Index of content:
Volume 93, Issue 4, April 1993

Ultrasonic scattering from spherical shells including viscous and thermal effects
View Description Hide DescriptionAn analysis for sound scattering and attenuation by shell structures immersed in fluids is outlined. While the form of the analysis permits easy comparison with previous results for simple spheres, it includes novel features of viscoelasticity in the shell and core materials (which may be fluid or solid), viscosity in the suspending fluid, and thermal effects. The computational procedure adopted is outlined with particular reference to the avoidance of numerical instabilities at very low and very high values of ‘‘k a.’’ The testing of the program by comparison with previous results on simpler systems is reported, and a limited selection of numerical results is given. The program has been used successfully over the range of k a values between 10^{−6} and 200, its robustness needs closer definition.

A gradient formulation of the Helmholtz integral equation for acoustic radiation and scattering
View Description Hide DescriptionA method of overcoming the problem of nonuniqueness in the discretized Helmholtz integral equation is described, based on a partial application of the Helmholtz gradient formulation of Burton and Miller [Proc. R. Soc. London, Ser. A 3 2 3, 201–210 (1971)]. The numerical implementation is designed to be compatible with a finite‐element structural analysis, and uses boundary elements of the quadratic isoparametric type. The method is illustrated for scattering from a sphere, and for radiation by a piston vibrating in the end of a cylinder, with consistent results being obtained across a wide frequency range. The additional computation is of the order of 35% of that required for the standard formulation.

Eigenfrequencies of an acoustic rectangular cavity containing a rigid small sphere
View Description Hide DescriptionAnalytical expressions are derived for the eigenfrequencies of any mode in a hard‐walled acoustic rectangular cavity, containing an acoustically small rigid sphere. A straightforward and simple approach is employed to obtain a first‐order perturbation of the cavity eigenfunctions and the corresponding eigenfrequency shifts. Rectangular wave functions, as well as their expansion in terms of cylindrical wave functions and finally in terms of spherical ones, are used. The results are useful in problems connected with acoustic levitation and with excitation or probing of resonant cavities. Some suggestions are made about the best positioning of the probe, for more exact measurement of the eigenfrequency of any mode in the unperturbed cavity. Graphical results for some of the lower‐order modes are given, for various values of the parameters.

Ray tracing in a moving medium with two‐dimensional sound‐speed variation and application to sound propagation over terrain discontinuities
View Description Hide DescriptionThe development of an efficient method for tracing rays and determining sound pressure levels in a moving medium is described. The medium is specified with horizontal and vertical variations in temperature and vectorial wind velocity. Reflections from a nonlevel ground surface of complex impedance are treated and an adaptive time step numerical integration is implemented for high accuracy. Two integration algorithms are compared for accuracy and computational efficiency. Recommendations are given for improved discretization of the medium and for accurate ray tracing through a ground reflection. This model is used to investigate the effect that terrain discontinuities, such as hills, have on atmospheric sound propagation. The importance of correct inclusion of the vectorial nature of wind is demonstrated.

On the validity of the heuristic ray‐trace‐based modification to the Weyl–Van der Pol formula
View Description Hide DescriptionThe Weyl–Van der Pol formula from electromagnetic wave propagation theory is known to give tolerable predictions of sound levels over a wide range of ground surfaces in a homogeneous atmosphere. A ray‐tracing technique is perhaps the simplest approach to extend the applicability of the Weyl–Van der Pol formula to the inhomogeneous situation. Essentially one uses ray theory to determine the path lengths and the angle of incidence at the ground. These modified parameters are then substituted into the Weyl–Van der Pol formula for the overall sound levels. This heuristic approach may be criticized for its lack of mathematical rigor even though it is physically reasonable. By using a ray theory approximation in the wave equation, a closed‐form solution has been obtained for the sound field over an impedance plane in the presence of a sound‐speed gradient. It has been shown that the heuristic formula represents an approximation of the solution. The theory also suggests that there is a strong interaction between the sound‐speed profile and the ground surface, in particular, when the impedance is high. The solution is limited to a single reflection.

A split‐step Padé solution for the parabolic equation method
View Description Hide DescriptionA split‐step Padé solution is derived for the parabolic equation (PE) method. Higher‐order Padé approximations are used to reduce both numerical errors and asymptotic errors (e.g., phase errors due to wide‐angle propagation). This approach is approximately two orders of magnitude faster than solutions based on Padé approximations that account for asymptotic errors but not numerical errors. In contrast to the split‐step Fourier solution, which achieves similar efficiency for some problems, the split‐step Padé solution is valid for problems involving very wide propagation angles, large depth variations in the properties of the waveguide, and elasticocean bottoms. The split‐step Padé solution is practical for global‐scale problems.

