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Volume 91, Issue 3, March 1992

Floquet waves in anisotropic periodically layered composites
View Description Hide DescriptionThe field equations governing the plane harmonic elastic motions of anisotropic stratified media are written in the form of a matrix system of differential equations, where the dependent variables are the displacements and tractions acting across planes normal to the direction of stratification. In the case of periodically layered media, Floquet’s theorem and the propagator matrix method can be applied to solve the governing sextic matrix equation. In the absence of sources or body forces, the general motion of the layered medium is described by a combination of six partial waves (Floquet waves). A closed‐form algebraic solution for the dispersionequation of such waves is derived. Numerical results describing the dispersion spectrum of a cross‐ply periodic laminated are discussed in detail.

Forward modeling and data inversion for beam propagation in a stratified medium. I: Theory
View Description Hide DescriptionA previously developed matrix Green’s function formulation [I. T. Lu and L. B. Felsen, Geophys. J. R. Astron. Soc. 8 4, 31–42 (1986)] for line‐source excited propagation in piecewise‐homogeneous stratified media is extended to allow arbitrarily inhomogeneous layers and beam source excitation. This formulation reveals dominant wave processes and allows simultaneous excitation and detection at arbitrarily specified locations. It provides a unified and systematic approach for deriving all alternative representations including mode, ray(beam), spectral integral, ray(beam) mode, collective ray(beam), etc. Both rigorous and practical aspects of the analytic continuation procedures in the complex source point technique are discussed in detail. An efficient beam source localization algorithm, which estimates the beam parameters and the beam location sequentially, is also presented. Numerical results are presented in a companion paper [I. T. Lu and H. Y. Chen, J. Acoust. Soc. Am. 9 1, ▪▪▪–▪▪▪ (1992)].

Forward modeling and data inversion for beam propagation in a stratified medium. II: Numerical results
View Description Hide DescriptionThe general theory of both forward and inverse modelings for beam propagation in stratified media developed in Part I of this presentation [I‐Tai Lu, J. Acoust. Soc. Am. 9 1, ▪▪▪▪–▪▪▪▪ (1992)] is examined in detail by numerical simulations. Modes, collective beams, and conventional beams have been employed to calculate the beam‐excited wave phenomena in both piecewise‐homogeneous and arbitrarily inhomogeneous layered media. These algorithms are formulated in the same framework and yield essentially the same results. Efficient data inversion procedures to minimize least‐square errors for localizing a beam source are also presented and examined by numerical simulations. The conventional modal‐filtering procedure using inner products of the received signals with the orthonormal eigenfunctions is proved to be a special case of a weighted least‐square scheme.

The influence of piezoelectricity on free and reflected waves from fluid‐loaded anisotropic plates
View Description Hide DescriptionA unified analytical treatment is presented supported by extensive numerical illustrations of the interactions of ultrasonicwaves with piezoelectricanisotropic plates. The plates are allowed to possess up to monoclinic anisotropic symmetry and associated piezoelectric coupling. The plates are also assumed to be immersed in water and subjected to incident acoustic beams at arbitrary polar and azimuthal angles. Simple analytical expressions for the reflection and transmission coefficients are derived, from which all propagation characteristics are identified. Such expressions contain, as a by‐product, the secular equation for the propagation of free harmonic waves on the piezoelectric plates. This equation is written in simple and completely separate terms pertaining to symmetric and antisymmetric modes. It is found that piezoelectric coupling, as well as water, influence both types of modes. Higher symmetry, such as orthotropic, transverse isotropic, and cubic, are contained implicitly in this analysis. It is also demonstrated that the motion of the sagittal (Lamb) and SH modes uncouple for propagation along axis of symmetry. For such cases, however, piezoelectric coupling can influence one of these types of modes depending upon the type of piezoelectric model adopted.

Reflection of sound waves by sound‐speed inhomogeneities
View Description Hide DescriptionAn analysis is given of the influence of the rate of change of sound speed on reflection and transmission in a perfect gas. Asymptotic formulas, valid for both low and high frequencies, are developed to compute the reflection and transmission coefficients for one‐dimensional waves propagating through variable‐speed layers. The sound speed may have a discontinuous first derivative. It is shown that local reflection effects are proportional to the square of the derivative of the logarithm of the sound speed. The method predicts reflections for gradually varying sound‐speed profiles having continuous derivatives of all orders. In the special case where the sound speed is a piecewise linear function, the general method produces an exact solution in addition to the asymptotic formulas. This exact solution is valid for arbitrarily large sound‐speed gradient and reduces to the classical result for discontinuous sound speed in the limit of an infinite gradient over an infinitesimal distance. The exact solution produces an effective numerical method when an arbitrary sound‐speed profile is modeled by a mixed sequence of discontinuities and ramp functions.

