Volume 89, Issue 2, February 1991

Analysis of boundary conditions for elastic wave interaction with an interface between two solids
View Description Hide DescriptionThe boundary conditions for an interface between two solids are analyzed to model a thin viscoelastic interface layer. Boundary conditions that relate stresses and displacements on both sides of the interface are obtained as an asymptotic representation of three‐dimensional solutions for an interface layer in the limit of small wavelength to thickness ratio. The interface boundary conditions obtained include interface stiffnesses and inertia and terms involving coupling between normal and tangential stresses and displacements. The applicability of such boundary conditions is analyzed by comparison with exact solutions for ultrasonic wave reflection. Fundamental boundary conditions are introduced where only one transverse or normal mass or stiffness is included. It is shown that the solution for more exact interface boundary conditions which include two inertia elements and two stiffness elements can be decomposed into a sum of fundamental solutions. The transition between welded and slip boundary conditions on an interface with a thin viscous layer is also analyzed as a function of interface thickness, viscousskin depth, and frequency.

Influence of voids in interface zones on Lamb wave propagation in composite plates
View Description Hide DescriptionAn attempt is made to develop a simple analytical model to study the influence of voids in the interface zones—leading to a weak adhesion—between the constituent laminas of a composite plate treating the thin adhesive layer as a linear elasticmaterial with voids (LEMV) [S. C. Cowin and J. W. Nunziato, J. Elasticity1 3, 125–147 (1983)]. The frequency equation for free harmonic wave motion in a sandwich plate with an LEMV adhesive layer at the core is derived and the dispersion curves for Al/adhesive/Al sandwich are compared for a thin adhesive layer core with those obtained from other available imperfect bond model studies.

A generalized diffraction tomography algorithm
View Description Hide DescriptionAvailable diffraction tomography algorithms are based on Fourier transform techniques and require either plane‐wave illumination in a uniform background medium or far‐field illumination combined with paraxial approximations. In this paper a generalized diffraction tomography algorithm is introduced that can handle both irregularly spaced measurement data, nonuniform background models, and general aquisition geometries. Using data from water tank experiments, the method’s ability to yield high‐quality reconstructions of geometry and velocity, as long as the weak‐scattering assumption is satisfied, is demonstrated.

An improved formalism for wave scattering from rough surfaces
View Description Hide DescriptionThe scattered field and its normal gradient obey a mutual linear relation at the scattering surface that is distinct from the physical boundary condition connecting either quantity to the incident radiation. On a moderately rough surface this relation can be represented by an operator whose series expansion in surface slope converges nicely even for large values of the Rayleigh parameter. This allows the Helmholtz integral for scattering amplitude to be written as a series of readily computable terms, one or two of which provide good approximations for surfaces too irregular for the usual Bragg expansion. This formulation reproduces the Bragg series for small surface elevations, and in the limit of low roughness wavenumber gives the Kirchhoff approximation with an explicit correction term in surface curvature. The usual results for a composite surface also emerge naturally. On several test profiles the method produces better overall accuracy than other multiscale approximations, at comparable efficiency.

Elastic wave propagation in fluid‐loaded multiaxial anisotropic media
View Description Hide DescriptionTheoretical investigations supported by extensive experimental comparisons are carried out on the interaction of ultrasonicwaves with multilayered media. It is assumed that each constituent of the plate can possess up to as low as monoclinic symmetry. The plate is assumed to be immersed in a fluid and subjected to incident acoustic waves at arbitrary angles from the normal as well as at arbitrary azimuthal angles. Reflection and transmission coefficients are derived from which all characteristic behavior of the system is identified. Solutions are obtained for the individual layers that relate the field variables at the upper and lower layer surfaces. The response of the total plate proceeds by satisfying appropriate interfacial conditions across the layers.

