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May 1990

Volume 87, Issue S1, pp. S1-S164

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back to top Session V. Physical Acoustics III: Chaos and Turbulence in Acoustics
Invited Papers
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Chaotic wave propagation (A)

A. Pidwerbetsky

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S53-S53 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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Over the past several years, the development of the science of chaos has led to new insights and understanding of nonlinear dynamics. Wave propagation can also become chaotic in both deterministic and stochastic (random media) environments. This problem has been treated by both ray and wave theory as well as through numerical simulations. In a ray description, chaotic propagation is characterized by exponential divergence of nearby rays with propagation distance (quantified by Lyapunov exponents). In a wave description, propagation is characterized by exponential divergence of narrow wave beams with range. In both cases, this divergence results in the tangling of the wave fields in phase space. In wave propagation through random media. the spreading due to chaotic wave propagation competes with spreading due to diffraction and scattering by small‐scale irregularities, leading to different parameter regimes where these different spreading mechanisms dominate. Environments that produce strong multiple scattering by irregularities with a large length scale are most likely to cause chaotic wave propagation. Finally, implications and applications of chaotic wave theory along with remaining questions and issues will be discussed.
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Nonlinear spatial evolution of forced instability waves in free shear layers (A)

M. E. Goldstein

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S54-S54 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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External forcing of free shear layers between parallel streams produces spatially growing instability waves that are initially governed by linear dynamics for sufficiently small excitation amplitudes. While the instability wave amplitude continues to increase in the downstream direction, its local growth rate must ultimately decrease due to viscous spreading of the mean shear layer. Nonlinear effects can then become important in a critical layer at the transverse position where the mean flow and instability wave phase velocities are equal. A disturbance evolving from the strictly linear, finite‐growth‐rate instability wave on a weakly nonparallel mean flow by using matched asymptotic expansions is considered. Expansions for the various streamwise regions of the flow are combined into a single composite formula accounting for both shear layer spreading and nonlinear critical layer effects. The two‐dimensional incompressible case is considered first. A comparison is made with some recent experimental data. This is followed by a discussion of the two‐dimensional supersonic case, ending with a discussion of the interaction between a pair of oblique waves with the same streamwise wavenumber and frequency, but with equal and opposite spanwise wavenumbers, representing a disturbance with a fixed spanwise disturbance pattern growing in the streamwise direction.
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Instability and chaos in acoustically driven flows (A)

K. Chandra, V. Mehta, A. Mulpur, and C. Thompson

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S54-S54 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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A number of inquiries into the mechanisms responsible for the instabilities observed in time‐periodic fluid motion have been made. However, only recently has it been theoretically shown that a fluid, driven by a time‐periodic acoustic wave, can bifurcate from stability at a fixed acoustic particle velocity amplitude. This departure from stability is the result of the action of the gradient of the Reynolds stress. Hence, the parametric interaction of acoustically driven vorticity with external disturbances is important. This basic result will serve as the foundation material. Sensitivity of the acoustic instability mechanism to temporal and spatial modulations generated by harmonics resulting from shock formation will be examined. A theoretical model will be presented to describe the nonlinear evolution of unstable disturbances, such as burst‐associated, Reynolds stress pulses near a rigid boundary wall. The relationship between unstable vortical disturbances and self‐modulation to experimentally observed low‐frequency noise will be discussed. Transient conditions for instability will be addressed.
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Using animation to visualize global stability properties of nonlinear systems (A)

J. P. Cusumano

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S54-S54 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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A salient feature of nonlinear systems is that several distinct steady‐state solutions can coexist: Different initial conditions can lead asymptotically to different attractors. The way in which parameter variations affect the geometry of the basins of attraction is central to an understanding of stability transition phenomena, including the onset of chaotic vibrations. A computer program is described that allows steady‐state solutions and their basins of attraction to be rapidly obtained and visualized for given parameter values. The code was implemented on a vector‐parallel architecture supercomputer. The results of analyses carried out on several mechanical systems are presented, including movies of basin‐boundary evolution as the parameters are varied over a curve in the parameter space. Of particular interest is the way in which animation reveals phenomena not readily seen in still images, such as rapid transitions between smooth and fractal basin boundaries, or the rapid creation and annihilation of entire basins of attraction.
Contributed Papers
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A finite element solution of the inverse scattering problem for determining acoustic bubble spectra (A)

Kerry W. Commander and Robert J. McDonald

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S54-S55 (1990); (2 pages)

Online Publication Date: 13 Aug 2005

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The determination of acoustic bubble spectra in the ocean from multi‐frequency attenuation measurements was improved by numerical solution of the inverse scattering problem [K. W. Commander and E. Mortiz, in Proceedings of IEEE Oceans '89, 1181–1185 (1989)]. Although this numerical solution offers an improvement over the resonance approximation (at small radii), it too suffers a loss of accuracy at the small radii end of the bubble spectrum. The numerical solution of this problem has been improved by replacing the Fourier series approximation to the unknown distribution by a piecewise linear polynomial constructed from a series of basis functions. The basis functions used were linear B‐splines or “hat” functions. The resulting coefficient matrices are much less ill‐conditioned than those from the previous method because of the local support of the basis functions. This improvement leads to a more accurate solution of the inverse scattering problem near the end points. Improvements in accuracy due to the finite element solution are quantified using calculated attenuations from several bubble distributions of interest. [Work supported by ONT.]
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When bubbles go bad…. Chaotic shape oscillations of air bubbles in water (A)