Computation of the signal distortion of reflections from specular reflectors by impulse ray modeling
View Description Hide DescriptionThe shape of an ultrasound pulse that is reflected by a specular reflector will be distorted because of differences in the length of the path traveled by the acoustical waves between different points on the transducer surface to different points on the reflector surface. This effect can be studied by calculating the radiation coupling function for the combination of a transducer and a reflector. The calculation is performed with a numerical method, basically a modification of ray models used in the calculation of heat transfer by radiation. The signal distortion can be described as a linear filtering process. Its transfer function elucidates the distortion of an emitted waveform. By way of example, the transfer function is calculated for various positions and diameters of a cylindrical reflector in front of a focused transducer. It appears that severe signal distortions can be expected in practical situations.

Solving knife‐edge scattering problems using singular boundary elements
View Description Hide DescriptionA multidomain boundary element method(BEM) is employed to study the scattering of time‐harmonic acoustic waves by knife‐edge structures in an acoustic field. Analysis shows that singularities exist in such structures, and that the behavior of the acoustic pressure and its derivative near a knife edge resembles that of the displacement and stress near a crack tip in an elastic body. Singular elements are employed to model the behavior on the part of the boundary in the vicinity of the singularity. The advantage of the singular elements is seen through several numerical examples involving mufflerlike structures containing baffles and one with a knife‐edge scattering in full space. For a given level of mesh refinement, both the near‐field and far‐field solutions are improved with the introduction of singular elements.

Analysis and computations of measurement system effects in ultrasonic scattering experiments
View Description Hide DescriptionA model that characterizes the effects of beams and waveforms on the measurement of ultrasonic scattering is analyzed in detail. The analysis obtains a wideband expression for the system function in terms of an integration over spatial‐ and temporal‐frequency variables. The temporal‐frequency integration is reduced to a convolution in the direction of the scattering vector when the temporal frequencies are concentrated in a narrow band around a central frequency. The spatial‐frequency integration is simplified to a straight line path when the spatial frequencies in the angular spectra of the emitter are concentrated around a point on the axis of the emitter and the spatial frequencies of the detector sensitivity pattern are similarly concentrated around a point on the axis of the detector. Expressions that result from the temporal and spatial approximations are evaluated analytically for circularly symmetric Gaussian spatial apertures and Gaussian temporal waveforms. In addition, numerical results are obtained to compare the effects of circularly symmetric Gaussian, exponential, and uniform spatial aperture functions on the weight that beam patterns have on measurements of scattering. The results may be used to design experiments from which intrinsic parameters of scattering media can be obtained by an appropriate normalization to remove measurement system effects from the data.

Effects of dispersion on the inference of metal texture from S _{0} plate mode measurements. Part I. Evaluation of dispersion correction methods
View Description Hide DescriptionUltrasonicS _{0} waves (fundamental symmetric Lamb modes) are being considered in several laboratories for the nondestructive characterization of the texture (preferred grain orientation) and formability of metal sheets and plates. In a typical experimental setup, the velocities of the S _{0} waves are measured as a function of wave propagation angle with respect to the rolling direction of the plate. However, the S _{0} waves are known to be dispersive, and that dispersion must be considered in order to isolate the small, texture‐induced shifts in the S _{0} wave velocity. Currently, there are two approximate dispersion correction methods, one proposed by Thompson e t a l.^{9} and the other introduced by Hirao and Fukuoka.^{2} ^{0} In this paper, these two methods will be evaluated using an exact theory for wave propagation in orthotropic plates. Through the evaluation, the limits of the current texture measurement techniques are established. It is found that when plate thickness to wavelength ratio is less than 0.15, both Thompson’s and Hirao’s methods work satisfactorily. When the thickness to wavelength ratio exceeds 0.3, neither Thompson’s nor Hirao’s dispersion correction method provides adequate corrections for the current texture measurement techniques. Within the range of 0.15–0.3, Thompson’s method is recommended for weakly anisotropic sheets and plates and Hirao’s method may be more appropriate for some strongly anisotropic cases.

