Index of content:
Volume 67, Issue 4, April 1980

The mean‐value method of predicting the dynamic response of complex vibrators
View Description Hide DescriptionThe mean‐value theory predicts the mean line through the logarithmically recorded frequency‐response curve of a complex vibrator, the height of the resonance peaks, and the height of the minima that occur between every two resonance peaks. It predicts the mean and the extremes in the frequency‐ response curve from the first resonance of the vibrator to very high frequencies. The computations are based on the mode masses and on the density of the resonances. They are simple and they give considerable insight into the dynamics of complex vibrators. Part I of this paper presents the basics of the mean‐value theory. Part II deals with the variation of the vibration velocity with the distance from the driver and with the frequency. Part III is concerned with shells without and with stiffeners and with rib‐stiffened plates. Because it is always the component of the characteristic velocity that is transmitted by discontinuities (as though the parts of the vibrator behind them were infinitely large) that determines the geometric mean values of the response, the effect of ribs and other appendages on the vibration can be predicted without much computation. As an example of the application of the theory, the vibration isolation that can be obtained by spring mounting a motor on a ring welded into a shell is computed in detail and the predictions for the ring excited shell are compared with the measurements.

Gaussian–Laguerre description of ultrasonic fields—Numerical example: Circular piston
View Description Hide DescriptionThe sound field produced by plane baffled transducers of high k a can be described by treating the radiated space as a waveguide of infinite size. Under the Fresnel approximation for cylindrical geometries, the modes of this waveguide are known to be the Gaussian–Laguerre functions. The general Gaussian–Laguerre formulation for expanding acoustic fields is presented and numerical results are shown for the circular piston. The Gaussian–Laguerre expansion coefficients for this example are obtained analytically, thus providing a new solution in series form.

One‐dimensional velocity inversion for acoustic waves: Numerical results
View Description Hide DescriptionWe consider the inverse problem of determining small variations in propagation speed from remote observations of signals which pass through an inhomogeneous medium. Under the conditions (1) that the variations can be written as a small perturbation from a known reference value and (2) that the medium of interest varies in one direction only, an integral equation has been developed for the variations which can be solved in closed form. Here, a technique is presented to obtain and process synthetic data from a scattering profile of arbitrary shape. The results of numerical testing show that, as long as a velocity variation is indeed ’’small,’’ both its size and its shape can be reproduced with negligible error by this method.

Caustics and the spreading of adjacent acoustic rays
View Description Hide DescriptionSome previous discussions of phase shifts at caustics have employed the WKBJ method of solution to the wave equation for a point source in a vertically stratified medium where a single sound‐speed minimum exists. Geometrical acoustic techniques are used in this paper to show for the same problem that a ray encounters only one caustic between turning points (starting with the first and second points), and that turning points and points of encounter with caustics approach one another with increasing range from the source.

Numerical integration method for reflected beam profiles near Rayleigh angle
View Description Hide DescriptionA numerical integration method is devoloped to calculate the intensity profile of an ultrasonic beam reflected from a liquid–solid interface. This numerical treatment is used to calculate nonspecular reflectivity at a range of angles of incidence near and at the Rayleigh angle. Calculations for a water–stainless steel interface are compared to a known approximate analysis for various beamwidths and frequencies. The theoretical predictions of the reflected beam profile near the Rayleigh angle of incidence are compared to experimental results.

Flow‐induced tones in side‐branch pipe resonators
View Description Hide DescriptionAcoustic tones generated by turbulent flows and shear‐layer‐instability interactions with side‐branch resonator pipes were investigated experimentally using fast Fourier transform techniques. The experimental values of resonant frequencies and instability frequencies were compared with predictions for two stages of shear‐layer interaction. Relative tonal amplitudes are shown to demonstrate cut‐in and cut‐out phenomena. The qualitative differences between turbulent‐flow‐generated tones and instability‐generated tones are also noted.

Diffraction of Lamb waves by a finite crack in an elastic layer
View Description Hide DescriptionThis paper analyzes the diffraction of Lamb waves by a finite crack situated on the plane of symmetry of an elastic layer. The surface of the crack and of the layer are assumed to be stress‐free. The problem is solved by the modified Wiener–Hopf technique. The field of the reflected and transmitted waves, and also the field in the vicinity of the crack, are given as an expansion in natural waves of the elastic layer. The amplitudes of these waves are expressed in terms of certain generalized quantities, which are found from exponentially converging infinite systems of equations. The solution converges at any value of the parameter L/H≳0 (where L is the length of the crack and H is the thickness of the layer) and is particularly effective as this parameter increases. The strongly resonant phenomena in the region of the layer occupied by the crack are identified and discussed.

Amplitude distribution of pulses produced by shock waves due to cavitation bubbles
View Description Hide DescriptionThe authors study the amplitudes of the pulses delivered by a piezoelectric sensor immersed in a cavitation field. A 1024‐channel analyzer shows that the amplitude distribution of these pulses includes peaks. They go on to show the pattern of evolution of this distribution as a function of the ultrasonic power and the distance between the piezoelectric sensor and the transducer.