G. Holt, J. Holzfuss, A. Judt, and A. Phillip

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S55-S55 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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Using optical scattering techniques, the nonlinear oscillations of single gas bubbles are observed. A single bubble is acoustically levitated and driven by a standing‐wave pressure field at kilohertz frequencies. The scattered light intensity from a single plane wave (TEM00) is observed as a function of time for both radial and nonradial oscillations. The resulting time series are analyzed using some techniques from dynamical systems theory. These measurements are accompanied by high‐speed photographs in an attempt to determine the presence and character of shape oscillation modes. [Work supported by ONR, NCPA, and DFG.]
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An experimental search for classical second sound in a system of nonlinear random waves (A)

A. Larraza, R. K. Yarber, S. L. Garrett, and Seth J. Putterman

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S55-S55 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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A nonlinear medium driven far from equilibrium by the input of energy in the form of mechanical wave motion becomes wave turbulent. The steady state corresponds to a power spectrum with a low‐frequency dependence that is characteristic of the particular system [A. Larraza and S. J. Putterman, in Irreversible Phenomena and Dynamical Systems Analysis in Geosciences (Reidel, Dordrecht, The Netherlands, 1987), p. 139]. It has been shown [A. Larraza and S. J. Putterman, Phys. Rev. Lett. 57, 2810 (1986)] that the wave turbulence can be elastic, so that its energy fluctuations, instead of diffusing, propagate as waves, in analogy to (thermal) second sound in superfluid helium. Experiments in a wind‐driven wave tank will be described that are designed to measure the theoretical predicted second sound mode for a random field of short, deep‐water gravity waves [Larraza et al., Phys. Rev. A (March 1990)], by use of dual‐channel capacitive wave staff and a “probe wave” generated by a hydraulically actuated plunger. [Work supported by the NPS Direct Funded Research Program.]
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Sound ray chaos induced by oceanic internal waves and mesoscale structure (A)

Michael G. Brown, Kevin B. Smith, and Frederick D. Tappert

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S55-S55 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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Ray dynamics in models of the ocean sound channel with periodic range dependence was investigated previously. Some rays were shown to exhibit chaotic behavior, i.e., exponential sensitivity to initial conditions. In the present study, ray dynamics in realistically range‐dependent ocean models is considered. Both internal wave and mesoscale structure are considered. The background and perturbation sound‐speed fields that are used are prescribed analytically and are derivable from a Brunt‐Vaisala frequency profile that decays exponentially in depth. A Garett‐Munk internal wave spectrum is simulated using 64 horizontal wavenumbers and 25 vertical internal wave modes. Mesoscale structure is simulated using four vertical modes of the linearized quasigeostrophic potential vorticity equation with varying frequencies and horizontal wavenumbers. In all calculations, the sound‐speed field is assumed to be frozen during the passage of the acoustic waves. The ocean surface and bottom are assumed to be flat. Numerical results, consisting of estimates of both Lyapunov exponents and power spectra, suggest that ray trajectories are predominantly chaotic. Internal wave and mesoscale structure are considered separately and together.
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A search for wave chaos in a simple range‐dependent underwater acoustic waveguide (A)

Frederick D. Tappert, Gustavo J. Goni, and Michael G. Brown

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S55-S55 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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Ray dynamics in a number of simple range‐dependent underwater acoustic waveguides was examined previously. It has been shown that, in such environments, at least some ray trajectories exhibit chaotic behavior, i.e., exponential sensitivity to initial conditions. This phenomenon is called ray chaos. In the present study, properties of the solution to the parabolic wave equation are examined in a bottom interacting shallow water environment in which ray trajectories are known to be predominantly chaotic. An attempt is made to determine whether the exponential sensitivity associated with ray chaos carries over to finite frequency wave fields. This phenomenon, whose existence is in question, is called wave chaos. In the search for wave chaos, 2‐, 4‐, 8‐, and 16‐kHz wave fields have been used to investigate the spreads in angle and depth of an initially narrow sound beam (these spreads grow exponentially in range under chaotic conditions according to ray theory if the initial beam is sufficiently narrow); the feasibility of back‐propagating sound fields (at ranges beyond some “predictability horizon” chaotic ray trajectories cannot be traced backwards to recover their initial conditions); and several measures of wave field complexity versus range (the complexity of a chaotic geometric wave field grows exponentially in range). In no case was exponential sensitivity, or any associated lack of predictability, observed in the finite frequency wave fields. In other words, no evidence that wave chaos exists was found.
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Bifurcation phenomena in nonlinear clarinetlike systems (A)

Klaus Brod

J. Acoust. Soc. Am. Volume 87, Issue S1, pp. S55-S55 (1990); (1 page)

Online Publication Date: 13 Aug 2005

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The physical processes leading to sound generation in reed instruments can be represented as an interaction between the linear sound propagation in the tube and the strongly nonlinear dependence between flow rate and pressure in the mouthpiece. Considering the model equations of this system as a function of the fraction ϵ = E/E0, E0 being the energy introduced into the tube and E being the energy reflected at the end of the tube, one finds bifurcational behavior for the sound pressure in the passive range (∣ϵ∣ ⩽ 1). In the active range (∣ϵ∣ > 1) the bifurcations continue, leading eventually to chaos. This behavior is also dependent on the shape of the reflection function; for very narrow ones the model equations simplify to a one‐dimensional iterated map that can be treated rather easily. The types of bifurcations occurring here as well as typical quantities (Lyapunov exponents, etc.) will be analyzed; similarities and differences between this model and experiments carried out will be discussed.
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