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Volume 81, Issue 6, June 1987

Diffraction of elastic waves by cracks or cavities using the discrete wavenumber method
View Description Hide DescriptionA new method to calculate the diffraction of elastic waves by cracks or cavities is presented. The method yields the complete diffracted wave field and is valid at any frequency. It is based on a discretization of the inclusion surface and on the assumption of a periodic repetition of the diffracting inclusion. Body forces are applied at the discretized points along the inclusion surface and the discrete wavenumber method is used to evaluate the resulting radiation. The application of the boundary conditions then leads to a linear system of equations, which is solved by matrix inversion. Examples of calculation and comparisons with results from other techniques are presented in the case of plane incident waves as well as in the case of a line source. The presence of a free surface is also considered.

General formulas for the low‐frequency acoustic scattering by a soft body or disk
View Description Hide DescriptionThe scattering problem of an incident plane sound wave by a soft arbitrarily shaped body as well as its allied diffraction problem by a disk is discussed. A first‐class Fredholm integral equation is posed which, for low frequencies, yields a potential equations sequence for a wavenumber power solution expansion. The power solution expansion coefficients of the scattering cross section and the farfield amplitude up to O(κ^{6}), where κ is the wavenumber, are found to be related through only a few low‐order solutions of the above equations. Reciprocity relations between them were exploited. The following new general formulas are obtained. The scattering cross‐section coefficient σ^{4} of O(κ^{4}) is found in terms of three solutions up to O(κ^{2}), rather than six solutions up to O(κ^{5}). As for the farfield amplitude, the sixth (fifth, fourth, and third) coefficient of O(κ^{5}) (κ^{4}, κ^{3}, and κ^{2}) is related to only six (five, four, and two) solutions up to O(κ^{3}) (κ^{2}, κ, and κ). As an application of the above results, the coefficient σ^{4} is explicitly calculated using newly found solutions for an elliptic disk. Also, the circular disk results are shown for an arbitrary angle of incidence.

On the transition matrix for acoustic waves scattered by a multilayered inclusion
View Description Hide DescriptionThe transition matrix (T matrix) for acoustic wavesscattered by a multilayered inclusion with arbitrary geometry is reformulated by applying the Green’s identity for scalar wavefunctions. The derivation of the T matrix is much simpler than previous formulations based on the Helmholtz integral formula. Further simplification is accomplished by a new combination of basis functions for refracted waves. The final answer is different but can be reduced to previous results.

The interior acoustic field of an automobile cabin
View Description Hide DescriptionThe acoustic field in an automobile cabin can be calculated using a Green’s function method. A method for calculating the acoustic field of an arbitrary volume bounded by an arbitrary surface is shown. The eigenvalues of the system, including the effects of arbitrary shape and surface impedance, are found. Alternatively, the acoustic response is calculated when one or more of the surfaces is in vibration without first calculating the eigenvalues. The method works equally well in one, two, or three dimensions.

Acoustic power output from moving point singularities
View Description Hide DescriptionThis article provides theoretical results for the acoustic power output from point sources moving with an arbitrary subsonic speed along a general trajectory. Applications to steady and nonsteady rotational motions and to rectilinear and elliptical motions are presented. A harmonic analysis of the power output from one or more point sources in steady rotational motion is given, which includes an account of the distribution among individual modes at different speeds.

Sum and difference frequency generation due to noncollinear wave interaction in a rectangular duct
View Description Hide DescriptionNoncollinear wave interaction in a rectangular duct is investigated both theoretically and experimentally. An inhomogeneous wave equation, exact to second order in the field variables, is derived for the sum and difference frequency pressure waves generated by noncollinear interaction of two finite amplitude plane waves in a lossless fluid. This equation is extended to the interaction of waves in higher‐order modes of a rectangular duct. Quasilinear solutions are obtained, and tube wall attenuation is included a d h o c. Experimental results are reported for the interaction of waves in the (0,0) and (1,0) modes of an air‐filled rectangular duct. Theory and experiment are in excellent agreement with regard to oscillatory structure of the sum and difference frequency wave fields. Although overall agreement between theory and experiment is reasonable (±2 dB), it is not within estimated experimental error (±1 dB). It is shown that because local rather than cumulative nonlinear effects dominate the interaction, knowledge of the proper second‐order source condition is of crucial importance. Discrepancies between the predicted and measured amplitudes are attributed to an inadequate description of the source condition.

Variation of the strain spectra of random waves in nonlinear elastic materials
View Description Hide DescriptionRandom waves in one‐dimensional nonlinear elasticmaterials are considered. An integrodifferential equation is derived that governs the variation of the strain spectrum during propagation. By discretizing the spectrum with respect to the frequency, the spectral equation reduces to a system of ordinary differential equations for the discrete values of the spectrum. Numerical results are given for plane random longitudinal waves in an aluminum alloy. It is found that harmonic sound waves are generated for a narrow‐band spectrum, the power spectrum grows faster for higher frequencies for a wideband spectrum, and secondary waves are generated due to the interaction of two bands of spectra.

