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Volume 97, Issue 3, March 1995

Generalization of the k‐space formulation to elastodynamic scattering problems
View Description Hide DescriptionA generalized k‐space (GKS) formulation is presented for vectorial elastodynamic scattering problems. It represents a generalization of Bojarski’s scalar k‐space formulation. From the basic second‐order partial differential equation or its integral representation in the space‐frequency (r‐f) domain, a local equation is derived for the displacement field in the spectral‐frequency (k‐f) domain. This equation, together with the constitutive equation in the r‐f domain, reduces the original scattering problem into two simultaneous local equations with two unknowns (displacement field and the induced source), which are then solved by the conjugate‐gradient (CG) method. The connection between the k‐f domain and r‐f domain is obtained by the spatial fast Fourier transform (FFT) algorithm. The number of complex multiply‐add operations per CG iteration is O(N log_{2} N), and the storage requirement is only O(N), where N is the number of spatial discretization points. This is much more efficient than the conventional method of moment combined with the CG procedures which requires O(N ^{2}) operations per iteration and O(N ^{2}) storage. In the spectral‐time (k‐t) domain, however, it is found that four simultaneous local equations have to be used to formulate the k‐space algorithm because of the existence of two wave speeds. By virtue of the causality, a new local time‐stepping algorithm is derived with the aid of two temporal propagators, i.e., the compressional and shear propagators. The connection between the r‐t domain and k‐t domain is again obtained by the spatial FFT algorithm. Therefore, in each time step, the number of complex multiply–add operations is O(N log_{2} N), and the storage requirement is O(N). Most importantly, for the same accuracy, N can be much smaller than that for the conventional finite‐difference method.

Dispersion of elastic waves in random particulate composites
View Description Hide DescriptionElastic wave propagation in a discrete random medium is studied to predict dynamic effective properties of composite materials containing spherical inclusions. A self‐consistent method is proposed which is analogous to the well‐known coherent potential approximation in alloy physics. Three conditions are derived that should be satisfied by two effective elastic moduli and effective density. The derived self‐consistency conditions have the physical meaning that the scattering of a coherent wave by the constituents in the effective medium vanishes, on the average. The frequency‐dependent effective wave speed and coherent attenuation can be obtained by solving the self‐consistency conditions numerically. At the lowest resonance frequency, the phase speed increases rapidly and the attenuation reaches the maximum in the composites having a large density mismatch. The lowest resonance is caused mainly by the density mismatch between matrix and particles and higher resonances by the stiffness mismatch. The dispersion and attenuation of longitudinal and shear waves are affected by the lowest resonance much more than by higher ones. The lowest resonance frequency of particles in the effective medium is found to be higher than that of a single particle embedded in the matrix material of composites due to the stiffening effect. The results obtained from the present theory are shown to be in good agreement with previous experimental observations of Kinra et al. [Int. J. Solid Struct. 16, 301–312 (1980)]. Part of the calculated results are compared with those computed by the Waterman and Truell theory. The present theory is in better agreement with the experiments for the examples dealt with.

Energy branching of a subsonic flexural wave on a plate at an air–water interface. I. Observation of the wave field near the interface and near the plate
View Description Hide DescriptionThe radiation of subsonic flexural plate waves due to a discontinuity in fluid loading is experimentally investigated. A tone burst of flexural waves propagates down a plate, the lower section of which is submerged in water. Measurements indicate that there occurs a branching of energy as the flexural wave passes through the air–water interface, with little reflected energy. A portion of the transmitted energy continues along the plate as a subsonic flexural wave with an associated acoustic evanescent wave. A second acoustic wave (which is termed transition radiation) originates at or near where the plate crosses the interface, and propagates in water to the far field. In the near field of the interface there exists an interference between the two acoustic waves in water that results in a series of pressure nulls. The pressure nulls are associated with a π phase change in the wave field and are indicators of wavefront dislocations. A numerical computation of the wave field in an unbounded fluid due to a line‐moment excitation of a plate has similar features as the null pattern observed but differs in certain details.

