Index of content:
Volume 91, Issue 5, May 1992

Asymptotic and numerical determinations of the complex eigenfrequencies and the real resonances of an acoustically insonified sphere
View Description Hide DescriptionVarious aspects of the scattering problem associated with the incidence of a plane sound wave on a sphere are studied. Various tables are numerically generated that exhibit the external eigenfrequencies of acoustically rigid, soft, and elastic spheres, as well as the related (electromagnetic) TM and TE modes of perfectly conducting spheres. Also generated are tables of the (complex) ‘‘composition’’ zeros of an elastic sphere in water and they are compared to the (real) physical natural resonances of this target. Some of these tables cover new expanded frequency ranges, and in some instances either correct some of our previous misprints/typos or are totally novel. An acoustical analog of the Bohr–Sommerfeld quantization rule of (the old) quantum theory is used to asymptotically evaluate the external eigenfrequencies of the sphere. The resonance condition that emerges from this phase‐matching principle is identical to the asymptotic expansion for the (complex) zeros of the spherical Hankel functions found by Olver in the fifties. This yields an asymptotic determination of the external eigenfrequencies of a sphere. This paper concludes with a study of the asymptotic s p a c i n g s between consecutive shape‐dependent zeros of the sphere, where a uniform value for one of these spacings is obtained (i.e., the one that results when the index n is kept fixed), which agrees with earlier numerical calculations of ‘‘background’’ curves. The zeros in these tables, which are analyzed here, are essential for the characterization of scatterers and for the solution of target identification problems. They are also indispensable for the determination of the dispersion plots for the phase (and group) velocities and attenuations of the surface waves that revolve around scatterers of any shape, and all these applications are described.

Application of the Sommerfeld–Watson transformation to scattering of acoustic waves obliquely incident upon cylindrical shells
View Description Hide DescriptionThe Sommerfeld–Watson transformation (SWT) is applied to the normal mode series for scattering of an obliquely incident plane acoustic wave from an infinite circularly cylindrical shell to express the scattered field in terms of its wave components. The ‘‘geometric’’ part of the scattered field is extracted and expressed as a line integral that is evaluated by shifting to a path of steepest descent (SDP). When deforming the path, one or more poles may be crossed and their residues must be accounted for, resulting in discontinuities in the computed field unless the effects of these poles are also included in the evaluation of the SDP integral. This correction, made in the analysis in this paper, is applied to the particular example of a thin steel shell in water, using thin‐shell theory. Very good agreement for scattering into the backward half‐space is obtained between the results of the normal mode series and the modified SWT calculation for frequency‐angle combinations such that k a cos φ≥2, where φ is the angle of incidence of the plane wave relative to the shell normal. Inclusion of the corrections to the solution also improves the calculation even when no poles are crossed by the path deformation.

Stable localized symmetric integral equation method for acoustic scattering problems
View Description Hide DescriptionAn energy‐based infinite boundary element integral equation method is developed for the solution of two‐ or three‐dimensional time harmonic fluid scattering problems. This method is essentially based on a domain decomposition that insures the validity for all frequencies, and uses a hypersingular operator that can be integrated readily by standard procedures for single layers. It leads to a set of sparse, symmetric discretized equations. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave confirm the stability of the new procedure, and serve to assess its accuracy for wave numbers ranging from 0 to 30, both directly on the scatterer and in the far field.

Dynamic internal response of fluid‐loaded multilayered anisotropic media
View Description Hide DescriptionTheoretical investigations are carried out on the interaction of ultrasonicwaves with multilayered plates separating a fluid from a vacuum, another fluid, or a solid half‐space. It is assumed that each constituent of the plate and the solid half‐space possesses no less than monoclinic symmetry. The plate is subjected to incident acoustic waves (propagated through a fluid) at arbitrary angles from the normal as well as at arbitrary azimuthal angles. It is further assumed that all solidinterfaces are either rigidly or smoothly bonded. Solutions are obtained for the individual layers that relate the field variables at the upper and lower layer surfaces. The response of the total system proceeds by satisfying appropriate interfacial conditions across the constituents. The internal response, consisting of complete descriptions of the variations of displacements and stresses, of both the layered plate and the half‐space is determined. Reflection and transmission coefficients are derived as by‐products of the analysis. Representative numerical results are given in order to delineate the influence of the various bounding media and the smooth bond assumption on the propagation process and internal response.