Scattering enhancement by supersonic resonances of cylindrical shells
View Description Hide DescriptionOff‐beam backscattering peaks associated with supersonic resonances of a simply supported unstiffened shell are evaluated by means of a relatively unfamiliar powerful analytical technique conceived by H. Lamb [London Math. Soc. Proc. 32, 11–20 (1900)] and further developed by P. W. Smith, Jr. [J. Acoust. Soc. Am. 34, 640–646 (1962)]. Boundary conditions leading to a simple solution have been selected because they have only a minor effect on coupling between the fluid and supersonic resonances. Cross‐section peaks are effectively the same for the shear and compression families of shell resonances. Frequency dependence is associated only with a sin θ_{ i } factor, θ_{ i } being the angle of incidence measured from the cylindrical axis, associated with trace matching of the incident acoustic wave number and modal structural wave numbers. The level of sin θ_{ i } for shells in water varies by no more than 1 dB with frequency. Within this range, the target strength maximum contributed by a shell resonance is TS_{ r }=20 log(2L)−9.4 dB±0.6 dB, where 2L is the shell length. Elastic parameters only affect the range of angles of incidence θ_{ i } matched to the resonant mode’s configuration. The level is independent of the cylinder radius, though of course resonance frequencies are not.

Experimental investigation of sediment effect on acoustic wave propagation in the shallow ocean
View Description Hide DescriptionIn shallow water the sediment layers have very strong effects on the propagation of acoustic waves. An effort to study the effects of the sediment has been made using 50‐ to 600‐Hz continuous waveacoustic propagation data taken by Carey [‘‘Experimental verification and application of bottom shear modulus profile (BSMP) method,’’ Oceans ’91 Proceedings (1991)] at the Atlantic margin coring project (AMCOR) borehole 6010 off the coast of New Jersey combined with sediment properties measured at that site by Yamamoto et al. of the University of Miami Geoacoustic Laboratory using the bottom shear modulus profiler (BSMP) method. Excellent agreement was found between the model and data indicating the acceptability of BSMP sediment values as input for acoustic propagation studies. The introduction of shear or tangential stresses in the model was found to have no effect upon which modes propagated but only on their modal intensity. The higher the order of the mode the greater the penetration in the seafloor and the stronger the shear effect on the intensity. A sub‐seafloor acoustic waveguide was investigated for the site and found to give possible explanation for enhanced intensity of modes that propagate strongly within that region. The intrinsic attenuation was determined using a method of matching modal intensity from model calculations with measured data. Biot theory was utilized and depth‐dependent intrinsic sediment attenuation profiles were found for seven frequencies between 50 and 600 Hz.
The depth‐averaged attenuation for the first mode at each frequency and the first mode that penetrates to 100 m was found. The frequency dependence of the mode attenuation was determined. The attenuation values agreed well with previous experimental acoustic reflection data taken by Mitchell and Focke [J. Acoust. Soc. Am. 67, 1582–1589 (1980)] but were much lower than values for attenuation in this frequency range determined by Hamilton’s [Geophysics 36, 266–284 (1971)] extrapolation of high‐frequency laboratory and field measurements to the ≤1‐kHz range. The effect of interface and volume scattering on transmission loss was determined using the acoustic model and a thin layer representation of the scatteringinterface. The result showed scattering to significantly contribute to energy loss beyond a kilometer or so and to have increased significance as frequency is increased.

Scattering by objects buried in underwater sediments: Theory and experiment
View Description Hide DescriptionThe scattering of sound by objects buried in underwater sediments is studied in the context of an exactly soluble model. The model consists of two fluid half‐spaces separated by a planar, fluid, transition layer of arbitrary thickness. Attenuation is included in any of these regions by using complex wave numbers. A directional source field, generated in the upper half‐space by a continuous line array, insonifies an object placed in the lower half‐space. The scattered field detected by another line array placed anywhere in the system may be calculated. The solution is determined from the T matrix for the bounded scatteringsystem and is exact (in linear acoustics) to all orders of multiple scattering among the interfaces and object. Numerical results are presented to investigate the effect of the local acoustic environment on the free‐field, in‐water scattering resonances of thin spherical shells. The field scattered by a shallowly buried object is discussed with emphasis on the importance of evanescent wavescattering in detection from above the sediment over an extended range. An initial set of experiments meant to verify the model are described. Results are presented and discussed for the measured scattering response of buried, spherical, evacuated, steel shells, that are 2.25% and 11% of the outer radius in thickness.