Synthesized wave packet basis for monostatic scattering from a randomly ribbed, finite cylindrical shell
View Description Hide DescriptionWave packets are synthesized from scattering highlights in the signal Wigner distribution of the monostatic form function of a randomly ribbed, cylindrical shell in water. It is argued that they, like the acoustic rays derived from partial wave series representations of form functions for canonical geometries, form an approximate basis for the monostatic scattering from this more complex structure. It is shown that, despite the presence of a small amount of structural disorder in the rib spacings, rib excitations such as Bloch waves and Bragg reflections still occur and are significant contributions to the form function.

Transport of energy across a discontinuity by fluid loading
View Description Hide DescriptionShort‐circuiting effects due to fluid loading have been investigated for a plane flexural wave obliquely incident upon a line discontinuity in an elastic plate. General expressions for the transmission and reflection coefficients have been derived and exact analytical formulas are given for the required Green’s functions. The transmission coefficient resulting from the fluid loading has been evaluated and found to have a strong, step‐function‐like angular dependence vanishing at large angles of incidence. In addition, the near‐field pressure and velocity fields have been numerically computed to directly examine the energetics of the transmission process. Transmission due to the fluid has been found to arise from propagating acoustic fields that must carry energy a distance of the order of a flexural wavelength. Accordingly, for angles greater than the critical angle, sin θ_{ c }=k _{0}/k _{ s }, transmission due to the fluid is strongly suppressed due to phase cancellation along the line discontinuity, thus explaining the angular dependence of the transmission coefficient.

Determination of structural impedance from scattering data
View Description Hide DescriptionKnowledge of the structural impedance is significantly important in the computation of acoustic scattering and radiation because it contains all the information about the structure which is necessary for the evaluation of the scattered or radiated field in any fluid medium. The inverse problem of determining, or reconstructing, the structural impedance from scattering measurements is treated. An algorithm is developed for the reconstruction of the structural impedance from knowledge of scattered field for a set of incident plane wave directions. An intermediate step includes the reconstruction of the surfacepressure and normal velocity fields using acoustical holography. The resolution limit of the reconstruction is found in the case of the infinite cylinder and the sphere. The case of the sphere is generalizable to other finite shapes. Problems caused by noise in the inversion of the data are discussed. Alternative reconstructions of wet impedance and wet admittance which can be performed for any scattering data are defined.

Numerical simulation of elastic wave propagation in granular rock with the boundary integral equation method
View Description Hide DescriptionTo better understand the effects of the microstructural properties of granular rocks such as grain packing geometry, grain shape and size, grain contact properties, and grain scale heterogeneity on the velocities and amplitudes of seismic waves, a numerical approach for modelingelastic wave propagation in granular rock has been developed. The numerical approach employs the boundary integralequation method to model the complete dynamic response of a packing of arbitrarily shaped grains. The effects of dry, fluid‐filled, and clay‐filled grain contacts are incorporated into the numerical formulation using a discontinuity boundary condition with a complex stiffness parameter. Numerical simulations for SH‐wave propagation in an idealized grain packing illustrate the applicability of this technique for investigating the effects of grain contact properties and source frequency on the amplitudes and velocities of elastic waves.

Low‐frequency propagation modes in a liquid‐filled elastic tube waveguide
View Description Hide DescriptionAxisymmetric propagation in a liquid‐filled elastic tube waveguide is considered, with emphasis on the two modes existing down to zero frequency. Previous work by Del Grosso is used as the basis of the theoretical description of modal phase velocities and particle displacement profiles in such waveguides. It is shown that certain combinations of material properties can produce a mode which, in the zero frequency limit, has plane‐wave motion in the liquid. Two examples of waveguides with very different wall compliance, aluminum/water and PVC/water, are studied numerically and experimentally. Numerical calculations are used to show the frequency dependence of phase velocity in all waveguide modes and the radial dependence of complex particle displacement amplitude in the two low‐frequency modes (ET0 and ET1). Contrasting behavior in the two waveguides is seen—approximate plane‐wave motion in the liquid occurs in the ET0 mode of the aluminum/water waveguide, but in the ET1 mode of the PVC/water waveguide. Experimental measurements of the frequency dependence of phase velocity in the ET0 and ET1 modes of these waveguides are also presented. Good agreement with numerical predictions is obtained in both cases, although experimental difficulties more severely limit the frequency range of measurement in the PVC/water waveguide.

Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators
View Description Hide DescriptionThis paper examines dispersion characteristics of sound waves propagating in a tunnel with an array of Helmholtz resonators connected axially. Assuming plane waves over the tunnel’s cross section except a thin boundary layer, weakly dissipative effects due to the wall friction and the thermoviscous diffusivity of sound are taken into account. Sound propagation in such a spatially periodic structure may be termed ‘‘acoustic Bloch waves.’’ The dispersion relation derived exhibits peculiar characteristics marked by emergence of ‘‘stopping bands’’ in the frequency domain. The stopping bands inhibit selectively propagation of sound waves even if no dissipative effects are taken into account, and enhance the damping pronouncedly even in a dissipative case. The stopping bands result from the resonance with the resonators as side branches and also from the Bragg reflection by their periodic arrangements. In the ‘‘passing bands’’ outside of the stopping bands, the sound waves exhibit dispersion, though subjected intrinsically not only to the weak damping due to the dissipative effects but also to the weak dispersion due to the wall friction. Taking a plausible example, the dispersion relation and the Bloch wave functions for the pressure are displayed. Finally the validity of the continuum approximation for distribution of the Helmholtz resonators is discussed in terms of the dispersion relations.

An impulse response function for a fuzzy structure
View Description Hide DescriptionThe authors are employing some of their previous works in structural acoustics to define, investigate, and interpret the response behaviors of so called fuzzy structures. A formalism is developed to describe the response behavior of a master structure to which an ensemble of appendages are attached. The impulse response function for the master structure is, ad hoc, assumed to be proper and known. The appendages are assumed to be described by an impedance operator composed of individual impedances that are localized; each impedance is stated in terms of a singular position and an associated impedance. An impulse response function is then derived and is shown to be proper. The propriety of the impulse response function demands that it be dependent only on quantities and parameters that describe the appended master structure; it needs to be independent of the in situ response of the structure and the external drive that generates the response. Models of naturally interacting and artificially noninteracting appendages on a master structure yield impulse response functions that obey the Le Chatelier’s principle. A model based on the Born approximation violates this principle and, therefore, its use is highly limited. All three models are employed to explain the notion of a configuration for the disposition of an ensemble of appendages. A configuration is shown to require the specification of a two‐vector set comprising the positions and impedances of the appendages. The notion of an ensemble of configurations is then proposed to define deviations from a prime configuration. Configurational averages of the equation of motion of the appended master structure can then be performed exposing the averaged response behavior for this structure. An ensemble of appendages that can be assigned an ensemble of configurational deviations that are defined statistically is dubbed fuzzy (structural fuzzy). In turn, a statistical ensemble of configurations about a prime configuration of appendages on a master structure defines a fuzzily appended master structure.

Impulse measurements of impedance and propagation constant compared to rigid‐frame and dual‐wave predictions for foam
View Description Hide DescriptionAcoustic impulse techniques are used to obtain measurements of propagation constant in three different foams and to determine both the characteristic impedance and the normal impedance at the surface of a hard‐backed layer over the frequency range from 200 Hz to 10 kHz. A comparison is made of existing rigid‐frame and dual‐wave models of sound propagation in porous media, and predictions from both models are compared with experimental measurements. Both models give good agreement with the lower flow resistivityfoammeasurements, however, the dual‐wave model better predicts the overall results for a foam with a flow resistivity of 25100 N s m^{−4}, implying the need of the slow wave mode. The position of the maxima in the layer impedance results are particularly sensitive to the correct choice of tortuosity. An emphasis is placed on how the two models compare as the frame material becomes rigid and these ideas are used to consider the implication of a previous claim about the existence of a slow wave in foams.

Generation of harmonics in a focused Gaussian sound field
View Description Hide DescriptionA general solution of the lossy nonlinear KZK (Khokhlov–Zabolotskaya–Kuznetsov) equation is derived for any order harmonic of a Gaussian beam. The result shows that in a weakly focused Gaussian field all harmonics maintain a Gaussian profile in the axial direction at any distance. Moreover, the Gaussian coefficient of the nth harmonic is exactly n times as great as that of the fundamental; i.e., the beamwidth of the harmonics decreases as n ^{−1/2}.

