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Volume 87, Issue 5, May 1990

Modeling of compliant tube underwater reflectors
View Description Hide DescriptionA theoretical method for determination of multiple scattering of an acoustical wave by a grating of elastic obstacles is presented. The main advantages of the method are: The same formulation works for finite arbitrary gratings as well as for infinite plane gratings; and the calculation of the properties of the whole grating is decoupled from the calculation of the scattering properties of the individual components set in free field. Some results on compliant tube gratings (plane gratings and a parabolic reflector) are shown and compared to experiments.

A variational principle for the scattered wave
View Description Hide DescriptionA Schwinger‐type variational principle is presented for the scattered field in the case of scalar wave scattering with an arbitrary field incident on an object of arbitrary shape with homogeneous Dirichlet boundary conditions. The result is variationally invariant at field points ranging from the surface of the scatterer to the farfield and is an important extension of the usual Schwinger variational principle for the scattering amplitude, which is a farfield quantity. Also, a generic procedure, physically motivated by the general principles of boundary conditions and shadowing, is presented for constructing simple trial functions to approximate the fields. The variational principle and the trial function design are tested for the special case of a spherical scatterer and accurate answers are found over the entire frequency range.

Analysis of the scattering of a plane acoustic wave by a periodic elastic structure using the finite element method: Application to compliant tube gratings
View Description Hide DescriptionA two‐dimensional mathematical model has been developed to analyze the scattering of plane acoustic waves from an infinite, uniform, plane grating of compliant tubes. It relies upon the finite element method and uses the atila code [J. N. Decarpigny e t a l., J. Acoust. Soc. Am. 7 8, 1499 (1985)]. To do this, only the unit cell of the periodic structure, including a small part of the surrounding fluid domain, has to be meshed, thanks to the Bloch–Floquet theorem, and the effects of the remaining fluid domain are accounted for by matching the pressure field in the finite element mesh with simple plane wave expansions of the ingoing and outgoing waves. This paper describes results obtained for the scattering of a plane wave from different tube gratings, including internal losses, at oblique incidence. Comparing finite element results to analytical or experimental results allows for the validation of the model. Then, various compliant tube gratings are considered to demonstrate the efficiency and versatility of this approach. Finally, the generalization to doubly periodic gratings is emphasized.

Sound scattering of a spherical wave incident on a cylinder
View Description Hide DescriptionThe problem of the scattering of a spherical acoustic wave by an elastic circular cylinder of infinite length is investigated theoretically in this paper. In a cylindrical coordinate system, the mathematical form of a spherical wave is similar to that of a plane wave. Thus it is not necessary to solve the relevant differential equations and boundary conditions. An exact solution to the present problem can be directly formulated by using the already known solution for plane‐wave incidence. This solution is expressed by an integral, which always involves a numerical treatment. The scattered pressure is calculated for an aluminum cylinder in water. A detailed discussion of the results reveals the features of the scattering of a spherical wave.

Analysis of echoes from a cylinder that includes the directivity of the transmitter and receiver
View Description Hide DescriptionAnalysis of an echo from a solid elastic cylinder had been limited to the case of a plane incident wave so far. Recently, plane wavetheory was modified to the case of a spherically incident wave. In this paper an arbitrary ultrasonic transmitter and receiver are considered as a set of point sources and point receivers, respectively. The analysis is modified to the practical case, that is, the cylinder is placed so that its axis coincides with the central line of curvature of a cylindrical shell transducer. The dependence of the echo from the cylinder on the cylinder material is examined. These results are compared with those experimentally obtained using metal and plastic cylinders and they show very good agreement.

High‐frequency reflection and scattering by multicomponent rough surface distributions
View Description Hide DescriptionEarlier forms for the coherent reflection and incoherent scattering by multicomponent mixtures of bosses on rigid or free base planes [V. Twersky, J. Acoust. Soc. Am. 2 9, 209–225 (1957)] are applied to recent high‐frequency results for aligned hemiellipsoidal bosses [R. J. Lucas and V. Twersky, J. Acoust. Soc. Am. 8 3, 2005–2011 (1988)] to investigate continuous distributions in boss size. Approximations for the coherent reflected intensity and incoherent differential scattering cross sections are obtained in terms of integrals of simple functions and a general probability density. To provide illustrations, numerical computations and graphical results are based on truncating the two‐parameter gamma probability density functionP(t;m,v) with t as a dimensionless variable that scales one or more boss dimensions, m as the mean value of t, and v as the normalized variance (ranging from zero to unity). For v small, P is Gaussian and reduces to a delta function as v approaches zero (to reproduce one‐component results). More generally, the curve of P is skewed, and as v approaches unity P reduces to the exponential for the simplest Poisson case. Graphs are shown for cases where one (e.g., keel depth), two (e.g., base axes), or all three dimensions of the protuberances are randomized. The essentials are indicated by plots versus angle of incidence, with v as the parameter. The coherent intensity and the associated forward and backscattered incoherent differential scattering cross sections per unit area are emphasized.