Acoustic scattering from elemental Arctic ice features: Numerical modeling results
View Description Hide DescriptionIn this paper acoustic scattering from Arctic ice is considered. No analytic scattering theories are able to explain the observed loss at low frequency (10–100 Hz) in long‐range propagation experiments. A finite difference method is used to solve the heterogeneous elastodynamic equations in two dimensions; this technique permits arbitrary roughness, unrestricted in slope, displacement, or radius of curvature and provides direct, physical insight into the rough icescattering mechanism. Broadband numerical scattering simulations are conducted on pressure ridges. The specular loss due to a ridge is affected by three parameters: cross‐sectional area or mass of the ridge, excitation of plate waves, and a material‐dependent power law. The first two affect the magnitude of the loss, while the last affects the frequency dependence. Multi‐year ridges are completely frozen and are best modeled as elastic structures yielding a loss frequency dependence of ≊f ^{9/2}. Observed loss in field data, with a frequency dependence of ≊f ^{3/2}, is not explained by scatter from multi‐year ridges. In contrast, young pressure ridges are modeled as fluid structures since they are loose aggregations of ice blocks and cannot support shear strain. Scatter from fluid ridges has a loss frequency dependence of ≊f ^{3/2} and yields a good match to the observed frequency dependence in field data. These results suggest that observed long‐range propagation loss is best explained by scatter from large, young pressure ridges.

Broadband source localization and signature estimation
View Description Hide DescriptionThe temporal signature and source location of a broadband impulse propagating in an oceanwaveguide are estimated using matched mode processing and illustrated using simulated data in an Arctic sound channel. Matched mode processing provides a simple method for broadband signal detection and localization; the peak‐to‐sidelobe ratio is significantly improved using broadband data compared with the narrow‐band case. The range and depth ambiguity function evaluated at the source location yields directly the sourcespectrum, which is inverse Fourier transformed to estimate the original signal waveform. For the simulated data, the reconstructed signal, compensated for spreading loss and array signal gain, agrees rather well with the original waveform. The estimated spectrum/waveform is weaker than the original spectrum/waveform due predominantly to the fact that the Arctic wave guide is lossy. A method for coherent broadband processing is proposed to improve signal localization and detection when the original sourcespectrum is known. Multiplying (correlating) the original spectrum with the range and depth ambiguity function, the product can be coherently summed over the frequency band of the signal. Near theoretical processing gain is achieved for the simulated data. Broadband signal detection is proposed using a frequency versus depth plot in addition to the commonly used frequency versus bearing plot.

Normal mode wave‐number estimation using a towed array
View Description Hide DescriptionIn the analysis of the null‐steering technique of normal mode filtering presented by the authors previously [H. M. Chouhan and G. V. Anand, J. Acoust. Soc. Am. 8 9, 735–744 (1991)], it was assumed that the modal wave numbers k _{ m } are known a p r i o r i. In practice, the waveguide parameters upon which the modal wave numbers depend are normally not known and are difficult to measure accurately. In order to use the null‐steering technique of mode filtering, or in applications such as source localization and waveguide characterization, it is necessary to estimate the modal wave numbers. An estimation procedure using a towed horizontal line array is presented in this article. The method entails eigendecomposition of the range‐averaged array signal correlation matrix. Range‐averaging effectively decorrelates the normal mode signals and enables the use of high‐resolution spectral estimation techniques such as MUSIC for estimating the modal wave numbers. Simulation results for the Pekeris model of the ocean are presented.

A two‐way parabolic equation for elastic media
View Description Hide DescriptionThe parabolic equation (PE) method is extended to handle wave propagation and scattering in range‐dependent elastic waveguides, which are approximated by a sequence of range‐independent regions. The two‐way elastic PE involves an efficient PE‐based scattering method for computing reflected and transmitted fields at the vertical interfaces separating range‐independent regions. The one‐way elastic PE is used to propagate the outgoing and incoming fields through the range‐independent regions with two‐way range marching. In addition to providing the backscattered field, the two‐way PE provides a correction to the outgoing field. The self‐starter is generalized to handle a source in a solid layer.

Wave solutions in three‐dimensional ocean environments
View Description Hide DescriptionA three‐dimensional wavesolution is developed for variable depth but isovelocity environments by transforming the wave equation into a suitable coordinate system. These solutions can be compared with existing known solutions derived from a ray point of view where the vertical modes propagate like horizontal rays. A general approach for arbitrary profiles is used to give explicit analytical solutions for various particular topographies including troughs and ridges. The behavior of these solutions is well known since they all have analogs in ordinary two‐dimensional refracting media. Because this approach does not invoke the adiabatic approximation or ray invariants but evidently has similar limitations it throws some light on the limits of validity of these alternative approaches. In passing, some interesting problems in normalization are encountered, and investigations into slope and curvature restrictions reveal some general relations between average slope, curvature or higher derivatives for realistic surfaces. Numerical modelers, who currently only have the ocean wedge as a 3‐D benchmark solution, now have an extended repertoire of benchmarks.