Production of structural and acoustic waves by dipole sources adjacent to an elastic plate with a pressure‐release coating
View Description Hide DescriptionModel problems are examined to estimate the influence of a pressure‐release coating on an elastic plate on the generation of sound and vibration by hydroacoustic dipole sources. Predictions for this idealized case provide useful first approximations to the behavior of real systems in which the fluid loading is large. The dipole and plate may be assumed to characterize, respectively, the unsteady lift exerted on a blade of a ducted rotor or stator during interaction with vorticity (occurring either naturally in a turbulent stream or as discrete vortices shed from structural appendages), and the adjacent duct wall. Such l o n g i t u d i n a l dipoles (oriented parallel to the duct axis) produce equal and opposite images in a pressure‐release coating, and the efficiency of sound generation is reduced to that of a quadrupolesource when the dipole is situated well within an acoustic wavelength of the wall. In practice, considerations of cost and weight require that only a finite section of the duct wall be coated. In this paper, the dependence of the structural and acoustic noise produced by a dipole on its distance from an edge of the coated region, and on its normal distance from the coating, is determined.

Local high‐order radiation boundary conditions for the two‐dimensional time‐dependent structural acoustics problem
View Description Hide DescriptionThe time‐dependent structural acoustics problem involving solution of the coupled wave equation over an infinite fluid domain is posed as a coupled problem over a finite fluid domain with local time‐dependent radiation boundary conditions applied to the fluid truncation boundary. The proposed radiation boundary conditions are based on an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a nondimensional wave number. A sequence of differential operators that match the leading terms of the asymptotic expansion provide boundary conditions that are of progressively higher order and increasing accuracy. Time‐dependent boundary conditions are obtained through an inverse Fourier transform. The relationship of these approximate local operators to the exact nonlocal Dirichlet‐to‐Neumann map is examined. To illustrate their effectiveness, the boundary conditions are employed in a finite element formulation for the time‐dependent structural acoustics problem. In contrast to nonlocal boundary conditions based on the Dirichlet‐to‐Neumann map or retarded potential integral formulations, the proposed local boundary conditions preserve the data structure of the standard finite element method and do not require storage of a large pool of historical data during the solution process. Numerical results illustrate the accuracy of the proposed boundary conditions as effected by the operator order, acoustic wave number, radiation directionality, and distance from the acoustic source.

Scattering of sound by atmospheric turbulence: Predictions in a refractive shadow zone
View Description Hide DescriptionAccording to ray theory, regions exist in an upward refracting atmosphere where no sound should be present. Experiments show, however, that appreciable sound levels penetrate these so‐called shadow zones. Two mechanisms contribute to sound in the shadow zone: diffraction and turbulentscattering of sound. Diffractive effects can be pronounced at lower frequencies but are small at high frequencies. In the short wavelength limit then, scattering due to turbulence should be the predominant mechanism involved in producing the sound levels measured in shadow zones. No existing analytical method includes turbulence effects in the prediction of sound pressure levels in upward refractive shadow zones. In order to obtain quantitative average sound pressure level predictions, a numerical simulation of the effect of atmospheric turbulence on sound propagation is performed. The simulation is based on scattering from randomly distributed scattering centers (‘‘turbules’’). The scatteringeffect of each turbule is calculated with the help of the first Born approximation to scattering, using independently measured micrometeorological variables such as the variance 〈μ^{2}〉 and correlation length L of the refractive index fluctuations. Sound pressure levels are computed for many realizations of a turbulent atmosphere. Predictions from the numerical simulation are compared with existing theories and experimental data.

Normal mode solution for low‐frequency sound propagation in a downward refracting atmosphere above a complex impedance plane
View Description Hide DescriptionThe development of the fast field and parabolic equationsolutions to the wave equation has made it possible to solve for the combined effects of refraction in a layered atmosphere and the interaction of sound with a complex impedance ground surface. In many respects the numerical methods have advanced beyond our understanding of the basic phenomena. In an earlier study [J. Acoust. Soc. Am. 8 9, 107–114 (1991)], the residue series solution for upward refraction was investigated and provided insight into the nature of the interaction of refraction and ground reflection. In this paper results are presented of a similar normal modesolution for downward refraction above a complex impedance ground surface. This model is used to investigate when the surface wave is excited for downward refraction conditions and to develop criteria for the maximum range of cylindrical decay as a function of phase and magnitude of the ground impedance and the magnitude of the sound velocity gradient.

Effect of surficial sediment layering on high‐frequency seafloor reverberation
View Description Hide DescriptionPerturbation theory is applied to predict the effects of surficial sediment layering on high‐frequency acoustic backscatter from the seafloor. The case of a thin rough surface layer over a rough homogenous half‐space is considered in which the surface and half‐space roughnesses are partially correlated. The theory predicts strong interference effects near the critical angle similar to the anomalous scatter observed in recent field exercises.

