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Apr 1985

Volume 77, Issue S1, pp. S1-S108

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back to top Session BB. Physical Acoustics V: Theoretical and Computational Problems
Contributed Papers
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An efficient computational scheme for evaluating the Helmholtz integral (A)

K. Brod and G. H. Koopmann

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S60-S60 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Numerical solutions to the Helmholtz integral provide a means of computing the surface pressure on an arbitrarily shaped body with a prescribed surface velocity. When using the “exterior” form of the integral, i.e., restricting the source and field coordinates of the Green's function to the surface of the body, singularities within the Green's functions are encountered along with uniqueness problems for the solutions at certain eigenfrequencies of the interior space. In this study, we show that both of these problems can be circumvented by using the “interior” form of the integral, i.e., locating all of the field points in the interior space where a zero pressure condition is prescribed. Criteria for choosing the optimum set of interior field points are described along with a general discussion of the accuracy of the method. Comparisons of numerical and closed form solutions are presented for the monopole and dipole radiators. The method has been developed for use on a personal computer. [Work supported by NSF.]
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Accuracy of infinite elements in structural acoustics modeling (A)

Henno Allik and Robert C. Haberman

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S60-S60 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Infinite elements are used in conjunction with finite elements to solve a number of structural‐acoustics problems. When finite elements alone are used to model an infinite fluid domain, a truncation of the model is necessary at a “sufficiently large” distance from the area of interest. In addition, steady state problems require the imposition of a boundary condition to absorb outgoing waves. A common practice in fluid‐structure interaction problems is to model approximately one‐and‐a‐half acoustic wavelengths of external fluid and to impose a plane‐wave absorbing boundary condition. Although workable, this approach can result in very large models that are costly to solve. An alternate way to treat the infinite domain and significantly reduce the number of equations is by using infinite elements [e.g., P. Bettess, Int. J. Numer. Methods, Eng. 11, 53–64 (1977)] alone, or in conjunction with finite elements. This is possible because infinite element shape functions contain terms that describe an outward traveling and exponentially decaying wave. The derivation of the element is reviewed and its accuracy in problems involving infinite cyclinders is presented. Results from two‐dimensional scattering off rigid and elastic cylinders are compared to theoretical and previously published results. The fluid impedance for waves on a cylindrical boundary is also calculated and compared with closed form solutions. In all instances considered, accurate answers were obtained with the use of no more than two layers of finite elements, plus a layer of infinite elements. In many instances, the infinite elements alone produced excellent results.
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Rayleigh‐Ritz analysis and finite element descriptions derived from variational principles of diffraction by a circular disk (A)

J. H. Ginsberg, A. D. Pierce, X.‐F. Wu, and J. S. DiMarco

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S60-S60 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The Kirchhoff‐Helmholtz integral equation yields a variational principle for the pressure on the surface of a vibrating body [A. D. Pierce and X.‐F. Wu, J. Acoust. Soc. Am. Suppl. 1 74, S107 (1983)]. A comparable principle is derived here for wave diffraction by a rigid body. Both versions offer the prospect of a more efficient treatment of diffraction effects than other methods, such as T matrices and the doubly asymptotic approximation. As a first application, algorithms for the evaluation of the pressure on the surface of a vibrating disk are derived and developed for constant frequency excitation and an axisymmetric surface pressure. For radiation, this situation arises when the vibrational pattern is axisymmetric. Axisymmetry for the reflection problem is obtained for low values of ka at arbitrary incidence, or else for normal incidence at arbitrary ka. The variational principles are used to derive system equations associated with (1) assumed mode functions and (2) finite elements. In the former, a sequence of admissible functions are selected to represent the surface pressure, and the variational principle provides the best combination of (complex) amplitudes describing the overall contribution of each mode. The finite element version yields simultaneous equations for the pressures at the nodes of the finite element mesh. In both techniques the system equations are simultaneous linear algebraic equations. The only difference in the treatment of the radiation and reflection problems is the inhomogeneous terms in the equations. [Work supported by ONR.]
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Long wavelength acoustic properties of laminated plates (A)

