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Volume 91, Issue 4, April 1992

On the asymptotic solution to the integral equations corresponding to the scattering by metallic disks
View Description Hide DescriptionFor the steady‐state problem of the scattering of plane electromagnetic waves by metallic disks, the leading order term of the high‐frequency asymptotic expansion of the surface current density is found. For the far‐field coefficient, good agreement is found with numerical results for both amplitude and phase.

Three‐dimensional source field modeling by self‐consistent Gaussian beam superposition
View Description Hide DescriptionA self‐consistent superposition of Gaussians on a discretized (configuration)‐(wave number) phase space lattice has recently been applied to two‐dimensional wave propagation into an elastic solid [Felsen e t a l., J. Acoust. Soc. Am. 8 9, 63 (1991)]. It is here generalized to three dimensions by modeling the radiation from planar finitely extended two‐dimensional source distributions into an elastic solid. A distribution of forces over the source aperture is expanded self‐consistently into Gaussian basis elements, which are then propagated into the unbounded medium. Numerical results are presented for simple smoothly tapered and abruptly truncated source profiles. As for the two‐dimensional case, the validity of complex‐source‐point modeling of the Gaussians is explored by comparing the fields obtained from them with those generated by an independent numerical reference solution. Moreover, it is demonstrated how different self‐consistent choices of beams affect the convergence of the Gaussian series expansion.

Source signature and elastic waves in a half‐space under a semicircular source of a nonuniformly spatially distributed impulsive load
View Description Hide DescriptionThe half‐space elastic response to a source of a nonuniformly spatially distributed impulsive load is presented in this paper. Specifically, an infinite semicircular canyon of a finite radius is embedded in the surface of the half‐space where an impulsive load is radially distributed over the entire surface of the canyon. The spatial distribution of this load is such that the load is maximum at the axis of symmetry of the half‐space and then descends smoothly to zero at the stress‐free surface of the half‐space. In an earlier work, the author gave an analytical‐numerical method for transient multicurvilinear dimensional boundary‐value problems and its employment to the present problem. Here, the resultant transient deformation of the half‐space is described and interpreted. In particular, a detailed discussion is devoted to the appearance of spatially stationary strong discontinuity fronts in the interior of the deformed half‐space. These fronts, which disclose the nature of the prescribed source of disturbance, are called here the source signature. A complete account is then given of the waves emitted from the source signature and those generated by boundary conditions, all revealing themselves explicitly by the solution method.

Sound scattering by a cylindrical shell reinforced by lengthwise ribs and walls
View Description Hide DescriptionAn analytical solution is derived for the acoustic response of submerged thin‐walled ring cylindrical shell containing lengthwise stiffening members: internal stringers and walls. On the basis of the analysis of the acoustic pressure versus time diagrams the stiffener‐borne wave‐generation mechanisms are traced. Shown is that the shell/stiffener junctions act as additional entry and exit points of circumferential waves circulating in the shell and the fluid. The stiffening members cause transformations of circumferential waves from one propagation type to another.

Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions
View Description Hide DescriptionThe method of fundamental solutions (MFS) is a boundary method for the numerical solution of certain elliptic boundary value problems. In the MFS, the approximate solution is a linear combination of fundamental solutions of the governing partial differential equation, with singularities placed outside the domain of the problem. In the present paper, the MFS is applied to acoustic scattering in fluids. The singularities are allowed to move during the solution process from arbitrary locations to more optimal locations. Numerical results demonstrate that the ‘‘fictitious eigenfrequency’’ difficulty encountered with the boundary element method(BEM) is not present in the MFS. In addition, MFS results obtained by the use of fixed singularities are presented for scattering of waves in elastic solids.

A method to overcome computational difficulties in the exterior acoustics problem
View Description Hide DescriptionWhen the boundary integralequation method is applied to exterior acoustics problems, singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem. These frequencies are referred to as ‘‘forbidden,’’ and various methods have been devised to overcome the computational difficulties presented at these frequencies. The work presented here is an extension to the CHIEF method in that higher derivatives, in addition to the function itself, are constrained to be zero at selected points in the interior domain. Whereas the relative success of either method depends on the quantity and selection of interior points, the SuperCHIEF method requires fewer interior points and is less sensitive to point selection, resulting in improved robustness without a significant increase in computational complexity.