The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross‐sectional shape
View Description Hide DescriptionThe general Kirchhoff theory of sound propagation in a circular tube is shown to take a considerably simpler form in a regime that includes both narrow and wide tubes. For tube radii greater than r _{ w }=10^{−3} cm and sound frequencies f such that r _{ w } f ^{3/2}<10^{6} cm s^{−3/2}, the Kirchhoff solution reduces to the approximate solution suggested by Zwikker and Kosten. In this regime, viscosity and thermal conductivityeffects are treated separately, within complex density and complex compressibility functions. The sound pressure is essentially constant through each cross section, and the excess density and sound pressure (when scaled by the equilibrium density and pressure of air, respectively) are comparable in magnitude. These last two observations are assumed to apply to uniform tubes having a r b i t r a r y cross‐sectional shape, and a generalized theory of sound propagation in narrow and wide tubes is derived. The two‐dimensional wave equation that results can be used to describe the variation of either particle velocity or excess temperature over a cross section. Complex density and compressibility functions, propagation constants, and characteristic impedances may then be calculated. As an example, this procedure has been used to determine the propagation characteristics for a tube of rectangular cross section.

Signal pressure received by a hydrophone placed on a plate backed by a compliant baffle
View Description Hide DescriptionAn investigation is made of the signal reception of a hydrophone placed in front of an elastic plate backed by a compliant baffle layer. The baffle layer is the compliant tube array modeled by Junger [J. Acoust. Soc. Am. 7 8, 1010 (1985)] to represent a homogeneous, dispersive fluid layer. Noise reduction baffles such as compliant tubes are acoustically soft and thus tend to degrade the signal received. This paper describes the development of a theoretical model for evaluating the signal reception, and a comparison of theoretical results with experimental data is presented.

An eigenvector method to determine the transient response of cylindrical shells in a fluid with uniform axial flow
View Description Hide DescriptionA general time domain approach is presented to evaluate the vibratory response of cylindrical shells in time‐invariant axial flows to broadband time‐dependent excitations. The approach is based on the use of an i n v a c u oeigenvector expansion with time‐dependent coefficients for the velocity field of the shell. A set of convolution integral equations are developed for the eigenvelocities that are coupled due to the fluid loading. These equations are similiar to those developed for planar vibrators and are readily solved by marching forward in time. The known terms in the equations are the ρc fluid‐loaded admittances of each eigenvector and the radition impulse responses. Since these are easily obtained, a large number of eigenvectors can be included in the analysis. Numerical results illustrating the responses of a simply supported cylindrical shell to an impulsive ring excitation are presented.

Ray tracing for reconstructive tomography in the presence of object discontinuity boundaries: A comparative analysis of recursive schemes
View Description Hide DescriptionA comparative analysis of recursive ray tracing strategies for tomographic reconstruction from projections with diffracting sources is presented. One algorithm employs ray tracing in reprojection toward a correction of the true projection values for subsequent straight‐ray reconstruction. The other algorithm performs curved‐ray reconstruction along the retraced rays by algebraic methods for each recursive step. The performance for reconstruction of objects exhibiting discontinuity boundaries is studied. Within the geometrical optics approximation, the forward process is also shown to lead to multiple linking refracted rays. In an object of low velocity, there may be ranges of receive positions over which no linking refracted rays exist. Geometrically diffracted rays are introduced to describe the signal actually received over the ‘‘forbidden’’ range.

Noncollinear interaction of a tone with noise
View Description Hide DescriptionAn experiment was performed to investigate the noncollinear interaction of a high‐frequency tone with low‐frequency noise. The nonlinear interaction took place in an air‐filled rectangular duct. A low‐frequency band of noise was transmitted in the (0,0) mode together with a high‐frequency tone in the (1,0) mode. The angle between the two interacting waves was 55°. Measurements of the frequency spectra were made at locations that were well within the shock formation distance of either primary wave. The spectral shapes of the sum and difference frequency sidebands of noise are scalloped in appearance, and the number of nulls in each sideband increases with range from the source. Theoretical predictions for the nonlinearly generated sidebands are in good overall agreement with experiment. The predictions are based on a quasilinear analysis developed previously to describe the noncollinear interaction of two tones in a rectangular duct [M. F. Hamilton and J. A. TenCate, J. Acoust. Soc. Am. 8 1, 1703–1712 (1987)]. Comparisons are made with the corresponding case of collinear interaction, in which both primary waves propagate in the (0,0) mode.