Information paths and the determination of state relations from displacement velocity measurements of elastic rods
View Description Hide DescriptionNew methods are developed for determining the nonlinear compressional state relation between stress and strain for one‐dimensional elasticmaterials from measurements of displacement velocity. An iterative procedure is proposed when the early displacement velocities are provided at both ends of the sample. This procedure is rapidly convergent and simple to implement in the weakly nonlinear regime. Noniterative procedures are developed for displacement velocity data provided at an end and a location in the interior of the sample or for displacement velocity data provided at an end and at a fixed laboratory coordinate.

Boundary layer attenuation and acoustic streaming accompanying plane‐wave propagation in a tube
View Description Hide DescriptionThe viscous and thermal boundary layer attenuation, and the acoustic streaming that accompanies the lowest acoustic mode propagating down a long, wide circular tube are calculated asymptotically. By fully including the effects of compressibility and heat conduction, corrections are made to earlier work. The leading order acoustic wave solution is analyzed using matched asymptotic expansions. The matching gives the dispersion relation and from this a model propagation equation is proposed. These results agree with earlier ones derived quite differently. The acoustic streaming driven by the Reynolds stress in the boundary layer is analyzed next. The streaming velocities are calculated and the corrections to earlier expressions, arising from the effects of compressibility and heat conduction, are expressed in terms of the Prandtl number and the specific heat ratio. Further, a second‐order time‐independent temperature field caused by the passage of the wave is found.

The effects of an elastic solid surface layer on the radial pulsations of gas bubbles
View Description Hide DescriptionMost previous theoretical investigations of gas bubbledynamics have assumed an uncontaminated gas–liquid interface. Recently, however, the potential importance of layers of surface active agents on bubbledynamics has been increasingly recognized. In this work it is assumed that a continuous layer of incompressible, solid elastic material separates the gas from the bulk Newtonian liquid. Elasticity is modeled to include viscous damping. A Rayleigh–Plesset‐like equation describing the dynamics of such surface‐contaminated gas bubbles is derived. The equation predicts that the surface layer supports a strain that counters the Laplace pressure and thereby stabilizes the bubble against dissolution. An analytical solution to this equation which includes both the fundamental and second‐harmonic response is presented. The dispersion relation describing the propagation of linear pressure waves in liquids containing suspensions of these bubbles also is presented. It is found that (1) the resonance frequencies of individual bubbles tend to increase as the modulus of rigidity increases; (2) the damping provided by the viscosity of the shell dominates thermal effects for bubble radii less than ∼10 μm; (3) the attenuation coefficient in a bubbly liquid decreases as either the rigidity or the viscosity of the surface layer increases; (4) encapsulated bubbles with shell rigidity greater than ∼85 MPa provide a greater total scattering cross section per unit attenuation in the lower biomedical frequency range than do free bubbles of the equivalent size.

Influence of mean shear on sound produced by turbulent flow over surface slots
View Description Hide DescriptionAn analytical investigation is made of the sound produced by low Mach numberturbulent flow over a slot in a thin rigid plane and over an array of parallel slots in the plane. The plane separates the turbulent mean stream from nominally stagnant fluid. The turbulent wall pressures excite unstable oscillations of the mean shear layer in the slots, and linear perturbation theory is used to estimate the influence of these instabilities on sound generation. For both isolated slots and parallel slots, it is shown that the growth of shear layer disturbances across a slot can greatly increase the contribution from the slot trailing edge to the volume flux through the slot, and produce a corresponding increase in the net radiation relative to that predicted by the usual ‘‘diffraction theory,’’ which neglects the presence of the shear layer. Previous analyses of this problem have ignored the influence of mean shear, and their predictions at higher frequencies are typically 20–40 dB smaller than those given here. At very low frequencies, additional vorticity shed from the leading edge of a slot effectively blocks unsteady motion through the slot and predicted sound levels are then very much smaller than those of diffraction theory. An extended slotted surface can support subsonic surface waves that decay with distance into the fluid; these are shown to be unstable within a certain narrow range of frequencies, and may well be associated with the generation of tonal noise by the boundary layer.