Midfrequency enhancement of the backscattering of tone bursts by thin spherical shells
View Description Hide DescriptionA broad enhancement of the form function f(k a) for steady‐state backscattering by thin spherical shells in the midfrequency range has been noted by several investigators. Consequences of this enhancement on the backscattering of tone bursts are investigated with a Fourier synthesis of the temporal response from the exact f(k a). The emphasis is on incident bursts that are sufficiently short so that scattering consists primarily of distinct echoes associated with the specular reflection and different circumnavigations of a dominant surface guided wave. That wave lies on the subsonic branch of the lowest antisymmetric Lamb mode for a fluid‐loaded shell designated as a _{0} _{−} by some authors. A ray method, previously verified for echo amplitudes from leaky (or supersonic) Lamb waves on thick shells [S. Kargl and P. Marston, J. Acoust. Soc. Am. 8 5, 1014–1028 (1989)], is generalized to subsonic waves. The ray method well approximates echo amplitudes from the Fourier synthesis provided the incident burst is sufficiently long that effects of dispersion are weak. Over a broad frequency range, the amplitude of the earliest a _{0} _{−}echo is enhanced relative to the specular echo. For the 2.5% thick stainless steel shell in water considered, the enhancement factor peaks near k a≊46 where the amplitude ratio ≊3.1. The ray theory suggests that the peak ratio depends only weakly on shell thickness and material parameters provided certain conditions hold. The a _{0} _{−} wave is also found to contribute a prominent wave packet to the far‐field impulse response of the shell. The radiation damping and velocity of relevant surface waves were computed from the complete elastic equations and those results may be helpful for testing thin shell approximations near the coincidence frequency.

Equivalent boundary conditions for thin orthotropic layer between two solids: Reflection, refraction, and interface waves
View Description Hide DescriptionBoundary conditions for an interface between two solids are introduced to model a thin orthotropic interface layer. The plane of symmetry of the layer material coincides with the incidence plane. Boundary conditions relating stresses and displacements on both sides of the interface are obtained from an asymptotic representation of the three‐dimensional solutions for an interface layer whose thickness is small compared to the wavelength. The results for anisotropicboundary conditions are a generalization of our previous results [S. I. Rokhlin and Y. J. Wang, J. Acoust. Soc. Am. 8 9, 505–515 (1991)] for an isotropic viscoelastic layer. The interface boundary conditions obtained contain interface stiffness 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 reflection. As in the isotropic case, fundamental boundary‐layer conditions are introduced containing only one transverse or normal mass or stiffness. It is shown that the solution for more accurate interface boundary conditions, which include two inertia elements and two stiffness elements, can be decomposed into a sum of fundamental solutions. Interface waves along such an interface are considered. Characteristic equations for these waves are obtained in closed form for different types of approximate boundary conditions and the velocities calculated from them are compared to the exact solution. It is shown that retention of the terms describing coupling between normal and transverse stresses and displacements is essential for calculating the velocity of an antisymmetric interface wave.

A computationally efficient representation for propagation of elastic waves in anisotropic solids
View Description Hide DescriptionA new closed‐form representation is developed for the exact solution of the Christoffel equation for wave propagation in solids. The new representation is numerically more efficient than the traditional representations based on the use of Fourier and Laplace transforms. Using the new representation, the retarded Green’s functions are derived for an infinite anisotropic solid and an anisotropic half‐space. The method is applied to calculate the elastic‐wave response of an anisotropic cubic solid to highly localized delta function and step function type impulses. Both surface and bulk wave responses have been calculated. The effect of anisotropy is discussed by considering cubic solids with different anisotropy parameters. Interestingly, it is found that, for certain values of the anisotropy parameter, two distinct longitudinally polarized components can be observed to propagate along an axis of cubic symmetry. One of the signals is the normal longitudinal wave signal while the other results from the concave shape of the transverse slowness surface.

The effect of irregularity on the scattering of acoustic waves from a ribbed plate
View Description Hide DescriptionThe scattered field from an irregularly ribbed fluid‐loaded plate is ‘‘measured’’ using numerical simulations. The diffuse field scattered by the irregularity exhibits strong coherent features in the far‐field scattering pattern associated with phase matching to locally propagating Bloch waves on the structure. These features are also apparent in the scattering from finite periodic arrays, but are strongly enhanced by irregularity. Some examples of the behavior of the system are presented and the scaling of the cross section with the number of scatterers and irregularity is determined.