Elastic wave radiation and diffraction of a piston source
View Description Hide DescriptionThe radiation of an elastic field from a plane piston source is formulated using the representation theorem, in which the Green’s function for an elastic half‐space is employed. On the basis of this formulation, the radiated elastic wave field for both compressional and shear cylindrical wave sources is derived. The diffraction of elastic waves incident on a receiver that has the same geometry as that of the source and is coaxially aligned with it is studied. The authors present a procedure in which both numerical and asymptotic techniques are employed to allow them to evaluate the diffraction effects in any frequency range of interest. The elasticdiffraction is compared with the acoustic diffraction and it is discovered that they differ only in the nearfield of the piston source because of the coupling between shear and compressional components in the elastic case. In the farfield, however, the elasticdiffraction approaches the acoustic diffraction. With the help of ultrasonic laboratory measurements, the authors test the theoretical results and find that the theory and experiments agree well for the elastic solution. An important application of the results of this study is in attenuation measurements using pulse propagation techniques, where spectral ratio of a sample relative to a standard sample, or ratio of samples of the same material but different length is used. In the former case, the attenuation can be overestimated. While in the latter case, the attenuation can be significantly underestimated, if corrections for diffraction effects are not made.

Analysis of echoes from a sphere which includes the directivity of a transmitter and a receiver
View Description Hide DescriptionAnalysis of an echo from a solid elastic sphere has been limited to the case of a plane incident wave so far. Plane‐wave theory is modified to the case of spherical incident wave first, and then it is modified to the case of an arbitrary ultrasonic transmitter and receiver by considering them as a set of point sources and point receivers, respectively. Using this relation, the general properties of the echo scattered from the sphere are analyzed first, and then the analysis is carried out in full for important practical cases; that is, the sphere is placed on the focal point of a circular concave transducer and in the focal plane of the transducer. It is applied to study the material dependence of the echo from the sphere placed at the focal point of the transducer as well as that of the sidelobe structure of the echo. These results are compared with those experimentally obtained using metal and plastic spheres of 3 mm in diameter, which show very good agreement.

Influence of anisotropy on the dispersion characteristics of guided ultrasonic plate modes
View Description Hide DescriptionDispersion curves are developed for elastic wave propagation in an anisotropic plate of monoclinic or higher symmetry. Emphasis is placed on analytic expressions for various features. Generalization of the isotropic Rayleigh–Lamb dispersion relations are derived for the cases of (a) propagation along a material symmetry axis and (b) propagation in a general direction. Examination of the high‐frequency limit of the lowest symmetric and antisymmetric mode dispersion curves yields expressions for the half‐space surface or Rayleigh wave velocity. It is shown that the dispersion curves for these modes can exhibit multiple crossings in approaching this limit, and an analytic solution is presented for the constant crossing interval that occurs for propagation along symmetry directions. The analytic results are illustrated by extensive numerical calculations for a variety of degrees of anisotropy with emphasis placed on the relationship between the slowness curves governing partial wave propagation and various features of the dispersion curves.

Reciprocity theorems for acoustic wave fields in fluid/solid configurations
View Description Hide DescriptionThe time Laplace‐transform domain convolution‐type reciprocity relation for acoustic waves in a fluid/solid configuration is derived. Arbitrary inhomogeneity, anisotropy, and loss mechanisms are taken into account. Reciprocity between transmitting and receiving transducers located in either the fluid or the solid parts of the configuration is established. It is shown how the reciprocity relation leads to the source‐type wave field integral representations for direct source problems and to the integral‐equation formulation of inverse source, and direct and inverse scattering problems through the associated contrast source representations. Since neither a fluid inclusion in a solid nor a solid inclusion in a fluid leads to a regular perturbation problem in the integral‐equation formulation, the embedding must be adapted to the location of the contrast sources as far as the type of medium (fluid or solid) is concerned. Applications to acoustic emission and to acoustic imaging and profile inversion are briefly indicated.