Solitary waves on nonlinear elastic rods. II.
View Description Hide DescriptionIn continuation of an earlier study of propagation of solitary waves on nonlinear elastic rods, numerical investigations of blowup, reflection, and fission at continuous and discontinuous variation of the cross section for the rod and reflection at the end of the rod are presented. The results are compared with predictions of conservation theorems for energy and momentum.

Attenuation of intense sinusoidal waves in air‐saturated, bulk porous materials
View Description Hide DescriptionAs intense, initially sinusoidal waves propagate in fluids, shocks form and excess attenuation of the wave occurs. We present data indicating that shock formation is not necessary for the occurrence of excess attenuation in nonlinear, lossy media, i.e., air‐saturated, porous materials. An empirical equation is used to describe the excess attenuation of intense sinusoids in porous materials. The acoustic nonlinearity of and the excess attenuation in porous materials may be predicted directly from dc flow resistivity data. An empirical relationship is used to relate an acoustic nonlinearity parameter to the fundamental frequency and relative dc nonlinearity of two structurally different materials.

Wave propagation in media with three‐dimensional quadratic refractive index profile
View Description Hide DescriptionThis article presents a new method for studying wave propagation in an unbounded medium where the square of the refractive index is of a general three‐dimensional quadratic form. By this method, the Green’s function can be expressed exactly by a onefold integral, for which the integrand is derived from a path integral. The numerical results for one‐dimensional Green’s functions in linearly stratified media are first checked with those obtained from the well‐known Airy function expressions. The method is then applied to obtain the three‐dimensional Green’s functions for linear stratified media and for parabolic profiles with range‐dependent (horizontal) variations. Some interesting phenomena caused by the horizontal variations (i.e., range dependence) of the medium will be shown and interpreted by simple ray pictures whenever appropriate. For the special case of a parabolic profile with symmetric turning points, a wave duct is formed and the Green’s function can be expanded as a series of wave modes. It will be shown that an evanescent mode in a purely stratified medium may switch to a propagating mode due to the range dependence of the medium property. Application of the results to acoustic wave propagation in the atmosphere and in the ocean will be discussed.

Effects of bottom attenuation on acoustic propagation with a modified ray theory
View Description Hide DescriptionThe influence on ray propagation of including both attenuation and beam displacement at the bottom of a shallow‐water isospeed channel is determined. Attenuation is incorporated using the Mackenzie‐bottom model, rather than the commonly used Rayleigh reflection theory. Properties of displacement are studied as a function of launch angle and parameters such as bottom‐to‐water density, channel depth in wavelengths, and channel aspect ratio. Differences in beam displacement for the two bottom models are shown to affect the number and geometry of rays between surfaced source and receiver. Formulas for per‐ray amplitude and phase shift and for incoherent total‐field intensity are developed. Comparisons are made between these quantities for both modified and classical ray theory with a Mackenzie bottom, and also for both Mackenzie‐ and Rayleigh‐bottom models using modified rays. Numerical computations show significant phase and amplitude differences arising in both types of comparisons.

Shallow water time‐series simulation using ray theory
View Description Hide DescriptionRay theory with beam displacement is used to simulate the propagation of pulses in shallow water. Comparisons show very close agreement between simulated time series and time seriesmeasured in an indoor tank having a sand bottom. The six normal modes present are successfully extracted from both the simulated and the experimental sets of depth‐dependent time series. The ray model accounts for the frequency dependence of beam displacement by interpolating eigenray characteristics at three frequencies in the spectrum of the source pulse. Also, the model corrects ray theory when receivers are close to caustics formed by beam displacement, in which case eigenrays must be found at six frequencies. Mode theory phenomena such as mode cutoff and dispersion can be observed in the simulations produced by this method.

The effects of boundary surface inhomogeneities on acoustic scattering. I. Theory
View Description Hide DescriptionAnalytical methods of Marsh [J. Acoust. Soc. Am. 3 3, 330–333 (1961)] and Kuo [J. Acoust. Soc. Am. 3 6, 2135–2142 (1964)] are applied to high‐frequency acoustic wave scattering from an ocean bottom of roughness amplitude typically quite small in comparison with the acoustic wavelength of interest. The bottom surface variation in acoustic velocity is considered negligible but the bottom surface variation in density is considered appreciable and included in the model. The analytical result indicates the possibility of a new scattering effect in grazing incident directions due to correlated roughness and density variations. When there exists an identifiable dominant roughness wavenumber, the theory predicts discrete backscattering. This phenomenon was observed in an experiment reported by Roderick and Dullea (TD 7181, Naval Underwater Systems Center, 1984). When the dominant roughness wavenumber is zero, the result is that of Rayleigh’s flat boundary. When the dominant roughness wavenumber is low compared to acoustic wavenumber, the result is that of the Kirchhoff approximation.