Study of acoustic resonance in enclosures using eigenanalysis based on boundary element methods
View Description Hide DescriptionIt is well known that, from the modal theory of room acoustics,resonance will occur if the driving frequency of a sound source located in a room coincides with one of the natural frequencies of the sound field. In this study, an eigenanalysis technique based on the boundary element method(BEM) is developed for extracting eigenmodes of a sound field in an enclosure. A method of singular value search, in conjunction with a golden section optimization algorithm, is utilized for efficient calculation of eigenmodes. In particular, modes associated with repeated eigenvalues can be well resolved by the technique developed in this research. Enclosures of various geometries have been analyzed by using the developed algorithm in a numerical simulation. Satisfactory agreement has been achieved between the BEM results, the FEM results, and the analytical solutions if available.

Sonic band structure in fluids with periodic density variations
View Description Hide DescriptionIn direct analogy to the electronic band structure found in semiconductors and the photonic bands for light in a medium with a periodic dielectric constant, a periodic density variation in a fluid can give rise to sonic frequency passbands and band gaps. Hence, a fluid medium can be constructed that prohibits sound propagation at certain frequencies while allowing practically free propagation at others. The effect of a sonic band medium on a monopole acoustic source is discussed in a simple one‐dimensional model. In particular, the complete quenching of radiated power is seen for a harmonic radiator at a frequency that corresponds to a band gap—in analogy with a similar effect that is predicted for the atomic emission of electromagnetic waves in a photonicband structure. The ability to construct a medium that selectively prohibits sound propagation and emission in a certain range of frequencies, while allowing transmission and enhanced radiation rates at others, could have interesting practical applications.

Sound radiation from vibrating bodies in motion
View Description Hide DescriptionThis paper presents an integral formulation for prediction of sound radiation from a vibrating body in motion. The formulation consists of volume and surface integrals. The volume integral represents the effect of the turbulent stress field induced by the motion of the vibrating body, while the surface integral represents the combined effects of the gradient of the turbulent stress field and the unsteady force exerted on the fluid by the surface, as well as the surface displacement effect. It is shown explicitly that source cancellations take place within the surface integral when turbulence effects are taken into account. Moreover, dimensional analysis demonstrates that the volume integral is negligible compared with the surface integral when the source is in rectilinear motion at low Mach numbers. If turbulence effects are neglected at the outset of the derivation, then there is no interaction between the turbulent stress field and the vibrating surface in motion and the resulting integral formulation is different. The fundamental difference is whether or not the effect of the gradient of the turbulent stress field is included in the integral formulation. Numerical examples of sound radiation from the dilating and transversely oscillating spheres moving rectilinearly in space at constant subsonic speeds are demonstrated.

Finite‐amplitude wave propagation through a two‐phase system of particles in a viscothermal fluid
View Description Hide DescriptionThe propagation of finite‐amplitude, planar acoustic waves through a system of nonvolatile fluid particles dispersed in a fluid matrix has been examined theoretically by using the continuum volume‐averaged balance equations and linear constitutive equations for a two‐phase fluid system having a single variable temperature. These equations contain more information than those introduced by Marble [Ann. Rev. Fluid Mech. 2, 379–447 (1970)], and used by other workers to determine the attenuation and dispersion of sound propagating through a dilute dusty perfect gas. Specifically, terms that account for phoresis of the particles due to thermal and concentration gradients; Dufour heat transport; and the particles’ contribution to the viscous effects are included. We have manipulated the acoustic equations using the methods developed by previous investigators, and obtained coupled Burgers’ equations; one for each phase. In the limit of very small concentration of particles, these equations reduce to those obtained by G. A. Davidson [J. Sound Vibration 3 8, 475–495 (1975)]. Solutions to the equations were determined by expanding the velocity field in a regular perturbation series in terms of the acoustic Mach number. The attenuation and dispersion of the first‐ and second‐order harmonic terms are presented. These results are not restricted to a perfect gas and should be applicable to semi‐concentrated emulsions, suspensions, and aerosols.

Nonlinear propagation of plane and circular Rayleigh waves in isotropic solids
View Description Hide DescriptionNonlinear Rayleigh wave propagation in an isotropic solid is investigated theoretically. Hamiltonian formalism is used to derive a set of coupled equations for the harmonic amplitudes. Both plane and circular waves are considered. Numerical results are presented for an initially monochromatc wave that propagates in steel. It is shown that the horizontal component of the particle velocitywave forms a shock profile, while the vertical component forms a pulse. An evolution equation for the waveform is derived.

Fermi–Pasta–Ulam recurrence of bound solitary waves in a rectangular water trough
View Description Hide DescriptionThe solution of bound solitary waves of a NLS‐type equation governing the modulational evolution of standing gravity wave in a rectangular fluid trough is given. It is seen that the time evolution of this solution exhibits the Fermi–Pasta–Ulam (FPU) recurrence behavior that is in good agreement with the phenomenon observed by Wu e t a l. [Phys. Rev. Lett. 5 2, 1421–1424 (1984)] in a water trough experiment.