Modeling air‐to‐water sound transmission using standard numerical codes of underwater acoustics
View Description Hide DescriptionRecently, it was shown that a normal modeunderwater acoustic propagation code can be modified to model air‐to‐water sound transmission simply by altering the mode excitation coefficients [D. M. F. Chapman and P. D. Ward, J. Acoust. Soc. Am. 8 7, 601–618 (1990)]. Herein, it is shown that if the height of the source above the surface is sufficiently small, then the air‐to‐water transmission problem can be modeled by replacing the source in air with a source in water at a depth d≪λ below the surface, where λ is the acoustic wavelength in water. For a distant receiver in water, both the true source in air and the effective source in water exhibit nearly identical dipole radiation patterns, although the source strengths are different. The correct transmission loss from the source in air can be recovered by adding the quantity 20 log_{10}(k _{ a } d), where k _{ a } is the wave number in air. Using this approach, numerical results for three standard underwater acousticmodels(normal mode, multipath expansion, parabolic equation) are compared to benchmark results provided by the SAFARI model for a generic air‐to‐water transmission example.

Reflection and transmission coefficients in fluid‐saturated porous media
View Description Hide DescriptionUsing Biot’s theory to describe the propagation of elastic waves in a fluid‐saturated porous elasticsolid (a Biot medium), the reflection and transmission coefficients were computed at a plane interface between a fluid and a Biot medium and at interfaces inside a Biot medium defined by either a change in saturant fluids or in the intrinsic rock permeability. The reflection and transmission coefficients were computed with and without the inclusion of a frequency correction factor that according to Biot has to be introduced in the equations above a certain critical frequency (‘‘frequency‐dependent’’ versus ‘‘classic model’’). For a fluid–Biot medium interface and in the range 5 kHz–10 MHz for the example analyzed the two models show differences of the order of 11% for the reflection coefficients and between 11% and 31% for the type I, type II, and shear transmission coefficients. For the interfaces within a Biot medium, and for type II incident waves, in the same range of frequencies the cases examined showed differences in the reflection and transmission coefficients in the range 5%–80%. Because of the asymptotic properties of the frequency correction factor, the reflection and transmission coefficients coincide at very low and high frequencies. The analysis shows the importance of the inclusion of the frequency correction factor in analyzing wave propagation in Biot media for frequencies lying between the seismic and ultrasonic ranges.

Acoustically forced oscillations of air bubbles in water: Experimental results
View Description Hide DescriptionAn experimental technique for measuring the time‐varying response of an oscillating, acoustically levitated air bubble in water is developed. The bubble is levitated in a resonant cell driven in the (r,θ,z) mode of (1,0,1) at a frequency f _{ d }≊24 kHz. Linearly polarized laser light (Ar–I 488.0 nm) is scattered from the bubble, and the scattered intensity is measured with a suitable photodetector positioned at some known angle from the forward, subtending some solid acceptance angle. The output photodetector current, which is linearly proportional to the light intensity, is converted into a voltage, digitized, and then stored on a computer for analysis. For spherical bubbles, the scattered intensity I _{exp}(t) as a function of radius R and angle θ is calculated theoretically by solving the boundary value problem (Mie theory) for the water/bubble interface. The inverse transfer function R(I) is obtained by integrating over the solid angle centered at some constant θ. Using R(I) as a look‐up table, the radius versus time [R(t)] response is calculated from the measured intensity versus time [I _{exp}(R,t)]. Fourier and phase space analyses are applied to individual R(t) curves. Resonance response curves are also constructed from the R(t) curves for equilibrium radii ranging from 20 to 90 microns, and harmonic resonances are observed. Comparisons are made to a model for bubble oscillations developed by Prosperetti e t a l. [Prosperetti e t a l., J. Acoust. Soc. Am. 8 3, 502 (1988)]. Complex I _{exp}(t) behavior is also measured, with subharmonics and broadband noise apparent in the Fourier spectra. Possible explanations for this phenomenon are discussed, including shape oscillations and chaos.

Acoustic wave propagation in a fluid‐filled borehole drilled in an unbounded viscoelastic medium
View Description Hide DescriptionThe problem of acoustic wave propagation in a fluid‐filled borehole drilled in an unbounded viscoelastic medium excited by a point source on the axis is solved in the lowest order of approximation when the viscosity is much smaller than the elasticity. The fluid inside the borehole is assumed to be inviscid. Decay constants for various wave types are obtained. They are all proportional to ω^{2}. The full wavetrain of the Green’s function (the fundamental solution) is derived and the space‐time waveform plotted. The spectra of head waves are discussed in detail. The influence of the source spectrum on the full wavetrain is described only qualitatively. The spectrum of a full wavetrain observed by a long spaced logging tool in a production borehole is presented to show the general feature mentioned in the text.