Finite‐element solution of the inverse problem in bubble swarm acoustics
View Description Hide DescriptionThe bubble population near the ocean surface is of considerable interest. This population affects surface scattering strength, propagation near the surface, and the exchange of gases between the atmosphere and the sea. Both optical and acoustical means have been used to measure the bubble population with varying degrees of success. The acoustic method requires measurements at multiple frequencies and their subsequent conversion to bubble densities through either the resonancetheory approximation or numerical solution of the resulting integral equation. In this paper, a numerical solution to the integral equation is obtained using the method of weighted residuals with linear B splines as basis functions. A regularization technique is employed to stabilize the solution. A number of plausible bubble distribution functions are generated along with their acoustic properties to test the robustness of the technique. This approach is shown to yield very accurate bubble distributions from high‐quality attenuation data.

Continuous wave phase detection for probing nonlinear elastic wave interactions in rocks
View Description Hide DescriptionA new method that uses nonlinear elastic wave generation to produce a continuous wave (cw) phase measurement from which dimensions or velocities of a body can be obtained is described. Like the technique of standing waveresonance for obtaining sound velocities, this method takes advantage of the high accuracy characteristic of frequency measurements. In the experiment, two intersecting, high‐frequency primary signals f _{1} and f _{2} are mixed inside a sample, creating a directional beam at the difference frequency Δf=f _{1}−f _{2}. An externally generated, low‐pass‐filtered Δf signal is electronically mixed with the signal obtained from the sample. As either primary frequency is swept, the dc component from the mixer varies between relative maximum and minimum values at characteristic frequency intervals depending on the phase differences. The resulting interference signal can be used to calculate the distance from the mixing volume in the sample to the detector and to the two primary signal transmitters, providing that a single characteristic distance and wavevelocities are known. The reverse experiment is determining velocities from known dimensions.

Coherent propagation of sound in correlated distributions of resonant spherical scatterers
View Description Hide DescriptionExpressions for underwater propagation in bubble regions based on the index of refraction (η) in uncorrelated random distributions of monopole resonators are restricted to sparse bubble packing (very small volume fraction w). As w increases, correlations arise, and coupling with higher‐order multipoles is not necessarily negligible. To provide prototypes for data inversion, integral equations are analyzed for η in correlated distributions of spheres (including up to quadrupole coefficients) in terms of shell and moment expansions of the Percus–Yevick correlation function. Graphical results for w up to 20% indicate the decrease in magnitudes at resonance, the increase in resonance frequency, and the broadening of the resonance region relative to the uncorrelated case. A simple explicit three‐moment approximation is derived for distributions of monopoles plus dipoles that provides good accord with machine computations (based on ten‐moment or on eight‐shell expansions) to about w=7.5%, and also holds at least qualitatively for larger w. The explicit form may also be used with the first three moments of other correlation functions.

Echoes from vertically striated subresonant bubble clouds: A model for ocean surface reverberation
View Description Hide DescriptionA surface reverberation model for acoustic frequencies below several kHz is proposed based on weak scatter from inhomogeneities whose geometry is descriptive of recent ocean observation [e.g., Farmer and Vagle, J. Acoust. Soc. Am. 8 6, 1897 (1989)]. Scatterers in this model are vertical cylinders of elliptical cross section representing either filamentary‐ or sheetlike subresonant microbubble clouds whose population decreases exponentially with depth. This geometry approximates intermediate‐aged fossils of breaking waves and/or convective processes. Born approximation (weak scatter) results from this model show substantial agreement with observed surface backscatter cross sections as a function of wind speed, grazing angle, and acoustic frequency in the range 0.2–20 kHz. It is demonstrated that almost all the high‐frequency weak backscatter in the model is specular reflection from surfaces of volume scatterers. Some preliminary speculations involving Langmuir circulation are offered for the application of this model to surface reverberation at frequencies below a few hundred Hz, where data are scarce.

Simulations of rough interface scattering
View Description Hide DescriptionThis paper describes numerical simulations of rough interface scattering. Both Dirichlet and fluid–solid boundary conditions are treated. The Rayleigh–Fourier method is used to compute exact plane‐wave scattering amplitudes and results are compared to various approximations. The small‐slope approximation of Voronovich performs remarkably well, while the tangent‐plane approximation is shown to miss some essential physics of scattering. It is observed that at the Rayleigh angle there is a peak in the backscattering amplitude, even for plane‐wave incidence. It is argued that the statistics of plane‐wave scattering amplitudes are Gaussian.

