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Volume 69, Issue 5, May 1981

General theory of acoustic propagation through arbitrary fluid media—I. Propagation equations, conditions of the medium, and general dynamical solutions
View Description Hide DescriptionFrom first principles a general (nonlinear) differential equation is derived, applicable to acoustic propagation in an arbitrary fluid medium. This general dynamical equation is then formally solved exactly, where the medium may include (surface, bottom) boundaries, turbulence, body forces, dissipation, and other physical conditions known to affect propagation. Because the medium is basically stochastic, one must consider in addition the ensemble (or set) of the above dynamical equations, which together constitute the fundamental Langevin (or stochastic) equation governing propagation here. It is then the s t a t i s t i c s, namely the moments, probability densities, etc., of the resultant random pressure field, obtained from the corresponding ensemble of dynamical solutions generated here, which represent the ultimately desired solutions of the governing Langevin equation. In the present paper only dynamical solutions are derived; (later, in Parts II and III, stochastic solutions will be obtained from the former). Among the new results obtained in Part I, in addition to the general dynamical solutions, it is shown how the many forms of the ’’wave equation,’’ in past and current usage, are particular approximations of the general dynamical equation derived above, and what specific dynamical conditions must be quantitatively obeyed for the various approximate forms to be valid in applications.

Time domain integral equation solution for acoustic scattering from fluid targets
View Description Hide DescriptionThe responses of fluid targets are computed using space–time integral equations formulated in the time domain. The incident pressure wave is a ’’smoothed‐impulse’’ with a Gaussian shaped time dependence whose width is of the order of a target dimension. A space–time integral equation for the pressure field on the outside of the target surface and a space–time integral equation for the pressure field on the inside of the target surface are solved simultaneously for the pressure and the pressure gradient by stepping on in time and making use of the boundary conditions (continuity of pressure and displacement). The farfield is then computed from these source fields. The technique is applicable to targets of arbitrary contour and is demonstrated for a sphere and right circular cylinder at various angles of incidence. Fluid targets support interior compression waves; sound‐hard and sound‐soft targets are treated as limiting cases of this formulation. The technique has been verified for the case of a sphere by comparison with the response computed by classical expansions.

Sound diffraction around screens and wedges for arbitrary point source locations
View Description Hide DescriptionA development is presented for the Green’s function for a point source in the vicinity of a rigid wedge. The diffraction contributions to the Green’s function for arbitrary source and listener location is expressed in a form which can be readily evaluated using the Laguerre technique for numerical integration. The present approach offers the advantages of efficient numerical evaluation and of relatively straightforward reduction to well‐known analytical approximations in limiting cases. Comparisons with previously obtained experimental and numerical results obtained by Ambaud and Bergassoli [Acustica 27, 291–298 (1972)] are presented. The comparison with the experimental results is excellent; the advantages of the present numerical technique, v i s a v i s that of Ambaud and Bergassoli, are pointed out.

Case study using arrays of infrasonic microphones to detect and locate meteors and meteorites
View Description Hide DescriptionOn 22 April 1975, two infrasonicobservatories in Colorado detectedacoustic signals related to a fireball sighting. We use the infrasonic data in conjunction with surface and aircraft observations to investigate the signals. We deduce that the acoustic energy originated from an explosive interaction of the object with the atmosphere at an altitude of about 25 km at a distance approximately 250 km from the observatories.

Mixing of normal modes in a range‐dependent model ocean
View Description Hide DescriptionThe problem of calculating the horizontal propagation of sound along a range‐dependent underwater channel is solved in a model using an harmonic‐oscillator expansion for the vertical pressure distribution. This oscillator basis set is constant for all ranges. The local normal modes at any range may be recovered by a matrix diagonalization, but this is only strictly necessary at the range limits, to define the outgoing boundary conditions. The method is capable of handling arbitrary range dependencies, and is applied to a range of constant gradients to reveal three propagation regimes. Numerical criteria are found for distinguishing the three cases of (i) a d i a b a t i c p r o p a g a t i o n of modes when the channel changes are gradual; (ii) n o n a d i a b a d i c p r o p a g a t i o n when the channel shifts significantly within a ray period; and (iii) m o d e c u t o f f, when the energy in a mode is largely lost to the basement.