Errors in attenuation measurements due to nonlinear propagation effects
View Description Hide DescriptionAn investigation into the influence of finite amplitude distortion on narrow‐band ultrasonic attenuationmeasurements is described. Measurements have been made using a through‐transmission substitution technique in the nonlinear field of a 5‐MHz plane‐piston transducer driven under tone‐burst conditions. Various transducer excitation levels were used to generate a range of shock parameters σ at the position of the measuringhydrophone up to a maximum of 3. The influence of the resulting loss in amplitude at the fundamental frequency has been studied by measuring the transmission properties of reference attenuators consisting of Dow Corning‐710 fluid‐filled cells of various thickness. The presence of nonlinear distortion in the acoustic waveform produces overestimates of the measuredtransmission coefficients. The magnitude of the error has been shown to depend on the value of σ, the small signal transmission loss of the sample and its position in the acoustic field. In some situations, the error was as high as 100%. A theoretical model that describes the variation of the transmission coefficient with σ has been derived and good agreement demonstrated between theory and experiment. The implications for attenuationmeasurements, both narrow band and broadband, are discussed.

Computation of acoustic scattering from a moving rigid surface
View Description Hide DescriptionApplication of the Kirchhoff formula for subsonically moving surfaces to scattering of sound from a moving rigid body is described. The formulation leads to a singular integral equation that yields surface values of the field. The integral equation is solved using a boundary element technique in conjunction with the Galerkin method. Several checks on the accuracy and validity of the scheme are discussed. Numerical results are obtained for scattering of sound from a harmonic point source co‐moving with a thin, winglike body.

A physically motivated simulation technique for high‐frequency forward scattering derived using specular point theory
View Description Hide DescriptionA physically motivated simulation technique using specular point theory (SPT) is developed to examine high‐frequency signals forward scattered from a rough surface. This method is used to derive, for small receiver separations, the spatialcoherence for an infinite ensemble of signals forward scattered from random surfaces with Gaussian wave‐number spectra. The results are compared with those of previously published methods. An SPT single realization procedure is formulated and implemented for frequencies of 30 and 300 kHz using a Gaussian spectrum approximation to the random rough surface. Some SPT predictions for Gaussian and Pierson–Moskowitz surfaces are compared. Both the spatialcoherence and the temporal structure of finite ensembles of forward scattered signals are compared for the Gaussian case. The relation between the results for infinite and finite ensembles is then examined, and the practical implications and limitations of SPT for simulating forward scattering are discussed.

Experimental detection of a slow acoustic wave in sediment at shallow grazing angles
View Description Hide DescriptionFollowing recent experimental results at sea [N. P. Chotiros, Proceedings of Oceans ’89] that suggest the existence of a previously undetected type of acoustic wave in sandy sediments, an experiment was designed to detect and measure the speed of acoustic waves in an isolated environment. The experiment was conducted in a laboratory tank containing 1 m of unwashed river sediment under a 3‐m water column. Observations were made of the travel time and attenuation of a pulse from an acoustic source located above the water–sediment interface to a set of probes below the interface. It was observed that, at normal incidence the pulse traveled at about 1675 m/s, while at shallow grazing angles, the pulse traveled through the sediment at close to 1200 m/s. An interesting possible explanation exists in the Biot theory for acoustic propagation through fluid‐saturated porous media, which predicts a slow acoustic wave in porous materials.

An improved formalism for rough‐surface scattering. II: Numerical trials in three dimensions
View Description Hide DescriptionThe recently described operator‐expansion formalism can in theory accurately compute the wave scattering from arbitrary rough surfaces so long as the Fresnel number, the Rayleigh height times the surface slope, is small at all roughness scales. A limited number of tests have previously confirmed this for simple one‐dimensional profiles. Because of its efficiency the method can be implemented readily on two‐dimensional surfaces of substantial size as well. Trial computations on doubly periodic ‘‘eggcrate’’ gratings, for which exact solutions are available, yield accuracies as good as in one surface dimension; for example, cross‐section errors are 1% or 2% for Fresnel numbers of 0.3. A simulated random oceansurface on a 256×256 square grid produces bistatic scattering amplitudes that can be seen to combine Bragg diffraction at short scales with geometric shadowing at long scales.