Acoustic propagation through baffles with rectangular compliant tubes
View Description Hide DescriptionCompliant tube baffles have been designed as effective barriers for acoustic propagation underwater over wide bands of frequencies. Acoustic energy reflection occurs due to compliant cross‐sectional resonances in the tubes that exhibit significant volume deformation. Additional bandwidth is achieved with multiple layers of compliant tubes tuned to different frequencies. The design challenge is to provide sufficient baffle insertion loss levels and bandwidth while limiting adverse transmission peaks associated with interactions between noncompliant tube resonances and, tube and fluid layer system resonances. This paper discusses the acoustic performance of baffles with compliant tubes of rectangular cross section as opposed to the conventional designs with elliptical cross section. The tubes consist of flat face plates held apart along their edges by spacers. The knifelike edge support constitutes a simple support boundary condition for bending deformation of the face plate. This compliant tube configuration allows added design flexibility and degrees of freedom in minimizing the adverse effects of transmission peaks in a baffle’s insertion loss. Unequal face plate thicknesses and varied mass of the edge spacer have been effectively utilized in this regard.

Modeling the acoustic scattering by under‐ice‐ridge keels
View Description Hide DescriptionIn this paper, the exact solution of scattering off a flat surface with a single semielliptical cylindrical boss of infinite extent is developed. The cylindrical boss approximates the geometric shape of an ice‐ridge keel. A far‐field approximation and the results to a grid of randomly distributed scatterers are applied. First, the case where the simulated keels have a single, constant depth is examined. In the frequency range below 200 Hz, energy is scattered mostly into steep angles, where it is eventually lost to bottom absorption. Above 200 Hz, a significant fraction of the scattered energy is directed into shallow angles, where it becomes a propagating incoherent component. Statistics are developed that describe a Rayleigh distribution of keel depths. A formula is developed that relates the standard deviation of surface roughness to mean keel depth and mean ice‐ridge spacing. Then the model is applied to find that for a given ice roughness, a Rayleigh distribution yields less coherent reflection loss, and slightly less scattering loss, than a field of constant keel depths.

Modeling high‐frequency vertical directional spectra
View Description Hide DescriptionA measurement of the acoustic ambient arriving from a horizontal direction along with total sound‐pressure‐levelspectrum allows one to infer directional spectra and some physical characteristics of sea surface‐generated sound. A 1‐year measurement of these two quantities was made at high frequency, i.e., 8–32 kHz, in The Tongue of the Ocean, The Bahamas. The horizontally directed ambient was measured using vertically oriented line arrays (broadside beam only available) and was observed at wind speeds ranging from 1–30 kn. The resulting database was used to estimate the statistics of anisotropic “noise gain.” The measured statistics showed two distinct cases: whitecaps present and not present. The relationship of these results to the total vertical directional spectrum is investigated by comparisons with a model that includes spatially diffuse near‐surface sources, media scattering, and thermal noise. A comparison of the model and experiment results quantified an “effective” thickness of an acoustically active sublayer of monopole sources and a frequency‐dependent media smearing of the source‐to‐receiver ray paths.

Stabilized high‐resolution beamforming with horizontal arrays: Two experimental trials in shallow water
View Description Hide DescriptionHigh‐resolution algorithms are potentially able to outperform conventional linear methods at the task of beamforming, i.e., spatial‐spectrum estimation, and in controlled experiments or simple computer simulations they often do prove superior. In marine environments, however, high‐resolution techniques frequently fall short of these expectations. Their outputs can take on an unstable appearance characterized by fading of the true target peak and by the occurrence of multiple false targets. In recent years a class of stabilization techniques has evolved to moderate these effects by suppressing contributions from the smaller eigenvalues of the cross‐spectral density matrix. While such techniques generally produce good results in simulations with the maximum likelihood estimator and even with the more sensitive maximum entropy estimator, they have seen little use on sea data. Here, results are presented of two sea trials of a simple eigenvalue stabilization procedure in a relatively demanding class of environments—shallow water. The technique provided what is considered to be marginal stabilization for the maximum likelihood estimator and none at all for maximum entropy.