Energy transmission into shadow zone by rough surface boundary wave
View Description Hide DescriptionGiven a rough surface characterized by roughness elements of size d and spacing h, then, for a source on or near this surface and k h<1, Biot‐type boundary conditions predict the existence of a subsonic boundary wave which is f i r s t o r d e r in k d—a prediction which has been verified experimentally in detail for the case of a rough rigid wall and a homogeneous fluid. This paper examines the case of a stratified fluid in contact with a rough surface. It is shown that, for a soundvelocity decreasing away from the surface—i.e., under shadow‐zone conditions for a source on the surface—this boundary wave is attenuated; in the farfield it behaves like r ^{−1/2}exp(i k r)exp(−δ_{ B } r). The attenuation is due to a tunneling effect. The standard acoustic (volume) wavediffracted field is, under the same conditions, represented by a sum of terms r ^{−1/2}exp(i k r)exp(−δ_{ m } r). It is demonstrated that, for many cases of interest in ocean acoustics, δ_{ B }<δ_{1} where δ_{1} corresponds to the least attenuated of the diffraction terms. The surface roughness boundary wave can then transmit substantially more energy into the shadow zone than is allowed for by ordinary diffractioneffects. Under certain conditions its amplitude will even exceed that of a direct acoustic arrival in the absence of a soundvelocity gradient—i.e., near a rough surface the shadow‐zone effect can be more than offset by the roughness boundary mode.

New algorithm for relaxation spectra in ultrasonic spectroscopy II: Multiple relaxation analysis
View Description Hide DescriptionA previously reported algorithm for a single relaxation based on a modified least‐square method has been extended to analyze general ultrasonicabsorptions with multiple relaxation mechanism. The estimated errors of the experimental data for i‐steps (i = 1,2,⋅⋅⋅) multiple relaxations can be evaluated. This technique is successfully applied to the observed ultrasonicabsorption of an aqueous solution system of copper and sodium acetates (with each 0.2 mol) with 2‐steps relaxations.

Bleustein–Gulayev surface modes in elastic ferroelectrics
View Description Hide DescriptionOn the basis of a nonlinear theory of elasticferroelectrics linearized about a fundamental ferroelectric state, it is shown that the symmetry breaking caused by the initial intense polarization orthogonal to the sagittal plane allows for the existence of Bleustein–Gulayev surface modes in ferroelectrics of the bariumtitanate type. The two cases of a free boundary and a grounded boundary are examined. In the first case it is shown that above a certain critical wavenumber (interaction of elastic and ferroelectric modes) the exponential decrease with depth of the amplitudes is sinusoidally modulated while there exists a minute dispersion of the mode. In the second case the Bleustein–Gulayev mode which is exhibited is strongly dispersive in the neighborhood of this critical wavenumber (which then corresponds to a unilateral repulsion of dispersion branches). All the features of this surface‐wave propagation problem are examined both analytically and numerically (for BaTiO_{3}) with a particular emphasis on second‐order effects (corrective effects) and the appearance of boundary‐layer effects connected with the influence of polarization gradients which account for the ordering proper to ferroelectrics.

Piezoelectric Rayleigh waves in elastic ferroelectrics
View Description Hide DescriptionFollowing a companion paper devoted to Bleustein–Gulayev surface modes in elasticferroelectrics, the present paper examines the surface‐wave problem associated with the surface propagation of the remaining components of the elastic displacement and the dynamical polarization. This study yields the notion of strongly dispersive piezoelectricRayleigh waves in ferroelectrics, which results from an interaction between a classical Rayleigh mode and a ferroelectric mode in the form of a unilateral repulsion of dispersion curves by means of the piezomagnetic coupling induced during the symmetry breaking caused by the bias polarization field. This theory is applied to the case of bariumtitanate and the behavior of the various amplitudes with depth in the semi‐infinite space is exhibited with the allied second‐order and boundary‐layer effects.

Vibrations of circular plates with variable profile
View Description Hide DescriptionGalerkin’s variational approach is invoked in analyzing transverse vibrations of circular plates where the thickness varies according to the functional relation h(r) = h _{0}[1+γ(r/a)^{ n }] where n is an integer. The displacement function is approximated in terms of simple polynomials which identically satisfy the boundary conditions. The two lowest frequency coefficients of axisymmetric modes are determined in the present paper for several combinations of the governing mechanical parameters.

Free vibration of regular polygonal plates with simply supported edges
View Description Hide DescriptionThe free vibration of regular polygonal plates with simply supported edges is studied by the dynamical analogy with membranes. A regular polygonal membrane is formed on a rectangular membrane by fixing several segments. With the reaction forces acting on all edges of an actual polygonal membrane regarded as unknown harmonic loads, the stationary response of the membrane to these loads is expressed by the eigenfunctions of the extended rectangular membrane without internal supports. The force distributions along the edges are expanded into Fourier sine series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual membrane are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to an equilateral triangular through a regular decagonal membrane, the natural frequencies and mode shapes are calculated numerically and the effect of the shape of membrane is discussed. The numerical values obtained for polygonal membranes are immediately converted into those of simply supported polygonal plates.