Sea ice elastic moduli: Determination of Biot parameters using in‐field velocity measurements
View Description Hide DescriptionElastic moduli of sea ice have been determined from high‐frequency velocity measurements on small samples extracted from the arctic pack ice. The experimental apparatus allowed quick in‐field observations of the velocities. The experimental procedure included measurement of the ice samples’ salinity and temperature; the relations derived here can be used to predict elastic moduli from knowledge of these more easily obtainable physical parameters. The elastic moduli determined are those appropriate for a specialized Biot model in which the ice is viewed as an isotropic material with closed pores. For this model, seven parameters are needed to predict acoustic velocities, all of which are easily interpreted physically and are given here in terms of their salinity and temperature dependencies.

Ultrasonic spectroscopy of metallic spheres using electromagnetic‐acoustic transduction
View Description Hide DescriptionAn ultrasonic technique for studying vibrational resonant modes of metallic spheres is presented. The technique employs electromagnetic‐acoustic transduction with a configuration consisting of a sample surrounded by a coil in a static magnetic field.Resonance spectra from 0.5 to 4.5 MHz with the coil axis parallel and perpendicular to the magnetic field are measured for a 3.145‐mm‐diam sphere of polycrystalline 2024 aluminum.Elastic constants calculated from the resonant peak frequencies are consistent with results obtained using an ultrasonic pulse‐echo system. This new technique has advantages over pulse‐echo and conventional resonance techniques for experiments where high absolute accuracy is necessary or where samples are heated far above room temperature.

Ultrasonic measurement of stress in weakly anisotropic thin sheets
View Description Hide DescriptionA membrane theory is developed for measurement of stress in weakly anisotropic thin sheets. The thin sheets are modeled as prestressed planar membranes, and the application in stress evaluation of horizontally polarized quasishear waves (which correspond to the S H _{0} mode in a plate theory) is studied. In principle, the acoustoelastic theory presented here would be valid so long as the superimposed small‐amplitude stress waves could be taken as hyperelastic, irrespective of the origin of the prestress and the thermomechanical history of the specimen in question. A series of experiments was performed to examine the validity of this theory for thin aluminum sheets that had undergone plastic deformations. By using a probe that consisted of three electromagneticacoustic transducers (one transmitter and two receivers that were 30 mm apart), velocities of horizontally polarized quasishear waves were measured for various directions of propagation at various places of each sample sheet. There were indications that the present theory and measurement system delivered at each place a good estimate of the local principal surface‐stress directions and difference in principal surface stresses. Some of the samples were annealed after the aforementioned experiments were completed. Ultrasonic measurements were repeated for the annealed samples; they showed that residual stresses were relieved by the annealing.

Free vibration of a sagged cable supporting a discrete mass
View Description Hide DescriptionA continuum model is presented that describes the nonlinear and three‐dimensional response of an elastic cable that supports a single attached mass. Two asymptotic forms of this model are derived for the free, linear response of sagged suspensions having small equilibrium curvature (sag) and level supports. The first model, which is valid for relatively small attached masses, assumes that the cable stretches quasi‐statically and results in uniform dynamic cable tension. The quasi‐static stretching assumption is partially relaxed in the second model, which accounts for spatially varying dynamic tension in an approximate manner. In particular, the second model captures the discontinuous change in dynamic tension across the attached mass and the resulting tangential mass acceleration. The eigensolutions governing free response are compared for the two models. The comparison reveals that the first (simpler) model provides excellent approximations to the natural frequency spectrum for all cable modes having natural frequencies less than that of an (approximate) elastic mode.

Flexure waves in elastic rods
View Description Hide DescriptionThe order‐1 theory of the dynamics of homogeneous elastic rods is treated with the emphasis on twist‐free bending motions of inextensible rods. Various forms of the governing equations for such planar motions are presented, and their traveling wave solutions are shown to result in curves of the same form as those in Euler’s theory of equilibrium configurations. Traveling waves are called subsonic or supersonic in accord with whether their speed is less than or greater than (E/ρ)^{1/2} with E the tensile modulus and ρ the density. The solitary traveling waves are loops and are given by elementary functions. At each prescribed level of tension below a critical value, for both subsonic and supersonic waves, the larger the loop the faster the wave. The periodic traveling waves, both with and without inflexion points, are given by elliptic functions and integrals. Small amplitude sinusoidal waves are a limiting case of inflexional waves. The solitary waves are obtainable as appropriate limits of both inflexional and noninflexional waves. Although, in general, traveling waves are motions of rods of infinite length, there are traveling waves that are possible planar modes of motion for rods of finite length with joined ends. A rod with such a traveling wave forms a figure eight in the inflexional case and a circular ring in the noninflexional case.