Coupled finite element/boundary element approach for fluid–structure interaction
View Description Hide DescriptionA new computational capability is described for calculating the sound‐pressure field radiated or scattered by a harmonically excited, submerged, arbitrary, three‐dimensional elastic structure. This approach, called nashua, couples a nastranfinite element model of the structure with a boundary element model of the surrounding fluid. The surface fluid pressures and normal velocities are first calculated by coupling the finite element model of the structure with a discretized form of the Helmholtz surfaceintegral equation for the exterior fluid. After generation of the fluid matrices, most of the required matrix operations are performed using the general matrix manipulation package available in nastran. Farfield radiated pressures are then calculated from the surface solution using the Helmholtz exterior integral equation. The overall capability is very general, highly automated, and requires no independent specification of the fluid mesh. An efficient, new, out‐of‐core block equation solver was written so that very large problems could be solved. The use of nastran as the structural analyzer permits a variety of graphical displays of results, including computer animation of the dynamic response. The overall approach is illustrated and validated using known analytic solutions for submerged spherical shells subjected to both incident pressure and uniform and nonuniform applied mechanical loads.

Modal contributions of a finite plate to power spectra
View Description Hide DescriptionAn investigation is made of the modal contribution of a finite elastic plate, simply supported at both ends, to the power spectrum. Theoretical calculations were made for a finite steel plate submerged in a fluid and excited by two types of forcing functions: (1) a line force and (2) a statistically uniform force. Numerical results calculated for transfer functions and power spectra were presented. A comparison of power spectra calculated for different loading conditions is presented.

Scattering of sound by sound from two Gaussian beams
View Description Hide DescriptionThe scattering of sound by sound from Gaussian beams that intersect at small angles is investigated theoretically with a quasilinear solution of the Khokhlov–Zabolotskaya nonlinear parabolic wave equation. The analytical solution, which is valid throughout the entire paraxial field, is a generalization of a result obtained previously for parametric receiving arrays [Hamilton e t a l., J. Acoust. Soc. Am. 8 2, 311–318 (1987)]. Significant levels of scattered sum and difference frequency sound are shown to exist outside the nonlinear interaction region. An asymptotic formula reveals that sound is scattered in the approximate directions of k _{1}±k _{2}, where k _{ j } is the wave vector associated with the axis of jth primary beam. Computed propagation curves and beam patterns demonstrate the dependence of the scattered radiation on interaction angle, source separation, ratio of the primary frequencies, and source radii. Comparisons are made with the farfield results presented by Berntsen e t a l. [J. Acoust. Soc. Am. 8 6, 1968–1983 (1989)], which are valid for arbitrary interaction angles, source separations, and amplitude distributions.

Comparison between two approaches for solving nonlinear radiations from a bubble in a liquid
View Description Hide DescriptionTwo different approaches have been used in the literature to discuss the nonlinearity of the scatteredwave from a pulsating bubble in a liquid: The radial displacement approach and the volume displacement approach. These two treatments have been compared and it has been pointed out that there is no theoretical basis for the conclusion reached from the volume displacement approach that the second‐harmonic amplitude of the sound emission is zero at a specific frequency.

Application of the SAFARI model to sound propagation in the atmosphere
View Description Hide DescriptionThe SAFARI (seismo acoustic fast‐field algorithm for range‐independent environments) wave propagationmodel is applied for the first time to low‐frequency atmospheric sound propagation and tested by means of synthetic and real‐world data. Problems experienced with the use of SAFARI are discussed in detail. Good agreement of the model output with theoretical predictions is achieved for two half‐spaces (air/ground). Available meteorological data up to 5 000‐m height are applied to the model in the next step in order to compute sound transmission loss versus range and receiver height. Results are given in transmission loss contour plots covering a field of 1 000 m in height and 30 000 m in range. The influence of typical sound velocity profiles including strong gradients close to the ground is investigated. It turns out that knowledge of meteorological data is most relevant for heights of up to about 200 m for sound propagationmodeling within a field of interest as above. SAFARI is finally compared with a fast‐field program developed at the U.S. Army Construction Engineering Laboratory. This test yields very good agreement between both model predictions on a test case in which a real‐world sound profile is used. It has to be emphasized that all results are for very low frequencies where finite impedance ground effects are minimal. One would expect to see differences at higher frequencies. These encouraging results give rise to recommending the use of the models in a number of interesting applications such as various types of acoustic detection system and civil noise control.

