Index of content:
Volume 89, Issue 4, April 1991

Impulse response operators for complex structures
View Description Hide DescriptionThe response of a structure may be stated in terms of an impulse response function integrated over the external drive or equivalently in terms of an impulse response operator acting on the external drive. A complex structure usually consists of either several elemental structural components that are interconnected (e.g., two welded plates) each admitting to a single response type (e.g., either longitudinal or flexural), an elemental structural component that admits to several response types (e.g., longitudinal and flexural), or a combination of both. To derive the impulse response operator directly is not a simple task, even in these primitive examples. The boundaries between elemental structural components and/or the various response types in each component render the response of the complex structure orderly but multifaceted. Therefore, it is suggested that one may advantageously analyze the response, at a given spatial position in the structure and at a specific instant of time, by superposing the contributions to the response of a number of paths. A path defines a unique type of an impulse response operator. The impulse response operator of a path may be simpler to derive than that derived directly for the complex structure as a whole. In this multipaths analysis the complex structure is described by an impulse response vector operator, each element is an impulse response operator that is associated with a unique path. Correspondingly, the external drive is a vector, and the response is the scalar product of the impulse response vector operator and the external drive vector. A sequential procedure is introduced in which the complex structure and the external drive are further decomposed in the hope of achieving further simplifications. In this procedure one recognizes that a dynamic system may be associated with each type of response in an elemental structural component. With this recognition the complex structure is modeled by a set of dynamic systems. The formalism is then stated in terms of matrices and vectors, e.g., the response is a vector (each element represents the response of a specific dynamic system), the impulse response operator is a matrix (the offdiagonal elements describe the couplings between the dynamic systems), etc. If the dynamic systems are chosen so that each, in isolation, can be described in terms of an eigenimpedance operator, then, in addition, a modal analysis can be applied to the multiple dynamic systems that compose the model of the structure. In the modal analysis, however, the ranks of the impulse response matrix, the response vector, and the drive vector are swollen by the modal count, usually rendering the matrix equation unwieldy. In the parallel waveanalysis, the propagations in the dynamic systems are described by impulse response operators that are commensurate with those pertaining to boundlessly extrapolated dynamic systems. The finiteness of the dynamic systems is accounted for by junction matrices; a junction defines the boundaries through which dynamic systems interact either with each other (transmissions) or with self (reflections). As in the modal approach, in this wave approach, the resulting formalism is, again, rather unwieldy. It is shown that considerable reductions and simplifications are attained if the structure can be modeled by spatially onedimensional dynamic systems.

The general problem of elastic wave propagation in multilayered anisotropic media
View Description Hide DescriptionExact analytical treatment of the interaction of harmonic elastic waves with nlayered anisotropic plates is presented. Each layer of the plate can possess up to as low as monoclinic symmetry and thus allowing results for higher symmetry materials such as orthotropic, transversely isotropic, cubic, and isotropic to be obtained as special cases. The wave is allowed to propagate along an arbitrary angle from the normal to the plate as well as along any azimuthal angle. Solutions are obtained by using the transfer matrix method. According to this method formal solutions for each layer are derived and expressed in terms of wave amplitudes. By eliminating these amplitudes the stresses and displacements on one side of the layer are related to those of the other side. By satisfying appropriate continuity conditions at interlayer interfaces a global transfer matrix can be constructed which relates the displacements and stresses on one side of the plate to those on the other. Invoking appropriate boundary conditions on the plates outer boundaries a large variety of important problems can be solved. Of these mention is made of the propagation of free waves on the plate and the propagation of waves in a periodic media consisting of a periodic repetition of the plate. Confidence is the approach and results are confirmed by comparisons with whatever is available from specialized solutions. A variety of numerical illustrations are included.