Analysis of the amplification and convolution of reflected bulk acoustic waves in a piezoelectric/semiconductor structure
View Description Hide DescriptionThe amplification of reflected bulk acoustic waves (RBW) and the convolution of two oppositely traveling incident bulk acoustic waves is analyzed. The case of a normal field degenerate convolver is considered. It is shown, on an orientation of the bismuthgermanium oxide (BGO) crystal in which only one mode is excited and in which no mode conversion occurs upon reflection, that RBWs can be amplified by the drifting carriers in a semiconductor. It is also shown that convolution resulting from the mixing of two incident bulk acoustic waves can be obtained. The overall behavior of the amplified RBW and of the convolver open‐circuit voltage with the semiconductor parameters and the applied drift field and air gap is found to have some similarities to those of observed results in SAWs. A dependence of the acoustoelectric interactions on the angular distribution of the bulk wave radiation, hence the applied signal frequency, is shown. There may, however, exist some critical angles for the RBW for which no interaction exists due to the absence of an electric field in the gap. Results are presented for a BGO/Si structure and characteristics of the proposed RBW acoustoelectric devices are discussed.

Numerical solution for the dynamic moduli of a viscoelastic bar
View Description Hide DescriptionThe dynamic Young’s modulus and loss factor of a viscoelastic material may be calculated as functions of frequency from data on the relative motions of the two ends of a bar of the material that is in harmonic oscillation at that frequency. Most investigators using this technique have confined their measurements to resonant frequencies, but it would be useful to find the moduli of the material at regular, narrowly spaced intervals of frequency. The characteristic equation, from which the moduli of the material are calculated, is investigated with respect to numerical solvability and stability. It is shown that this equation has infinitely many solutions. An apparently effective method of choosing the physically relevant solution is developed. It is found in the case of low‐loss materials that at certain frequencies solutions to the characteristic equation lie near zeros of the Jacobian, which are the solutions in the limiting case of a perfectly elasticmaterial. Consequently, low‐loss viscoelastic materials will exhibit great sensitivity to errors in measurement at these frequencies. An error analysis is developed in order to estimate the magnitude of this instability. A microcomputer program written to solve these equations is applied to various simulated samples to illustrate the effects of this instability.

The acoustic radiation damping of the modes of a rectangular panel
View Description Hide DescriptionThe acoustic damping for single modes of a finite rectangular panel, simply supported in an infinite baffle, is theoretically determined from the ratio of the acoustic energy radiated per cycle to the vibratory energy of the panel. Asymptotic solutions for the low‐frequency region are presented for a panel mode driven at an arbitrary frequency and for a panel mode vibrating at its natural frequency. Curves of acoustic damping for a panel mode vibrating at resonance, as a function of the panel thickness‐to‐length ratio, are presented for various panel aspect ratios. For panels vibrating below the critical frequency, the damping depends on the aspect ratio with square panels developing the smallest value. For panels vibrating above the critical frequency, the damping is nearly independent of the aspect ratio.

Free vibration of unsymmetrically joined shell structures with a closed member
View Description Hide DescriptionFree vibration is analyzed for unsymmetrical joined plate or shell structures with a closed member by means of the transfer matrix method. For this purpose, the continuity and equilibrium relations for the displacements and forces at the joints are written with use of the joint matrices. The connection matrix between the closed member and other members at the joints is derived by introducing the structure matrix of the closed member, and the entire structure matrix is obtained by the product of the connection matrix of the closed member and the transfer matrices of other members. This method is applied to a plate structure with an unsymmetrically sited duct and a box‐type structure with an aslant interior plate, and the natural frequencies are calculated numerically together with the mode shapes of vibration giving the results.

A fundamental problem with mobility analysis of vibration isolation systems
View Description Hide DescriptionThe standard approach to vibration isolationanalysis represents the source, isolator, and receiver by their mobilities. However, the choice of appropriate descriptor for each element is not arbitrary. Causality arguments show that whether the isolator is treated as a one‐port or a two‐port element, at least one of the system components must be represented by its impedance.

Wheel/rail rolling noise, I: Theoretical analysis
View Description Hide DescriptionA comprehensive analytical model of the wayside noise generated by a railroad wheel rolling on straight track is presented. The model assumes that the small‐scale roughness on the running surfaces of the wheel and rail is the primary mechanism for the noise generation. Included in the model are such effects as the spatial filtering of the roughness due to the finite area of contact between the wheel and rail; the interaction between the wheel and rail, including local contact stiffness; axial response of the wheel; the radiation efficiencies of the wheel and rail; and the influence of sound propagation, including finite ground impedance.

Wheel/rail rolling noise, II: Validation of the theory
View Description Hide DescriptionAn analytical model for the prediction of wheel/rail noise is validated through comparison with measurements made using the state‐of‐the art car (SOAC) on a test track. Predictions of wheel and rail vibrations and wayside noise are seen to agree reasonably well with measurements although areas of uncertainty remain. Model improvements to reduce discrepancies between theory and measurement are discussed.