Nonstationary and nonuniform oceanic background in a high‐gain acoustic array
View Description Hide DescriptionNonstationary and nonuniform distributions of acoustic background in the ocean are important statistically when detection performance predictions are carried out for high‐gain receiving arrays. A high‐gain acoustic array, ADA (Acoustic Distribution Array), designed for measurement of high‐order temporal and spatial distributions of acoustic backgrounds in the ocean, was used to obtain envelope spectra of directional noise in octave bands centered at 1130 and 2260 Hz. The directional noise was measured through an infinitely clipped DIMUS beamformer and thus, because of the clipped normalization, the envelope spectra represent, solely, variations in the directional structure of the background noise, not its absolute level.

Calculations of sound propagation through an eddy
View Description Hide DescriptionBy means of a corrected split‐step parabolic‐equation numerical algorithm,acoustic propagation through an ocean region characterized by a sound‐speed distribution produced by an analytic model of an eddy is investigated. Parameter values for a moderate sized, cyclonic, Gulf Stream eddy are used. It is found that the presence of an eddy causes significant changes, both in nature and level, in the received acoustic field produced by an omnidirectional cw source. The eddy causes major changes in the arrival structure seen by a vertical array. The percentage of energy arriving at angles less than 5° from broadside increases from 5% to 50% in the presence of the eddy under consideration at 100 Hz. Greatly increased energy corresponding to horizontal arrival is noted. In addition, the effects of an eddy progressing through the region between an acoustic source and receiver is studied. The position of the eddy relative to the source causes changes in transmission loss of as much as 20 dB.

A theoretical model of ambient noise in a low‐loss, shallow water channel
View Description Hide DescriptionA theoretical model of ambient noise in an isovelocity, shallow water channel with independent, randomly distributed surface sources is presented. Bottom losses in the channel are assumed to be sufficiently small for the modal energy from distant sources to predominate over the continuous radiation from nearfield sources. For frequencies at which the channel can support about ten or more modes the predicted noise field is essentially homogeneous over a large proportion of the water column away from the boundaries. This allows the vertical spatial coherence of the noise to be expressed in terms of a plane‐wave directional density function. The properties of this function are shown to be characteristic of the modal structure of the noise field.

Simplified calculation of ray‐phase perturbations due to ocean‐environmental variations
View Description Hide DescriptionA simplied approach is described for determination of phase perturbations produced by variations in sound speed and current in the ocean. It is shown that corresponding perturbations of the ray geometry may be ignored in determining the phase perturbations, when the former are regular in a specific sense. Principal advantages in the procedure include its efficiency in calculation of phase variations and its indication of situations when ray‐geometric perturbations may significantly influence phase. The method is demonstrated for both shallow‐ and deep‐ocean examples, when the phase perturbation arises from weak horizontal deviations from a horizontally uniform sound‐speed structure. It is also illustrated for current in the deep ocean, considering cases with and without horizontal variations in current speed. Examples for both types of horizontal variations are shown in which they make significant contributions to ray phase. Finally, the procedure is applied to some single‐path acoustic observations in which the time series of travel time is dominated by tidal variations. Ranges of possible environmental changes in sound speed and current, leading to the observed travel‐time variations, are indicated.

Consistent environmental acoustics: Application to stochastic internal‐wave models
View Description Hide DescriptionA consistent environmental‐acoustic model for a deep moving ocean is formulated. The acoustic model for regularly perturbed SOFAR rays is approximately solved using a type of WKB (J) expansion. Interfacing conditions between the hydrodynamics and acoustics are developed which lead to constraints on acoustic frequency and transmission range. As an application, transmissions are considered through stochastic internal‐wave fields, which have been modeled in a previously published paper by the authors. Formulas for ray phase variances are derived. These formulas are asymptotically evaluated for rays with relatively significant depth variation, using the stationary phase method. New results are obtained for the dependence of the variances on internal‐wave primitives, such as energy spectra. Expected multipath intensity is calculated for transmission through an ocean with static state modeled by a bilinear sound‐speed profile. The effects of the internal‐wave field and of varying internal‐wave parameters on the expected intensity are shown to be significant.

Vibration of a clamped circular plate driven by a noncentral force
View Description Hide DescriptionExpressions are stated for the transmissibility and for the driving‐point impedance of an internally damped circular plate of radius a with a clamped boundary that is driven by a vibratory point force at an arbitrary distance μa from the plate center. Expressions are also stated for the plate transmissibility and driving‐point impedance when the plate is loaded at the arbitrary driving point either by a lumped mass, by a dynamic vibration absorber, or simultaneously by a lumped mass and a dynamic absorber. In all cases, representative calculations of transmissibility and impedance are plotted versus the square root of frequency. These curves clearly show the dependence of transmissibility and impedance on the plate damping factor, the value of the parameter μ, and the extent of the mass loading. They also show the effectiveness of the dynamic absorber, which varies with the value assigned to μ.

