial filter. Experimental results are also presented from a 2.5‐MHz transducer radiating into distilled water which confirm the theoretical predictions. It is possible to use this technique to assess the frequency response of an optical system over a large frequency range since frequencies as high as four times the fundamental are generated in the optical intensity variation. For small values of the phase parameter, the approach should be useful for observing the temporal waveforms of broadband ultrasonic pulses of the type frequently encountered in medicaldiagnostic systems.

The steady‐state response of an internally damped double‐beam system interconnected by several springs
View Description Hide DescriptionThe steady‐state response of an internally damped double‐beam system interconnected by several springs to a sinusoidally varying force is determined by the transfer matrix technique. For this purpose, the Timoshenko equations of transverse vibration of an internally damped beam are written as a coupled set of the first‐order differential equations by using the transfer matrix of the beam. Once the matrix has been determined, the response of a double‐beam system is obtained by the product of the transfer matrices of each beam and the point matrices at each connecting point. In this case, the elastic moduli of internally damped beams and springs are assumed to be complex quantities. The method is applied to double‐beam systems interconnected by several springs of the same stiffness located at equal intervals, and the driving‐point impedance, transfer impedance and force transmissibility of the systems are calculated numerically.

An improved piezoelectric acoustic emission transducer
View Description Hide DescriptionA piezoelectric transducer has been designed and developed that has promise of being a high fidelity acoustic emission(AE)transducer [T. M. Proctor, Jr., J. Acoust. Soc. Am. Suppl. 1 68, S568 (1980)]. Small transducer contact area, elimination of acoustical interference effects associated with certain geometries, and redistribution of the arrival times of reflected signals originating from various elements of the transducer were the guiding criteria in the design. This transducer consists of a conical active element and an extended backing. The transducer’s performance has been compared to a line capacitancetransducer using surface wave signals. These comparisons indicate an amplitude response which is flat within ±3 dB for the frequency range of 50 kHz to 1 MHz. The over‐all displacement sensitivity is nominally 2×10^{8} V/m. Factors that influence frequency response such as backing geometry and aperture size have been experimentally investigated and results are reported.

Acoustical holography with an annular aperture
View Description Hide DescriptionA synthetic‐aperture imagingsystem using an annular array of transducer elements is analyzed. The aperture is assumed to consist of N elements, where each element serves both as a source and receiver of sound, giving rise to N ^{2} amplitude and phase measurements around the annular circumference. Because of source–receiver reciprocity, however, (1/2)N(N−1) of these measurements [where (1/2)N(N−1) is the number of element pairs on the annulus] are redundant, giving a total of N ^{2}−(1/2)N(N−1) = (1/2)N(N+1) independent measurements. It is shown how suitable processing of these measurements can yield a high‐resolution image of a reflecting object in a plane parallel to the annulus and located within its Fresnel region. Moreover, the resultant resolution is shown to be equivalent to that of a f u l l circular aperture twice the diameter of the annulus. This approach differs from the J ^{2} synthesis of Wild [Proc. R. Soc. London Ser. A 286, 449–509 (1965)] in that the annular array acts as a source as well as a receiver and that no assumptions regarding the spatial coherence of the reflecting object are required. Numerical reconstructions based on simulated data are presented. Possible areas of application of the annular imagingsystem include medicalultrasonic imaging, underwater acoustic imaging, and microwave imaging.

Nearfield inverse scattering formalism for the three‐dimensional wave equation: The inclusion of a p r i o r i velocity information
View Description Hide DescriptionWe address the problem of identifying a local velocity, c(r), in a three‐dimensional scalar wave equation from nearfield data. The implicit model assumptions of previous nearfield inverse scattering procedures are examined. These assumptions cannot be reasonably assumed in certain applications. Therefore, a method for extending the range of validity of nearfield inverse scattering procedures is presented. In addition, we present a technique for incorporating a p r i o r i information or estimates into this formal wave equation inversion procedure. An approximate perturbative solution for this formalism has the ability to accommodate selective and potentially troublesome multiples within the Born model.

IHC–TM connect–disconnect in relation to sensitization and masking of a HF‐tone burst by a LF tone. IV
View Description Hide DescriptionEvidence continues to accumulate that although the outer hair cells (OHCs) of the cochlea are firmly bonded to the tectorial membrane (TM), the inner hair cells (IHCs) are not. This is the fourth in a series of papers that explores how the idea of a set of disconnected hair cells that ’’impact’’ the TM is consistent with psychophysical data. The paper extends the exploration to the masking of high‐frequency (HF) tone bursts by low‐frequency (LF) tones and shows that the model can explain important features of these complex data.

Implications of causality, time‐translation invariance, linearity, and minimum‐phase behavior for basilar‐membrane response functions
View Description Hide DescriptionSeveral implications of the assumptions of causality (C), time‐translation invariance (TTI), linearity (L), and minimum phase behavior (MPB) for basilar membrane (BM) frequency‐response functions are derived. They are then used to: (1) test the consistency of calculated results for a two‐dimensional cochlear model [S. T. Neely, J. Acoust. Soc. Am. 69, 1386–1393 (1981)] and (2) check experimental data on the BM displacement/malleus displacement [W. S. Rhode, J. Acoust. Soc. Am. 67, 1696–1703 (1980)] for approximate consistency with these assumptions. Both the theoretical model results and experimental cochlear‐partition response data are in fairly good accord with these assumptions.

The phases of basilar‐membrane vibrations
View Description Hide DescriptionExperimental phase data from the Mössbauer‐effect measurements of basilar‐membrane vibration are given. The phase lag increased with intensity for frequencies less than best frequency, but was a decreasing function of intensity above best frequency. A series of linear minimum‐phase representations of the basilar membrane’s motion were also calculated; they show similar trends.