Joseph F. Lingevitch

Quasi-steady monopole and tripole attractors for relaxing vortices

Using fully nonlinear simulations of the two-dimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasi-steady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasi-steady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing.

A wide angle and high Mach number parabolic equation

Various parabolic equations for advected acoustic waves have been derived based on the assumptions of small Mach number and narrow propagation angles, which are of limited validity in atmospheric acoustics. A parabolic equation solution that does not require these assumptions is derived in the weak shear limit, which is appropriate for frequencies of about 0.1 Hz and above for atmospheric acoustics. When the variables are scaled appropriately in this limit, terms involving derivatives of the sound speed, density, and wind speed are small but can have significant cumulative effects. To obtain a solution that is valid at large distances from the source, it is necessary to account for linear terms in the first derivatives of these quantities [A. D. Pierce, J. Acoust. Soc. Am. 87, 2292-2299 (1990)]. This approach is used to obtain a scalar wave equation for advected waves. Since this equation contains two depth operators that do not commute with each other, it does not readily factor into outgoing and incoming solutions. An approximate factorization is obtained that is correct to first order in the commutator of the depth operators.

Parabolic equations for gravity and acousto-gravity waves

Parabolic equations for gravity and acousto-gravity waves are derived and implemented. The wave equations for these problems contain singularities at depths at which the buoyancy frequency equals the forcing frequency. One of the advantages of the parabolic equation solution is that it is easy to avoid numerical problems associated with the singularities. Some problems involve an infinite number of propagating modes. This artifact of neglecting viscosity is handled by including stability constraints in the rational approximations used in the implementation of the parabolic equation. The parabolic equation is tested for idealized problems involving surface, internal, and interface gravity waves. Parabolic equation solutions are also presented for range-dependent problems involving internal waves in the ocean and acousto-gravity waves in the atmosphere.

Wave propagation in range-dependent poro-acoustic waveguides

The parabolic equation method is extended to handle range-dependent poro-acoustic waveguides. A poro-acoustic medium is the limiting case of a poro-elastic medium in which the shear wave speed vanishes. Recent experiments indicate that this is a relevant limit [Chotiros, “Biot model of sound propagation in water-saturated sand,” J. Acoust. Soc. Am. 97, 199–214 (1995)]. Energy-conserving and single-scattering techniques are developed for handling vertical interfaces. The single-scattering solution is extended to problems involving fluid layers above poro-acoustic sediments. Improved rational function approximations are developed by rotating the branch cut [Milinazzo et al., “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997)].

Demonstration at sea of the decomposition-of-the-time-reversal-operator techniquea)

This paper presents a derivation of the time reversal operator decomposition (DORT) using the sonar equation. DORT is inherently a frequency-domain technique, but the derivation is shown in the time-frequency domain to preserve range resolution. The magnitude of the singular values is related to sonar equation parameters. The time spreading of the time-domain back-propagation image is also related to the sonar equation. Noise-free, noise-only, and signal-plus-noise data are considered theoretically. Contamination of the echo singular component by noise is shown quantitatively to be very small at a signal-to-noise ratio of 0dB. Results are shown from the TREX-04 experiment during April 22 to May 4, 2004 in 94m deep, shallow water southwest of the Hudson Canyon. Rapid transmission of short, 500Hz wide linear frequency modulated beams with center frequencies of 750, 1250, 1750, 2250, 2750, and 3250Hz are used. Degradation caused by a lack of time invariance is found to be small at 750Hz and nearly complete a...

A two-way parabolic equation that accounts for multiple scattering.

A two-way parabolic equation that accounts for multiple scattering is derived and tested. A range-dependent medium is divided into a sequence of range-independent regions. The field is decomposed into outgoing and incoming fields in each region. The conditions between vertical interfaces are implemented using rational approximations for the square root of an operator. Rational approximations are also used to relate fields between neighboring interfaces. An iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces. The approach is useful for solving problems involving scattering from waveguide features and compact objects.

Geoacoustic inversion using a rotated coordinate system and simulated annealing

An efficient method for geoacoustic inversions in a range-dependent ocean waveguide is implemented and tested with synthetic data. This method combines a simulated annealing search with an optimal coordinate rotation that increases the efficiency of navigating parameter landscapes for which parameter coupling is important. The coordinate rotation associated with the parameter couplings also provides information about which parameters are resolvable for a particular inversion frequency and array geometry. Using this information, results from several single-frequency inversions can be combined to obtain an estimate for the sediment parameters.

Parabolic Equation Simulations of Reverberation Statistics From Non-Gaussian-Distributed Bottom Roughness

In this paper, a two-way parabolic equation (PE) method is developed for modeling rough interface reverberation. The model is employed to estimate the reverberation envelope probability density function from bottom roughness with Gaussian and exponential height distributions. For Gaussian-distributed roughness, the PE gives envelope statistics that closely conform to the expected Rayleigh distribution. However, for non-Gaussian-distributed roughness heights, heavy-tailed reverberation envelope statistics are observed. The PE simulation results are compared to the analytical model of K-distributed reverberation by Abraham and Lyons [IEEE J. Ocean. Eng., vol. 29, pp. 800-813, 2002] for discrete scatterers and to numerical predictions of the first and second moments of the reverberation intensity estimated with a coupled mode reverberation model for the multipath insonification of rough surfaces.

WAVE PROPAGATION IN PORO-ACOUSTIC MEDIA

Abstract Some ocean sediments may be modeled as poro-elastic media with relatively high slow-wave speeds and relatively low shear-wave speeds [N.P. Chotiros, “Biot model of sound propagation in water-saturated sand”, J. Acoust. Soc. Amer. 97 , 199–214 (1995)]. This singular limit may be handled efficiently by allowing the shear modulus to vanish so that shear waves are ignored. This approach reduces the number of equations and permits a relatively coarse numerical grid. The equations of poro-acoustic media are remarkably similar to the equations of acoustic media. The equations of motion are a vector generalization of the variable density wave equation of acoustics [P.G. Bergmann, “The wave equation in a medium with a variable index of refraction”, J. Acoust. Soc. Amer. 17 , 329–333 (1946)]. The interface conditions resemble the acoustic conditions for continuity of pressure and particle velocity. The energy-flux integrals of poro-acoustics and acoustics are also similar.

Two-way parabolic equation techniques for diffraction and scattering problems

Abstract Two-way parabolic equation techniques are based on replacing normal derivatives with one-way operators on scattering surfaces and iterating for the solution. This approach is extended to diffraction and scattering problems involving perfectly reflecting boundaries. Accuracy is demonstrated for the Sommerfeld diffraction problem. Two-way parabolic equation solutions are also presented for problems involving a diffraction grating, scattering in a waveguide, and diffraction of surface gravity waves by a seawall.