Vortex sound theory: Direct proof of equivalence of ‘‘vortex force’’ and ‘‘vorticity‐alone’’ formulations
View Description Hide DescriptionIn the theory of vortex sound the far‐field compact source strengths are the volume integrals of the time differential of the vortex force=−ρ(ζΛu)’, [ρ=(constant) fluid density, u=flow velocity,ζ=∇Λu=vorticity, ( )’≡∂( )/∂t], for dipole sound and of y _{ x }(ζΛu)_{ x } ^{‘} for quadrupole sound (y=local coordinate, y _{ x }=y⋅x/x with x=far‐field observation point). Möhring’s method, using a vector Green’s function, produces instead the vorticity‐alone volume integrals of 1/2(yΛζ)‘ and of 1/3y _{ x }(yΛζ)_{ x } ^{‴}, respectively, that do not explicitly involve the velocity of the inviscid incompressible sound‐generating flow. For two‐dimensional flows, the corresponding integrands can be written as (ζΛu) and y _{ x }(ζΛu)_{ x } ^{’} in the former formulation, and (yΛζ)’ and 1/2y _{ x }(yΛζ)_{ x } ^{‘} in the latter. Direct proofs that these different integral results are in fact exactly equal to each other are offered: Given one form, the other can be found with no appeal to acoustical theory whatsoever. However, the integrands are not equal, and while −ρ(ζΛu)’ can be interpreted uniquely as the dipole strength per unit volume, this is not so for ρ(yΛζ)_{ x } ^{‘}. The implications of these integral formulations is discussed.

A derivation of three‐dimensional ray equations in ellipsoidal coordinates
View Description Hide DescriptionConventional three‐dimensional (3‐D) ray equations ignore the effect of Earth flattening, whereas this effect, as has been shown by Munk et al. [J. Phys. Oceanogr. 18, 1876–1898 (1988)], cannot be ignored for long‐range transmissions. To take into account the earth flattening, new 3‐D ray equations are derived, in this paper, in terms of the ellipsoidal coordinates: Geographic latitude, longitude, and depth. The new 3‐D equations account for both Earth curvature and Earth flattening, and thus are more accurate than those conventional 3‐D ray equations. It is shown that under certain circumstances, the new equations reduce to the horizontal ray equations of Munk et al., the Aki‐Richards 3‐D spherical ray equations, and the conventional two‐dimensional ray equations, respectively. The advantages of using the new equations in ocean acoustics are discussed. Numerical examples are presented.

Interpretation of Sea Beam backscatter data collected at the Laurentian fan off Nova Scotia using acoustic backscatter theory
View Description Hide DescriptionThe purpose is to use acoustic scatteringtheory and Sea Beam measurements to estimate seafloor roughness parameters. The Sea Beam backscatter data are from a test area in the Laurentian fan, a relatively flat region. Sidescan sonarlike images were reconstructed from the multibeam data. These images in the test area show two distinctly different types of areas (A) and (B). The backscatter model uses the Helmholtz–Kirchhoff formulation of scattering theory and correlation functionC(r)=exp[−‖r/l‖^{ n }], where r is the displacement, l is the correlation length, and n is the exponent. A single rough interface model fits the backscatter data in (A). The root‐mean‐square roughness was 6–8 cm and the correlation lengths were 140–270 cm. The exponent n ranged from 0.95 to 1.5. The type (B) areas required a two‐layer model: the interfaces in the first type of areas (A) is covered by a sediment having a smoother surface. The rms roughness of the covering sediments were about 3 cm, the exponent n was nearly 2 and correlation lengths l were 90–120 cm. These acoustic models are consistent with the geological setting and processes.

Modeling the spatial and frequency distribution of narrow‐band acoustic signals scattering from the ocean surface
View Description Hide DescriptionEnergy from a narrow‐band acoustic signal, interacting with a time‐varying ocean surface, will be scattered in the spatial and frequency domains. This scattering can be modeled using range‐independent normal‐mode theory and a coupling scheme based on first‐order perturbation theory. Simulations will be performed using a two‐dimensional surface wave spectrum.