Ground characterization by short‐range propagation measurements
View Description Hide DescriptionIn recent years, short‐range measurements of excess attenuation from a point source have been used together with semiempirical formulas for frequency dependence of impedance to enable the acoustical characterization of ground surfaces, including snow, in terms of a single parameter. This has then been advocated as a basis for predicting ground and propagation effects at longer ranges. An alternative method is described for determining acoustical properties of ground surfaces including sands, soils, and snow from iterative least‐squares fitting of the level difference spectrum obtained between a pair of vertically separated microphones within 2 m of a broadband point source. The method is based on a three‐parameter model for the surface normal impedance as a function of frequency, together with well‐established formulations for propagation from a point source above either local or extended reaction surfaces. The three parameters are porosity, effective flow resistivity, and tortuosity. Independent (nonacoustic) measurements of porosity compare tolerably well with the acoustically determined values for soils that are homogeneous to several centimeters depth. For such soils, fitting comparisons reveal the superiority of the three‐parameter impedance model to the single‐parameter semiempirical model. Where there are obvious surface crusts, a double‐layer model based upon a two‐parameter approximation for the characteristic impedance of each layer is found to give better agreement with short‐range propagation measurements than the three‐parameter homogeneous approximation.

The influence of wind and temperature gradients on sound propagation, calculated with the two‐way wave equation
View Description Hide DescriptionA method is introduced to calculate the influence of wind and temperature gradients in stratified media on sound propagating above an absorbing ground surface. It is based on the ‘‘two‐way wave equation’’ for the Fourier transforms of the sound pressureP and its derivative V. The vector containing P and V is stepwise extrapolated through the medium in the direction perpendicular to the ground surface, fulfilling the boundary conditions at the ground surface, at a top level, and at the source height. The propagation equations for P and V appear as simple plane‐wave equations, and computer (CPU) time within each layer is very low. Therefore, many thin layers (in the order of centimeters if desired) can be applied, and any complicated gradient can be used. Calculations for a homogeneous atmosphere, with a computer program based on this model, show an excellent agreement with previous models. When a wind profile is present, results are mainly compared with measurements by Parkin and Scholes [P. H. Parkin and W. E. Scholes, J. Sound Vib. 1, 1–13 (1964); 2, 353–374 (1964)]. They show very good agreement in the no‐wind and downwind cases. In the upwind situation, agreement is very good below 500 Hz. Above this value the model does not predict sound to penetrate into the shadow region, while Parkin and Scholes found (low) sound levels.

A numerical approach to rough‐surface scattering by the parabolic equation method
View Description Hide DescriptionThe parabolic equation method is an effective approach when the acoustic wave field is incident at low grazing angles onto a rough surface. The method consists of an integral equation and an integral, the first of which yields the surface field derivative. The main part of this paper is concerned with an approximation to this equation, valid when wavenumber times surface height is up to order unity. The approximation has several advantages. First, it allows a decomposition of the equation into deterministic and stochastic components. The stochastic part depends only locally upon the surface in certain regimes, and this can give rise to a great reduction in computational expense. Some basic statistical moments of the stochastic component are also considered. These are nonstationary, but for the incident field a simple stationary transformation is found, which can therefore be compared with Monte Carlo simulations using far fewer realizations. These results are demonstrated computationally. The final part of the paper describes the numerical implementation of the full parabolic equation method. Both of the integrals contain singularities, and these are treated semianalytically.

Ocean‐bottom ultralow‐frequency (ULF) seismo‐acoustic ambient noise: 0.002 to 0.4 Hz
View Description Hide DescriptionObserved spatial and temporal characteristics of ultralow‐frequency (ULF) ocean‐bottom seismo‐acoustic ambient noise are required in order to construct realistic quantitative predictive models of the phenomena involved. Few such data exist or have been studied, especially for frequencies below about 0.1 Hz. Analysis of noise data is presented in the band 0.002 to 0.4 Hz from a 2‐week period, 11/28–12/12/67, recorded from long‐period, three‐component seismometers and a hydrophone of the Columbia‐Point Arena ocean‐bottom seismic station (OBSS, 38° 09.2’N–124° 54.4’W, 3903‐m depth). Two intense NE Pacific storms with hurricane force winds occurred during the emphasized time period. Time variations of spectra and of amplitude and phase coherencies of the four‐component OBSS data are related to the storm histories and to local weather/wave conditions and are used to identify motion (seismic wave) types and directions of propagation.

The time‐marched fast‐field program (FFP) for modeling acoustic pulse propagation
View Description Hide DescriptionFast‐field programs (FFPs) have emerged as an important tool for predicting transmission loss in an oceanwaveguide. Such models have been primarily used for time‐harmonic sources; however, pulses or other broadband sources may be treated by Fourier synthesis. A new technique is developed that provides a d i r e c t solution by marching the solution forward in time. As an example of the method, a pulse incident on an interface between two homogeneous half‐spaces is considered. Snapshots of the pulse in time illustrate graphically the effects on the reflected and transmitted waves. Second, an interesting hyperbolic cosine profile is considered that leads to repeated focusing as the pulse propagates out in range.