A two‐way parabolic equation for acoustic backscattering in the ocean
View Description Hide DescriptionThe parabolic equation (PE) method is generalized to handle backscattered acoustic energy in the ocean. The two‐way PE is based on the single‐scattering approximation and the approach of two‐way coupled modes in which range‐dependent environments are approximated by a sequence of range‐independent regions. At the vertical boundaries between regions, the solution of the two‐way PE is required to satisfy two continuity conditions. The range derivative in one of the conditions is replaced by a higher‐order PE depth operator. The reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme. The outgoing and incoming fields are propagated by two‐way range marching. The two‐way PE, which is presently implemented for two‐dimensional problems, is a practical method for solving large‐scale reverberation problems. The accuracy of the two‐way PE is demonstrated by comparisons with reference solutions. The two‐way PE is applied to simulate the localization of a source of backscattering using the method of back propagation.

Note on averaged horizontal refraction for long distance propagation in an ocean sound channel
View Description Hide DescriptionLong distance propagation in an ocean sound channel is considered, in the adiabatic limit. It is shown how the vertical oscillation of rays within such a channel can be averaged out so that one is left with equations describing the horizontal refraction. A modified Fermat’s principle for the averaged, horizontally projected rays is formulated. These results are also generalized to take into account: First, the effect of the Earth’s curvature, and then the effect of the phase shifts suffered by sound waves on hitting an external surface, or a caustic.

Axisymmetric monostatic and bistatic resonance excitation mechanisms of large aspect ratio targets: Impulse excitations and bipolar couplings
View Description Hide DescriptionAn experimental/theoretical study of the scattering from large aspect ratio, finite elastic cylinders is presented. The study focuses on the excitation mechanisms of the axisymmetric resonances. It is found that significant coupling of the incident field to these resonances occurs in the region of geometric shadow at low k D/2 (D is the cylinder diameter). The implications of this coupling for the relative excitation strengths of the resonances and for the associated bistatic beam pattern are examined. It is argued on physical grounds that the results must apply to all large aspect ratio targets (both solids and shells).

Pulse length effects on the transmissivity of bubbly water
View Description Hide DescriptionThe passage of sound through bubbly water is strongly attenuated by scattering and absorption. Such attenuation is most severe around the frequency of resonance of individual bubbles. A bubble takes a finite time to ring up to steady‐state conditions and continues to oscillate for a finite time after the driving pressure ceases. Low backscatter for short pulse lengths has been observed in near‐surface seawater [Akulichev e t a l., Sov. Phys. Acoust.3 2 (3), 177–180 (1986)]. An experiment is described that looked for a corresponding enhancement in transmission. Comparisons were made between the attenuations of brief waveform bursts and longer bursts. The center frequency of this experiment was 120 kHz. The bubbles were made by the electrolysis of fresh water in a small laboratory tank. For bursts from 6 to 20 cycles in duration, no difference in the attenuations was discerned in comparison with a 2.6‐oscillation burst.

Scattering of an obliquely incident acoustic wave by an infinite hollow cylindrical shell
View Description Hide DescriptionAcoustic scattering from an isotropic elastic hollow cylindrical shell of infinite length excited by an obliquely incident plane acoustic wave is investigated. The form functions of an aluminum cylindrical shell immersed in water have been calculated by the direct summation of the Rayleigh series. Computations are made at angles (with the normal to the cylinder axis) between α=0° and α=35°. The results of the theoretical calculation are in good agreement with the results of experiments. The experimental results have shown in a frequency range of k _{1}a=0 –20 that the resonances are related to three wave families: the circumferential wave (l=2) detected for angles smaller than the ‘‘angle of longitudinal wave in thin rods’’ (α_{ l }), the guided wave (p=1) detected for angles smaller than the second critical angle (α_{ T }), and the Scholte–Stoneley wave (l=0). The evolution of the resonance frequencies is followed for different angles and one can note experimentally, that at an angle superior to the Rayleigh critical angle (α=30.3°), resonances of the Scholte–Stoneley wave have been observed.

Ray synthesis of leaky Lamb wave contributions to backscattering from thick cylindrical shells
View Description Hide DescriptionThe ray representation of the scattering from an empty cylindrical shell is compared with the exact results of the partial‐wave series. The specific example considered is a thick stainless steel shell having the same parameters as a previous synthesis for spheres illustrated by Kargl and Marston [J. Acoust. Soc. Am. 8 9, 2545–2558 (1991)]. The present computation demonstrates a general agreement with the exact result for k a down to 7 and confirms the applicability of a simple approximation for the complex coupling coefficient.