Plane acoustic waves in linear viscoelastic porous media: Energy, particle displacement, and physical interpretation
View Description Hide DescriptionThe poroviscoelastic model, which is asynthesis of the wellknown viscoelasticmodel and Biot’s poroelastic model, is presented in the context of the propagation of plane acoustic waves in linear viscoelasticporous media. Except for the propagation of a second compressional wave in such media, results concerning particle displacement, maximum attenuation, and direction of maximum energy flow are very similar to that obtained by Borcherdt [J. Geophys. Res. 78, 2442–2453 (1973)] in simple viscoelastic media. The energy conservation relation is given explicitly in the case of time harmonic radiation field. From this, expressions of the energy flux, energy densities, dissipated energy, and are derived. Furthermore it is demonstrated that the total attenuation is equal to the sum of the viscoelasticattenuation and Biot’s poroelastic attenuation This allows a direct comparison between energy dissipated by viscoelasticity and that dissipated by Biot mechanism. For propagation of acoustic waves in infinite natural porous media the latter is always negligible compared to the former. Computed curves and physical interpretations are proposed to illustrate the theoretical derivations.

A twodimensional curvilinear wavegrid for integration along traveling waves
View Description Hide DescriptionA curvilinear grid is constructed for the numerical integration of a wave characteristics formulation, where the independent variables are the twospatial polar coordinates and the time dimension. The grid is formed by the bicharacteristic curves which represent the spacetime location of the wave surfaces. This scheme of integration paths allows the solution to propagate along all existing waves that traverse the solution domain. Without detriment to generality, the wavegrid construction method is developed by considering longitudinal and shear waves with constant speeds of which the speed of the shear waves is the smallest.

A halfspace response to a finite surface source of an impulsive disturbance
View Description Hide DescriptionThe response of an ideal elastic halfspace to an impulsive load is obtained by a computational method based on the theory of characteristics. The impulsive load is distributede unevenly but radially over the entire boundary of a semicircular infinite canyon embedded in the surface of the halfspace. For this configuration, a wavecharacteristics formulation is presented where its differential equations are extended to accommodate strong discontinuities which occur in the material motion of the halfspace. A stepbystep numerical integration of these extended differential equations is then carried out in a twodimensional curvilinear wavegrid, formed by the bicharacteristic curves of the wavecharacteristics formulation. The resultant transient deformation of the halfspace is shown by plots which give the time history behavior of the dependent variables at a specific spatial location. In these plots, the various wave fronts traversing the halfspace explicitly reveal themselves.

Scattering of sound from concentric cylindrical shells
View Description Hide DescriptionScatteredsound field from three concentric shells ensonified by a plane wave is treated. Formulation of the problem is carried out using impedance expressions for the shells and the acoustic media interior and exterior to the shells. Results of exact equations of elasticity have been used to obtain the shell impedances, making the results applicable to arbitrary thicknesses. The results show that in a target made up of multiple shells and layers, the first elastic continuum that the incoming wavesinteract, which most closely resembles an impenetrable target dominates scattering. For example, a very thick shell or a very lightdensity fluid layer effectively insulates scattering from the layers or shells in its interior. When locally resistive compliant coatings are placed on the surfaces of the shell, they alter the scattering response. A very compliant layer on the exterior of the target changes it to a pressurerelease type of boundary. © 1991 Acoustical Society of America.

A new efficient algorithm to compute the exact reflection and transmission factors for plane waves in layered absorbing media (liquids and solids)
View Description Hide DescriptionThis paper describes a matrix method for computing the exact reflection and transmission coefficients for harmonic plane waves within a stratified medium of homogeneous, isotropic, and absorbing plane layers. The new feature is that each layer can be either liquid or solid, whatever their successive order. Furthermore, this algorithm takes into account evanescent waves, but also applies whatever the thickness of each layer. A numerical example is shown.