Spatial concentrations of random response in point‐excited waveguides
View Description Hide DescriptionA recent modal analysis [J. Appl. Mech. 46, 417 (1979)] shows that a marked spatial concentration of the mean‐square response (larger than the spatial average by a factor 4) may exist in a dynamical system excited by a wideband, localized random force. Response is largest near the source. The system has distributed damping and a cutoff frequency below which waves could not freely propagate in an infinite, undamped system; the finite system has large modal density and strong modal overlap near the cutoff frequency. The same system is examined here by a multiple‐scattering analysis. It is shown that the spatial concentration, ascribed in the modal analysis to cross correlation between the responses of many modes, is largely described by the direct field (the first term of the scattering series) and is associated with a narrow band of response near the cutoff frequency, where free waves are strongly attenuated. Closed‐form analytical approximations are developed in terms of the fundamental dynamical parameters. In a general discussion, it is pointed out that the extent of modal overlap may be an unreliable basis for predictions of spatial concentrations of response, especially in two‐ or three‐dimensional systems or in systems whose dissipation is concentrated on the boundary.

The effects of noise upon human hearing sensitivity from 8000 to 20 000 Hz
View Description Hide DescriptionHigh‐frequency (8 to 20 kHz) hearing sensitivity was compared in thirty‐six, 20 to 29‐year‐old military veterans with histories of steady‐state or impulsive noise exposure. Threshold shifts were prominent for the steady‐state noise subjects from 13 to 20 kHz. Mean thresholds from 8 through 12 kHz were maximally 20 dB poorer than a sample of young adult normals. Audiometric configurations for this group were generally smooth and symmetrical above 8000 Hz. For the impulsive noise group, substantial shifts in sensitivity were seen from 2 to 20 kHz and the high‐frequency audiometric configurations were often jagged and/or asymmetrical. The variability of subjects in this group was greater than that seen in the steady‐state noise exposed sample. Several case studies are presented to illustrate these characteristics. Measurement of auditory sensitivity from 8 to 20 kHz extends the mapping of basal cochlear function, providing information which often is not predictable from conventional audiometric measurement. This additional information provides for more comprehensive inter‐ and intra‐subject comparison of the degree and extent of threshold changes present.

Spectral variance of aeroacoustic data
View Description Hide DescriptionAn asymptotic technique for estimating the variance of power spectra is applied to aircraft flyover noise data. The results are compared with directly estimated variances and they are in reasonable agreement. The basic time series need not be Gaussian for asymptotic theory to apply. The asymptotic variance formulae can be useful tools both in the design and analysis phase of experiments of this type.

Complementarity of the Reddi method of source direction estimation with those of Pisarenko and Cantoni and Godara. I.
View Description Hide DescriptionIt is shown that the methods of plane‐wave signal direction estimation of (a) Reddi, (b) Pisarenko, and (c) Cantoni and Godara (CG), i n t h e c a s e o f a n o n s i n g u l a r s i g n a l s o u r c e c o v a r i a n c e m a t r i x P, constitute orthogonal projection of the candidate signal direction vectors onto the right eigenspace corresponding to the eigenvalue σ^{2} of the complex covariance matrix R. Case (a) centralizes on the subspace spanned by the right eigenvectors of R corresponding to eigenvalues larger than σ^{2}, whereas cases (b) and (c) to the orthogonal complement of this subspace being the subspace (right eigenspace) spanned by the right eigenvectors of R corresponding to smallest eigenvalues equal to σ^{2}. It is indicated that Reddi’s and CG’s approach, when restriction is made in the case of CG to equispaced collinear arrays of sensors, are noneconomized versions of Pisarenko’s algorithm. A simplistic beamformer is defined herein which, unlike those derived in the work of Reddi, Pisarenko, and Cantoni and Godara, requires calculation of a minimal eigenvalue, σ^{2}, but no eigenvectors. A maximal linearly independent set of column vectors is selected from the received signal covariance matrix, (R−σ^{2} I _{ N }), and these are employed to define a simplistic signal (interference) nulling or preserving beamformer. Finally, it is shown that the case of P non‐negative definite, viz., P?0, led CG to a flawed conclusion.