Detection, estimation, and classification with spectrograms
View Description Hide DescriptionA locally optimum detector correlates the data spectrogram with a reference spectrogram in order to detect (i) a known signal with unknown delay and Doppler parameters, (ii) a random signal with known covariance function, or (iii) the output of a random, time‐varying channel with known scattering function. Spectrogram correlation can also be used for maximum likelihood parameter estimation, e.g., estimation of delay or center frequency of a signal. To estimate an analog input signal from its spectrogram, a modified deconvolution operation can be used together with a predictive noise canceler. If no noise is added to the spectrogram, the mean‐square error of this signal estimate is independent of the window function that is used to construct the spectrogram. When estimates of specific signal parameters are obtained directly from the spectrogram, these estimates have mean‐square errors that depend upon both signal and window waveforms. Spectrogram correlation can be used for classification as well as for estimation and detection. Parameter estimators and detectors are, in fact, specialized kinds of classifiers.

Resolving the directions of sources in a correlated field incident on an array
View Description Hide DescriptionAdaptive beamforming algorithms are often designed under the assumption that the sources illuminating the array are mutually uncorrelated. When the assumption is not valid, the performance of the adaptive beamforming algorithm in certain applications is severly limited. A simple example is included which illustrates how suppression of a look direction signal occurs when a power minimizing adaptive algorithm is used in the presence of correlated sources. This paper establishes certain properties of the eigenvectors of the correlation matrix of an array illuminated by a field of correlated discrete sources. On the basis of these properties an adaptive algorithm is proposed for resolving the direction of all discrete sources, even if they are mutually correlated. Simulation results are presented showing the performance of the algorithm proposed in this paper.

Degradation of the signal‐to‐noise ratio of Gaussian signals in Gaussian noise transmitted over a digital channel
View Description Hide DescriptionThis paper presents a mathematical analysis of the signal‐to‐noise power ratio degradation incurred when Gaussian signals in Gaussian noise are level‐quantized, time‐sampled, transmitted over a noisy digital channel, and then reconstructed by a low‐pass filter. Application of this analysis occurs when very weak acoustic or seismic signals in strong white noise are collected and transmitted over a digital channel to an analysis location remote from the collection point, and where detectability of the information‐bearing signals at the analysis location depends on the original signal‐to‐noise power ratio in small frequency bands. Therefore, it is important to understand the degradation of the original signal‐to‐noise ratio occurring during transmission. Curves are presented which show bounds on the degradation as a function of the quantizer step size and the channel bit error probability. Also considered is a scheme for transmitting a low‐rate digital sequence by periodically interrupting the level‐and‐time‐quantized analog signal. Three schemes are proposed for doing this, and one scheme is mathematically analyzed. It is shown that this scheme increases the degradation of the reconstructed analog signal by about 0.15 dB over that of the uninterrupted case.

Reconstruction of a two‐dimensional reflecting medium over a circular domain: Exact solution
View Description Hide DescriptionIf a two‐dimensional acoustic reflectivity function is defined within the interior of a circle, reflectivity data may be acquired by transmitting acoustic pulses from isotropic elements distributed around the circumference of the circle and recording the resulting backscatteredsound as a function of time. If the reflectivity function fulfills the conditions of an ’’idealized’’ weakly reflecting medium, the resulting pulse–echo data may be regarded as the line integrals of this function defined over circular arcs centered at points lying on the circumference of the enclosing circle. In this paper we show that on the basis of these line integrals the unknown reflectivity in the interior of the circle can be expressed explicitly in terms of its line integrals defined over the set of paths consisting of all circular arcs whose centers lie on the circumference of the enclosing circle. We propose that the resulting reconstruction equations could provide the foundation for a new approach to reflectivitytomography. A numerical example is also given.

Influence of data window shape on detectability of small cw signals in white noise
View Description Hide DescriptionThe power spectrum of a white noise waveform modified by an arbitrary, real data window is derived. It is shown that the weight of a component at frequency f relative to the cumulative weights of all other components in estimates of the power density at this frequency is greatest when a rectangular data window is used. On this basis it is concluded that the detectability of a small cw signal in white noise is greatest if a rectangular data window is used. Some examples are worked out to show the estimated loss in detectability when various other windows are used.

Random and partially random acoustic arrays
View Description Hide DescriptionStatistical properties of the beampatterns of three‐dimensional random arrays of arbitrary geometry are derived. The class of random arrays studied differs from those previously analyzed in that it includes partially random arrays. A partially random array is defined, for purposes of this paper, as a random array in which the various element positions may have different probability density functions and may be interdependent. Results presented include the expected value and variance of the pattern function, expected array power and the associated directivity index, and the variance of the array power. The results are applied to random and partially random disk arrays. The partially random disk array is assumed to have elements arranged in vertical strings. It is found that an array with M‐element strings is characterized by a region corresponding in elevation to the main lobe and extending 360° in azimuth in which the side‐lobe variance is a factor of M greater than that in the rest of the side‐lobe region.