Broadband matched‐field processing of transient signals in shallow water
View Description Hide DescriptionRange and depth source localization in shallow water amounts to the estimation of the normal‐mode structure of the acoustic field. As ‘‘seen’’ by a vertical array, and from a modeling point of view, the normal‐mode structure appears as a set of nonplane coherent waves closely spaced at a vertical angle. This paper presents a full‐wave‐field narrow‐band high‐resolution technique that uses the spectral decomposition of the sample covariance matrix to resolve the vertical arrival structure of the harmonic acoustic field. The broadband processor is obtained by weighted averaging of the narrow‐band range‐depth ambiguity estimates within the source signal frequency band. Results obtained on synthetic data show that its performance is always better than or equal to that of the generalized minimum variance processor, which itself largely outperforms the conventional matched‐field processor. It is shown, using both simulated and experimental data, that the effect of the broadband processor is to increase the stability of the source location estimate. Results obtained with this processor on short transient pulses collected during the North Elba’89 experiment with a 62‐m‐aperture vertical array, showed stable and accurate localizations over long time intervals. It is also shown that the sound field, received over a given frequency band, is relatively stable over time and is in agreement with the predictions given by a standard normal‐mode propagation model.

Applications of optimal time‐domain beamforming
View Description Hide DescriptionApplications to ocean acoustic data from a towed array and to speech processing are presented for an improved optimal time‐domain beamformer, which involves optimizing over all possible source bearings and time series for multiple sources using simulated annealing. The convergence of the parameter search is accelerated by accepting time series perturbations only when the energy decreases. A comparison with the conventional delay‐and‐sum beamformer illustrates that the optimal beamformer handles larger receiver spacing and larger source‐to‐receiver ratio. Periodic ambiguities are essentially eliminated by using irregular receiver spacing and the improved search algorithm. Weak sources are handled with fractional beamforming.Noise cancellation is possible if the parameters of the noise are included in the search space. Two‐dimensional localization is performed for nearby sources.

Stable marching schemes based on elliptic models of wave propagation
View Description Hide DescriptionA marching numerical scheme is applied to a far‐field elliptic model of underwater wave propagation. General stability conditions are derived for the scheme in the case of varying parameter functions. Several examples are presented to demonstrate the capacity of the method to detect backscattered energy. In these examples the elliptic equation is cast as an initial value problem with assumed correct initial data. The ill‐posed nature of initial value problems for elliptic problems is discussed.

An experiment on matched‐field acoustic tomography with continuous wave signals in the Norway Sea
View Description Hide DescriptionAn acoustic tomography experiment in the Norway Sea with continuous wave (cw) signals is described. Measurements were made using a point source and a vertical 560‐m‐long array 55 and 105 km apart. The tomography scheme was based on matched‐field and matched‐mode principles and consisted in ‘‘fitting’’ the media to the measuredacoustic signal at the array. The sound‐speed profiles were described at some points by empirical orthogonal functions and linear interpolation between these points was applied. Two‐ and six‐parameter descriptions of the medium were in fact used. The results of tomography reconstruction of the sound‐speed field agreed well with the spot measurements and demonstrated the feasibility of the tomography schemes applied.

The use of inhomogeneous waves in the reflection–transmission problem at a plane interface between two anisotropic media
View Description Hide DescriptionA mathematical form is proposed for the study of the problem of reflection–transmission of monochromatic ultrasonic plane waves at the interface between two arbitrary anisotropic semi‐infinite media. The method used leads to a complete determination of all characteristics of the studied waves for all possible configurations, i.e., their directions of propagation, polarizations, and magnitudes. In particular, cases are taken into account where the waves generated by the existence of the interface are evanescent or more generally inhomogeneous. Applying this method to the calculation of several practical cases typically encountered in ultrasonicnondestructive evaluation fields, this paper points out the existence of phenomena that cannot be interpreted by classical methods usually used for the resolution of this kind of problem.

On the crossing points of Lamb wave velocity dispersion curves
View Description Hide DescriptionStandard dispersion curves relating the phase velocity of Lamb waves to the frequency‐plate thickness parameter fd indicate that in some cases symmetrical and antisymmetrical mode velocity curves cross each other. Viktorov’s equations are used to show that the crossing points are points where the dispersion curves are discontinuous so that no distinct Lamb mode exists for these particular velocity‐fd combinations. Procedures are given to predict how many such points exist and where they are located in a given range of fd.