Impact of beam steering errors on shifted sideband and phase shift beamforming techniques
View Description Hide DescriptionThe Shifted Sideband Beamformer (SSB) is a digital, time‐domain beamformer intended for bandpass applications. Computational efficiency results from the digital processing of the complex envelope of the sensor data at a rate proportional to the bandwidth of the data. In a previous paper, the spatial response of the SSB was examined and compared to that of a conventional bandpass digital beamformer by modeling the beam steering quantization error as a uniformly distributed random variable. This paper examines, in detail, the specific case of a line array of uniformly spaced sensors where beam steering quantization is shown to produce systematic time‐delay errors for both the conventional beamformer and SSB implementations. Expressions are presented for predicting the directions of grating side lobes in the spatial response pattern as a result of these errors. Also, the spatial response of the Phase Shift Beamformer (PSB), which is a limiting case of the SSB and intended for narrow‐band applications, is examined. The systematic phase errors of the PSB for a planar array are shown to produce errors in the beam pointing direction for frequencies other than band center. Examples demonstrate this effect for the special case of a line array. Finally, it is shown that the SSB can be interpreted as time‐domain beamforming with partial beam outputs formed from subclusters of adjacent sensors with a PSB.

Digital solution of cochlear mechanics problems
View Description Hide DescriptionSeveral advantages are incurred when the greater part of the solution to a cochlear mechanics problem is carried out with an analytical method. Many cochlear mechanics problems can be formulated in such a way that they require solution of a specific integral equation in terms of the Fourier transformW(k) of the BM velocity w(x). An analytical solution for W(k) is possible for a certain class of impedance functions. In this paper it is shown how the Fourier transformation to the x domain can be carried out without producing aliasing errors. The impedance function Z(x) must be made periodic in x. For a particular choice of this function, one which puts the main emphasis on BM resonance, the solution to the integral equation can be found from a simple recurrence relation between the spectral components W _{ n } of w(x). The method is illustrated for typical one‐, two‐, and three‐dimensional cochlea models. It is proven that in the region of resonance the (nondigital) solution method published earlier produces nearly equivalent results. The digital method has a much wider scope of application.

AP tuning curves from normal and pathological human and guinea pig cochleas
View Description Hide DescriptionMeasures of cochlear selectivity can be obtained from compound responses using tone‐on‐tone masking procedures [Dallos and Cheatham, J. Acoust. Soc. Am. 59, 591–597 (1976)]. For the normal guinea pig, cochlear fiber tuning is sharper by a factor of 1.8 than AP tuning curves using simultaneous masking (threshold criterion = 25% N1 amplitude reduction). Anesthesia does not appear to affect AP tuning. In pathological cochleas, AP tuning is broadened by a factor of 2–3, and differences between forward and simultaneous masking curves are reduced. Tuning changes can sometimes occur without threshold elevation. AP tuning curves were obtained from humans during transtympanic electrocochleography. For subjects with near normal thresholds, Q _{10 dB} values (simultaneous masking) are approximately 2.3 at 2 kHz, 3.6 at 4 kHz, and 4.7 at 8 kHz. Using the relationship between cochlear fiber tuning and AP tuning in the guinea pig, estimates of human cochlear fiber tuning are 4.2 at 2 kHz, 6.5 at 4 kHz, and 8.5 at 8 kHz. Patients with threshold elevations of more than 30 dB resulting from cochlear deafness have AP tuning curves less sharply tuned by a factor of 2–3.

Finite difference solution of a two‐dimensional mathematical model of the cochlea
View Description Hide DescriptionA current, linear, two‐dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the modelequations. The finite‐difference method is used to discretize Laplace’s equation over a rectangular region with specified boundary conditions. The resulting matrix equation for fluid pressure is solved by using a Gaussian block‐elimination technique. Numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes. The finite difference method is a direct, versatile, and reasonably efficient means of solving the two‐dimensional cochlear model.

Interaural correlation discrimination: I. Bandwidth and level dependence
View Description Hide DescriptionMeasurements of interaural cross‐correlation jnds from two reference correlations at several bandwidths were obtained for constant‐total‐power and constant‐spectral‐power Gaussian noise. At a reference correlation of 1, the results indicate that for bandwidths less than or equal to 115 Hz the jnd remains at a constant value of approximately 0.004, and monotonically increases (discrimination performance degrades) to approximately 0.04 as bandwidth increases above 115 Hz. At a reference correlation of 0, the jnd decreases (discrimination performance improves) from approximately 0.7 to 0.35 as the bandwidth increases from 3 to 115 Hz, and remains at a constant value of approximately 0.35 for bandwidths greater than 115 Hz. A decrease in the spectral level causes an increase in the jnds at a reference correlation of 1, and no change in the jnds at a reference correlation of 0. Of the three models tested, none is able to completely describe all of the empirical results.