Michael Schoenberg

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S60-S60 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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A propagator matrix formulation is used to study the elastodynamics of laminated plates. For a free plate in the low‐frequency limit, the only mode is an extensional wave. The speed of this wave is usually found by a quasistatic analysis (first the extensional modulus for uniaxial strain is found and then the speed is calculated using the thickness weighted average density of the plate). This quasistatic result may be obtained from the exact dynamic relation for waves in a laminated plate (found by using the propagator matrix approach) and then taking the limit as frequency tends to zero. This free laminated plate extensional wave speed is αpl  =  2[(〈μ〉 − 〈γμ〉)/〈ρ〉]1/2, where μ is the lamina shear modulus, ρ is the lamina density, and γ denotes the square of the ratio of shear speed to compressional speed for each lamina. 〈〉 signifies a thickness weighted average over the constituent laminas. The propagator approach can then be used to find the leading term (as wavelength becomes long) of the reflection coefficient of the plate immersed in an ideal fluid and to find the long wavelength slowness relation for compressional wave propagation in a system of periodically spaced laminated plates immersed in an ideal fluid. These results are shown to be identical with analogous results found for a homogeneous transversely isotropic plate with elastic moduli given by the set of effective moduli for layered medium [G. E. Backus, “Long‐wave elastic anisotropy produced by horizontal layering,” J. Geophys. Res. 67, 4427 (1962)] and density 〈ρ〉. This means that for long wavelengths, a laminated plate is indistinguishable from an equivalent homogeneous transversely isotropic plate.
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Evaluated Rayleigh integrals for delta‐function driven circular membranes (A)

Stephen I. Warshaw

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S60-S61 (1985); (2 pages)

Online Publication Date: 12 Aug 2005

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Analytic and semianalytic evaluations of the pressure field due to impulsively driven flat membranes imbedded in an infinite plane baffle by using Rayleigh's integral are discussed and presented for a variety of delta function type motion distributions across the membrane. We first review the field due to a circular membrane subject to a uniform delta function acceleration, and then treat more complex membrane motions composed of different spatially and temporally dependent delta function accelerations. These will include annular (ring) and uniformly expanding annular (moving ring) distributions. In some cases closed analytic forms for the pressure field can be obtained, while in others elliptic integrals result, which can be conveniently calculated by Gauss‐Chebyshev quadrature of the first kind. All evaluations are performed directly in the time domain. [Work performed under auspices of U.S. Department of Energy by LLNL under Contract W‐7405‐Eng‐48.]
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On the effective size of a point acoustic multipole (A)

Ian Roebuck

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The Green function approach has been of great benefit in linear acoustics. But there are occasions when an uncritical use of this conceptual picture of the source distribution as a collection of elementary point sources can lead to paradoxical and misleading conclusions, especially when the physical problem takes account of nearby boundary surfaces. Further, with the Lighthill acoustic analogy, in the vicinity of the point source the nonlinear terms may dominate. In this paper we determine the neighborhood of the point source within which, even in linear problems, the existence of boundaries and/or other sources will preclude the use of unmodified “point source” ideas. The derivation relies on the rigorous utilization of the mutual impedance concept, and gives an effective size which is multipole‐order and radiation wavelength dependent. For radiators at low frequency this may exceed the classical “nearfield limit.” Some of the problems referred to earlier—in particular for turbulent flow noise over rough boundaries—are shown to lie in domains where moderating influences exist in the “farfield” of a source but within its effective dimensions.
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Sound reflection from a rigid concave/convex disk and its application to a spherical concave source (A)

Masahiro Ukigai and Yoshiro Miida

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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This paper presents an analytical technique for calculating the amplitude of reflected sound waves from a rigid concave/convex disk. The formulas for the reflectivity are derived approximately under certain geometrical restrictions using Fresnel‐Kirchhoff's diffraction formulas. Moreover, we identify the reflected sound pressure from a concave disk when a point source is located at the center of curvature with the radiated sound pressure from a spherical concave source, whose surface vibrates at the same time. Then we analyze the radiated sound pressure from a spherical concave source under this condition. To verify the validity of the approach, some experiments were performed in air. Experimental results showed good agreement between the calculated and the measured values and this confirmed the usefulness of the present technique to calculate the amplitude of reflected sound waves from a concave/convex disk.
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The Green's functions for an acoustic source of arbitrary shape (A)