Leaky Lamb waves in an anisotropic plate. II: Nondestructive evaluation of matrix cracks in fiberreinforced composites
View Description Hide DescriptionThis paper is concerned with the use of leaky Lamb waves for the nondestructive evaluation(NDE) of damage in anisotropicmaterials such as fiberreinforced composites. Two fundamental acoustic properties of the material, namely, the wave speed and attenuation have been measured. Stiffness is deduced from the wave speed. The damage mode selected for this study is matrix cracking. As expected, the inplane stiffness decreases and the attenuation increases with an increase in the linear crack density.

Diffraction tomography and the stochastic inverse scattering problem
View Description Hide DescriptionThe Fourier diffractiontheorem is the basis of diffractiontomography. The theorem states that the field scattered by a semitransparent object maps onto arcs in the Fourier space of the object. In this paper, the stochastic analog is derived to the theorem for a general anisotropic, statistically homogeneous random continuum. By acoustically probing the random medium from various angles, the secondorder statistics of the medium can be recovered from the secondorder statistics of the perturbed field. The robust nature of the result is confirmed by numerical experiments using finite arrays and autocorrelation estimation theory.

The propagation of time harmonic Rayleigh–Lamb waves in a bimaterial plate
View Description Hide DescriptionThe time harmonic elastodynamic response of two semiinfinite elastic plates of dissimilar material properties perfectly bonded along their lateral faces is studied. The wave field in either halfplate can be written as a superposition of the socalled Rayleigh–Lamb eigenmodes of an infinite plate. The interaction of a time harmonic incident wave with the interface results in reflected and transmitted fields that contain contributions from all of the real, imaginary, and complex eigenmodes of an infinite plate. Attention is focused on the distribution of energy among the various reflected and transmitted eigenmodes over a range of frequencies. The fundamental symmetric and the fundamental antisymmetric Lamb modes are each used as input excitations. Such excitations can be approximately realized in experiments. It is assumed that the solution of such a canonical problem will facilitate the solution of problems with complicated timedependent sources.

Structural intensity in thin cylindrical shells
View Description Hide DescriptionA rigorous derivation of the structural intensity vector for a thin cylindrical shell is provided based on Flügge’s equations of motion and expressions for the total energy density in the shell. The shell is assumed driven by normal forces, loaded externally by a fluid, and to have no internal loss. It is confirmed that the structural intensity vector is composed of five terms, and simple physical interpretations of these terms are provided. The relationship between the structural intensity and the normal acoustic intensity is derived. The five structural intensity terms are investigated in some detail using results from a numerical experiment. The numerical model was a simply supported, pointdriven cylindrical shell with external fluid loading, assumed to exist in an infinite rigid baffle. A sum over in vacuo modes was used to solve the equations of motion. The results show, in the frequency range below the ring resonance of a steel shell with =0.01, that the divergence of the extensional power flow term dominates over the other four terms in the transfer of power to the fluid and, equivalently, into the far field.

Numerically efficient evaluation of intrinsic modes in wedgeshaped waveguides
View Description Hide DescriptionEvaluation of intrinsic modes in wedgeshaped oceans overlying a fluid bottom is via a spectral domain integral. Here, a numerical technique, centered around the FFT, is applied in the spectral domain, which generates a numerically efficient algorithm. This numerical efficiency permits a demonstration of the intrinsic mode conserving power across a local transverse cross section. Comparisons between the parabolic equation method and the beam propagation method, with respect to the intrinsic mode, are commented upon.

A boundary element approach to ocean seismoacoustic facet reverberation
View Description Hide DescriptionA numerically efficient, hybrid method is introduced for modeling of short and long range seismoacoustic facet reverberation in the ocean environment. The method combines the global matrix approach to the solution of the wave equation in horizontally stratified media with a boundary element formulation of the boundary conditions at a contour surrounding the facet. The present paper describes a twodimensional formulation for facets within an elastic seabed or an elasticice cover, but allows for simulation of the reverberant field within the water column as well. The approach is directly extendable to treat reverberation from seabedpenetrating facets as well as threedimensional elastic facets. In contrast to discrete methods such as the finite element and finite difference approaches, the solution obtained with the present hybrid approach is not only efficient for short as well as longrange reverberation, but inherently decomposes the total solution in the temporal and spatial spectral components, of importance to the basic physical understanding of the factors affecting seismoacoustic facet reverberation.