Ray chaos in underwater acoustics
View Description Hide DescriptionGenerically, in range‐dependent environments, the acoustic waveequation cannot be solved by techniques which require that variables be separated. Under such conditions, the acoustic ray equations, which have Hamiltonian form, are nonintegrable. At least some ray trajectories are expected to exhibit chaotic motion, i.e., extreme sensitivity to initial and environmental conditions. These ideas are illustrated numerically using simple models of the ocean sound channel with weak periodic range dependence. The use of Poincaré sections, power spectra, and Lyapunov exponents to investigate and characterize ray chaos are discussed. The practical importance of chaotic ray trajectories—a limitation on one’s ability to make deterministic predictions using ray theory—is emphasized.

Acoustic ray chaos induced by mesoscale ocean structure
View Description Hide DescriptionIt has previously been shown that some acoustic ray trajectories in oceanmodels with periodic range dependence exhibit chaotic behavior, thereby imposing a limit on one’s ability to make deterministic predictions using ray theory. The objective of the work reported here is to quantify the limitations of ray theory to make predictions of underwater sound fields in the presence of realistic mesoscale structure. This is done by numerically investigating sound ray propagation in an analytically prescribed sound speedmodel consisting of Munk’s canonical profile perturbed by a randomly phased superposition of several baroclinic modes of the linearized quasigeostrophic potential vorticity equation. The ray equations used are consistent with the parabolic wave equation. To investigate ray chaos, power spectra are calculated and Lyapunov exponents are estimated. For realistic strengths of the mesoscale field, near‐axial ray trajectories are found to be chaotic with characteristic e‐folding distances (inverse Lyapunov exponents) of several hundred km. Steep rays, on the other hand, are found to be stable. This suggests that, in the presence of mesoscale structure, deterministic predictions of near‐axial underwater acoustic energy using ray theory are limited to ranges of one to two thousand km, while steep rays should be predictable over longer ranges.

Scintillation index of the acoustic field forward scattered by a rough surface for two‐ and three‐dimensional scattering geometries
View Description Hide DescriptionScintillation indices of acoustic waves forward scattered by a rough surface are obtained from numerical techniques leading to both accurate and approximate results in two‐dimensional (2‐D) and three‐dimensional (3‐D) geometries. The Kirchhoff approximation with a Gaussian spectrum and an empirical model for ocean surface height fluctuations is used. It is found that in the case of a Gaussian spectrum, the 3‐D accurate values obtained are lower than the 2‐D ones when the focusing–defocusing phenomenon does not occur. However when this phenomenon occurs, the 3‐D results become higher than the 2‐D ones. In the case of the empirical spectrum, both 2‐D and 3‐D accurate scintillation indices decrease with the wind speed on the ocean surface. The 2‐D model predicts larger values than the 3‐D model. When an ensonification factor or finite incident beam size is introduced, the scattering moves from the saturation limit into the focusing–defocusing range.

Fluctuations of high‐frequency shallow‐water seafloor reverberation
View Description Hide DescriptionHigh‐frequency shallow‐water reverberation statistics were measured from a seafloor that was covered by a thick layer of coarse shells. The reverberation statistics are presented as a function of frequency (20–180 kHz), grazing angle (8.8°–28°), and source beamwidths (1.2°–16°). In most cases the reverberation statistics did not follow a Rayleigh fading model. The model dependence of the reverberation statistics ranged from near Gaussian (high grazing angle) to beyond log normal (low grazing angle). The results suggest that small scale variations in bottom scattering properties and system beamwidth will significantly change the model dependence of the reverberation statistics.

Sea surface dipole sound source dependence on wave‐breaking variables
View Description Hide DescriptionEstimates of the surface dipole sound source spectrum per unit watersurface area were obtained from 40–4000 Hz and related to environmental parameters characterizing the wind and wave fields. The results give additional credence to the role of entrapped air bubbles and bubble clouds in the generation of acoustic ambient and indicate a change in the entrainment mechanism when the watersurface transitions from aerodynamically ‘‘smooth’’ to ‘‘rough.’’ The measured dipole sound source spectrum, when scaled by the gravity wave equilibrium range wave‐breaking rate of energy dissipation appears to be a self‐similar function dependent only on the acoustic frequency nondimensionalized by the frequency of peak acoustic spectral level.

Determining the extinction cross section of aggregating fish
View Description Hide DescriptionWhen fish are aggregated over a flat bottom, and fish and bottom echoes can be distinguished, it is possible to determine the fish extinction cross section by a simple application of the echo integration method. The theory for this is developed. Measurements at 38 kHz are presented for aggregations of the same 1983‐year class of herring over flat‐bottomed fjord areas in 1988, 1990, and 1991. The ratio of extinction and backscattering cross sections is found to lie in the approximate range from 1.2–2.3, depending on fish size and time of day.