Anthony J. Rudgers

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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When either the velocity or the pressure is specified on the surface of an acoustic source, there is an appropriate Green's function that can be used to describe the radiated pressure field of the source. It is demonstrated how to construct an analytic expression for each of these two Green's functions for a source having an arbitrary shape. The distribution of velocity or of pressure need not be uniform on the source surface. The resulting Green's functions, which are expressed in terms of the geometry of the source and the free‐space Green's function for the Helmholtz equation, are calculable, in principle, by performing a series of elementary mathematical operations. An iterative‐operator technique, based upon the Neumann‐series method for solving linear integral equations, is used to construct the Green's functions.
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Radiation impedance for baffled rectangular pistons of arbitrary aspect ratio at any frequency (A)

G. Kirby Miller

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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An algorithm has been developed for calculating the complex farfield radiation impedance of a plane rectangular piston vibrating at any frequency in a coplanar infinite baffle for any ratio of length to width. The method is based on previously published expressions for the radiation from an infinite strip [J. W. Miles, J. Acoust. Soc. Am. 20, 652–1664 (1948)] and for the radiation from a long strip [O. A. Lindemann, J. Acoust. Soc. Am. 52, 1045–1048 (1972)], as well as on the expressions and calculated values found in the thesis [D. S. Burnett, “Radiation Impedance Functions of Rectangular Pistons and Their Application to Sound Transmission Through Finite Depth Apertures,” Ph.D. dissertation, University of California, Berkeley, CA (1969)]. Six different power and asymptotic series (some believed new) in combination with three empirically derived algebraic expressions are shown to be valid for each of four contiguous regions in the dimensionless length and width parameters, kl and kw. The boundaries of the regions are slightly different for the real and the imaginary parts. The algorithms are amenable to very fast computer calculation and produce results that are usually accurate to within a few percent.
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Edge diffraction of transient nondispersive wave—theory and experiment (A)

Michael G. Brown and Brian K. Emmett

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Edge diffraction of transient nondispersive waves in two dimensions is considered. The characteristic single‐edge diffraction pattern for time‐harmonic waves is described by the Fresnel function. The time domain counterpart of the Fresnel function is derived. We argue that the time domain wavefield representation is more insightful than its frequency domain counterpart. It shows, for example, that the diffracted pulse which originates at the shadow casting edge is of opposite polarity in the illuminated and shadow regions. Theoretical and experimental results are compared. [Work supported by ONR.]
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Reflection and refraction of ultrasonic waves on a plane interface between two generally anisotropic media (A)

S. I. Rokhlin, Ken Bolland, and Laszlo Adler

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S61-S61 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Based on the Fedorov theory, we developed a unified approach for numerical solutions for the problem of the reflection‐refraction of elastic waves on interface between two generally anisotropic solids. In contrast to other studies we consider Kristoffel equations and boundary conditions for both anisotropic media in coordinate systems formed by incident and interface planes rather than in crystallographic coordinates. This makes it possible to write simple unified algorithms for finding slowness refraction vectors and unit displacement vectors. Special care was taken to include in the general algorithm the cases of propagation in directions of acoustics axes. We illustrate our results by calculation of energy reflection and refraction coefficients for two cases: (1) anisotropic stainless‐steel weld and (2) nickel‐nickel bicrystal interface. [This work is supported by the Department of Energy.]
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One‐dimensional acceleration waves in a compressible fluid mixture (A)

G. Batra and A. Bedford

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S62-S62 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The method of singular surfaces is used to study the propagation of one‐dimensional acceleration waves in compressible fluid mixtures of the type described by Bedford and Drumheller [Arch. Ration. Mech. Anal. 68, 37–51 (1978)]. It is found that four wave speeds are possible for a mixture of two constituents. The temporal evolution of the amplitude of the wave is governed by a first‐order ordinary differential equation—the Bernoulli equation. Thus, depending on the initial amplitude, constitutive assumptions and the state of the material ahead of the wave front, the wave may grow or decay. The possibility of shock formation in finite time is not ruled out. These results are obtained without any specific representation for constitutive relations and hence are common to materials within a consecutive class and may contribute in the evaluation of constitutive functions for such mixtures through experimental study of wave behavior. As an extension to this work, preliminary results on the propagation of acceleration waves in bubbly liquids are obtained.
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