Finestructure in NAPOLI 85, an ocean/acoustic experiment
View Description Hide DescriptionNAPOLI 85 was an ocean/acoustic experiment carried out in the Tyrrhenian Sea to investigate acoustic propagation variability due to inhomogeneities in the ocean medium. The variability in the pulse arrival times has already been presented. This article presents the analysis of the environmental conditions. Data are available from various instruments, including a towed oscillating body (TOB), which was equipped with sensors that provide sound speed. The TOB allows oceanic features to be mapped in the vertical–horizontal plane. These maps show significant finestructure features (typically 4 km long by 40 m deep) that can be tracked through 10TOB casts over a 3day period. In contrast to an oceanographer’s viewpoint, the analysis of these features is carried out in terms of their impact on acoustic propagation. A procedure for separating finestructure into internalwave and nonwave components is extended from previous work and applied. This procedure provides a means to identify the mechanism generating the variability, without the uncertainty inherent in the analysis of onedimensional data. The ocean variability is found to have been almost entirely caused by advective intrusions, with very little internalwave activity. An internalwave model is applied to the separated wave field and is shown to agree moderately well. The model cannot be made to match the nonwave field, which has a horizontal wavenumber spectrum. Characteristic lengthscale definitions are developed and evaluated for the horizontal and vertical directions. The horizontal length scale is 2 km and is depth invariant. Vertical scales vary from 7–25 m over the depth range 50–250 m, respectively.

Lamb and creeping waves around submerged spherical shells resonantly excited by sound scattering. II: Further applications
View Description Hide DescriptionThe scattering of plane sound waves from an airfilled steel spherical shell submerged in water in the frequency band is studied. This analysis is based on a methodology [Ayres et al., Int. J. Solids Struct. 23, 937–946 (1987) and G. Gaunaurd and M. F. Werby, J. Acoust. Soc. Am. 82, 2021–2033 (1987)] proposed that uses the exact threedimensional equations of dynamic elasticity to describe the shell motions and to predict its sonarscattering cross section. This approach is valid at all frequencies, for shells of any thickness, of any (constant) curvature, and it accounts for their fluidloaded condition. The methodology is used to predict the cross sections, which are later interpreted on the basis of the various resonance features that manifest themselves in the frequency response. The spectral locations of these resonances depend on the various types of elastic waves propagating along the shell, or in the surrounding fluid. The exact plots are generated for the phase velocities of these (Lamb) waves always accounting for the curvature and fluidloading effects present on the shell, without appeals to plate waves or theories. Some of the dispersion plots were generated using the Donnell shelltheory approximation, which seems to yield accurate results up to the coincidence frequency. Aside from the broad resonance lobe present at the coincidence frequency, there is another highfrequency resonance lobe, due to a thicknessresonance effect, which was also predicted and displayed. A partialwave analysis of the resonance response curve for a thin shell, around its coincidence frequency, serves to identify the origins of the various types of observed resonance features and to relate them to the elastic and acoustic waves that propagate along the shell or the outer fluid. Many computergenerated graphs are displayed to illustrate the above points.

Observation and inversion of seismoacoustic waves in a complex arctic ice environment
View Description Hide DescriptionThe propagation of lowfrequency seismoacousticwaves in the arctic ice canopy is examined through the analysis of data collected during the 1987 PRUDEXice camp. Study of the geophonetime series generated by underice detonations reveals not only the expected longitudinal and flexural waves in the ice plate, but also unexpected horizontally polarized transverse waves. The travel paths of all three wave types are found to be refracted in the horizontal plane along a known ridge line locally separating the ice canopy into two distinct halfplates of thin firstyear ice and thicker multiyear ice. The need to determine both the origin of the wave and location of each detonation from the received time series highlights the dramatic superiority of geophones over hydrophones in this aplication. Inversion of the geophone data for the ice sheet’s lowfrequency elastic parameters is conducted initially by modeling the ice as a single homogeneous isotropic plate using SAFARI. A modified stationary phase approach is then used to extend SAFARI modeling to invert for the parameters of the two ice halfplates simultaneously. The estimated compressional/shear bulk wave speeds are comparable to previously obtained values; however, the compression/shear attenuation values are up to four times greater than earlier estimates.

Reflection of a plane wave from a fluid layer with continuously varying density and sound speed
View Description Hide DescriptionThis paper considers the transmission of an acoustic plane wave through a horizontally stratified fluid layer whose density and sound speed both vary continuously with depth. The stratified layer is of finite thickness and lies between two semiinfinite homogeneous fluids. The situation modeled is intended to be representative of an inhomogeneous marine sediment overlying a uniform substrate. Exact solutions of the Helmholtz equation, valid in the stratified layer, are derived for a class of density profiles and a number of classes of soundspeed profile. These include profiles that closely resemble measured density and sound speed variation in marine sediments. The solutions in the stratified layer are matched with solutions in the homogeneous upper and lower layers, to derive analytical expressions for the reflection coefficient of a plane wave, incident from the upper medium. Internal losses are modeled in both the substrate and the sediment layer. The solutions enable assessments to be made of the dependence of reflection coefficient on wavenumber and grazing angle for different shapes of density and sound speed profile in the sediment. The limiting cases of very low and very high frequency are examined, and both limits are shown to give reflection coefficients independent of the shape of the density or sound speed profile in the sediment layer. A number of numerical results are presented, and comparison is made with computed results from a general propagation model.

New estimates of sound speed in water
View Description Hide DescriptionMeasured travel times of acoustic pulses, propagated across a 3000km section of the North Pacific, are inconsistent with travel times predicted with the internationally accepted algorithm for the speed of sound in water. The soundspeed algorithm predicts a speed that is too fast at oceanic pressures found below about 1km depth. An accurate algorithm of sound speed is important for deriving temperature from measurements of acoustic travel time across oceans. Accurate estimates of the temperature field are probably important for better understanding the ocean’s role in determining weather, climate, and the distribution of marine organisms.

Reverberant phase in a room and zeros in the complex frequency plane
View Description Hide DescriptionTransfer function (TF) phase of finitedamped systems can be estimated from the distribution of poles and zeros in the complex frequency domain. This paper investigates the distribution of nonminimum phase zeros of TFs from impulse response data measured in a reverberant space for which the modal overlap is large. The number of nonminimum phase zeros is found to be inversely proportional to the damping of the system. This result is shown to be consistent with earlier work on the relation between phase and energy decay. The distribution of zeros and the phase of transfer function are also changed by windowing the impulse response. An exponential window is recommended in order to reduce the effect of truncation on the zeros, although such a window will have the effect of added damping.

Decibel annoyance reduction of lowfrequency blast attenuating windows
View Description Hide DescriptionIn this study, the acoustical benefits of improved, blast noise reducing retrofit windows are determined using the methods of pairedcomparison testing with panels of subjects. The results show that the retrofit windows reduce the received indoor Cweighted SEL by about 7 dB. The retrofit windows result in about a 14dB improvement in terms of community response. Further, a regression line is fit to the indoor measured blast CSEL and their correspondingly equivalent control noise ASEL. The slope of this line is 2.5, indicating that a 1dB change in CSEL corresponds to about 2.5dB change in control ASEL. This corresponds to the result one would get if one used loudness to describe both indoor signals instead of describing the control (indoors) using ASEL and the blast (outdoors) using a variety of descriptors such as peak, CSEL or FSEL, etc.