D. Keith Wilson

Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation.

Finite-difference, time-domain (FDTD) calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium (with an emphasis on the atmosphere) are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere.

Relaxation-matched modeling of propagation through porous media, including fractal pore structure

The viscous and thermal dissipation of an acoustic wave propagating through a porous medium is shown to be characteristic of a relaxation process. Based on this interpretation, a new model for the complex density (dynamic permeability) and bulk modulus, which describe the viscous and thermal processes, respectively, is proposed. The model is based on approximating the relaxational characteristic, as opposed to previous models based on matching low‐ and high‐frequency asymptotic behavior. An advantage of the relaxation model is that one fewer parameter is required. The relaxation model is also simpler than the commonly used Zwikker/Kosten/Attenborough or Biot/Allard models, in the sense that it contains no Bessel or Kelvin functions, and is not a modified form of the solution for uniform, circular pores. And unlike the Delany/Bazley empirical equations, the relaxation model is physically realistic for all frequencies. Extension of the relaxation model to fractal pore surfaces is also discussed.

Performance bounds for acoustic direction-of-arrival arrays operating in atmospheric turbulence

A method is described for assessing the precision of wavefront direction-of-arrival (DOA) estimates made by acoustic arrays. Arrays operating in turbulent atmospheric boundary layers are considered. The method involves calculating the Cramer–Rao lower bound, which describes the best obtainable DOA precision (performance) for a given array and environment. For simplicity, it is assumed that the source is monochromatic, and multipath effects are not considered. The predicted performance bounds are found to degrade with increasing propagation distance, increasing frequency, and increasing turbulence strength. Performance predictions using several different three-dimensional turbulence models are compared: isotropic and anisotropic Gaussian models, the von Karman model, and the Kolmogorov model, with the last in both intermittent and nonintermittent forms. When array performance is limited by turbulence (as opposed to background noise), the turbulence model strongly affects the calculated performance bounds. ...

An Alternative Function For The Wind And Temperature Gradients In Unstable Surface Layers

The function φ(ζ)=(1+γ|z/L|2/3)1/2,where z is the height, L the Obukhov length, and γ a constant,is proposed for the nondimensional wind speed and temperaturegradients (flux-profile relationships) in anunstable surface layer. This function agrees quite well withboth wind speed and temperature data,has the theoretically correct behaviour in convective conditions,and leads to simple results when integrated to produce the mean profiles.

A turbulence spectral model for sound propagation in the atmosphere that incorporates shear and buoyancy forcings

A three-dimensional model for turbulent velocity fluctuations in the atmospheric boundary layer is developed and used to calculate scattering of sound. The model, which is based on von Karmans spectrum, incorporates separate contributions from shear- and buoyancy-forced turbulence. New equations are derived from the model that predict the strength and diffraction parameters for scattering of sound as a function of height from the ground and atmospheric conditions. The need is demonstrated for retaining two distinct scattering length scales, one associated with scattering strength and the other with diffraction. These length scales are height dependent and vary substantially with the relative proportions of shear and buoyancy forcing. The turbulence model predicts that for forward-scattered waves the phase variance is much larger than the log-amplitude variance, a behavior borne out by experimental data. A new method for synthesizing random fields, based on empirical orthogonal functions, is developed to accommodate the height dependence of the turbulence model. The method is applied to numerical calculations of scattering into an acoustic shadow zone, yielding good agreement with previous measurements.

Acoustic propagation through anisotropic, surface‐layer turbulence

The effects of atmospheric wind and temperature fluctuations on acoustic signal variability is discussed, with emphasis on the effects of large‐scale turbulence (motions having size larger than or comparable to the integral length scale). Such large‐scale turbulence is anisotropic, is generated by both shear and buoyancy instabilities, and has structure that depends strongly on the meteorological conditions as well as the distance from the ground. Previous research in the atmospheric sciences literature regarding length scales and anisotropy is reviewed and incorporated into an acoustic propagation model. The model is based on a multiply scaled, six‐termed, sound‐speed correlation function. A second, simpler model, based on fluctuating curvature of the vertical wind profile, is also proposed. Both models are compared with experimental measurements of amplitude and travel‐time fluctuations obtained during the Rock Springs Tomography Experiment, which involved concurrent monitoring of acoustic fluctuations ...

ACOUSTIC SCATTERING AND THE SPECTRUM OF ATMOSPHERIC TURBULENCE

Some issues regarding atmospheric turbulence modeling and its role in acoustic scattering calculations are discussed. Discrepancies between turbulence spectral models appearing in the acoustical and in the atmospheric sciences literature are noted, and it is argued that these discrepancies can be understood by recognizing that the acoustic wavelength and scattering geometry combine to act as an “acoustic filter” which selects a specific part of the turbulence spectrum. A particular model spectrum can yield satisfactory acoustic scattering predictions if it fits the actual spectrum well at the acoustically filtered turbulence scales, even if the model is a poor overall representation of the turbulence spectrum. Proper interpretation of length scales determined by fitting two-point correlation functions, and the importance of averaging times in estimating variances, are also discussed in relation to the action of the acoustic filter.

Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere

Stochastic inversion is a well known technique for the solution of inverse problems in tomography. It employs the idea that the propagation medium may be represented as random with a known spatial covariance function. In this paper, a generalization of the stochastic inverse for acoustic travel-time tomography of the atmosphere is developed. The atmospheric inhomogeneities are considered to be random, not only in space but also in time. This allows one to incorporate tomographic data (travel times) obtained at different times to estimate the state of the propagation medium at any given time, by using spatial-temporal covariance functions of atmospheric turbulence. This increases the amount of data without increasing the number of sources and∕or receivers. A numerical simulation for two-dimensional travel-time acoustic tomography of the atmosphere is performed in which travel times between sources to receivers are calculated, given the temperature and wind velocity fields. These travel times are used as da...

Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion.

Acoustic travel-time tomography allows one to reconstruct temperature and wind velocity fields in the atmosphere. In a recently published paper [S. Vecherin et al., J. Acoust. Soc. Am. 119, 2579 (2006)], a time-dependent stochastic inversion (TDSI) was developed for the reconstruction of these fields from travel times of sound propagation between sources and receivers in a tomography array. TDSI accounts for the correlation of temperature and wind velocity fluctuations both in space and time and therefore yields more accurate reconstruction of these fields in comparison with algebraic techniques and regular stochastic inversion. To use TDSI, one needs to estimate spatial-temporal covariance functions of temperature and wind velocity fluctuations. In this paper, these spatial-temporal covariance functions are derived for locally frozen turbulence which is a more general concept than a widely used hypothesis of frozen turbulence. The developed theory is applied to reconstruction of temperature and wind velocity fields in the acoustic tomography experiment carried out by University of Leipzig, Germany. The reconstructed temperature and velocity fields are presented and errors in reconstruction of these fields are studied.

Time-domain equations for sound propagation in rigid-frame porous media (L)

A general set of time-domain equations describing linear sound propagation in a rigid-frame, gas-saturated porous medium is derived. The equations, which are valid for all frequencies, are based on a relaxational model for the viscous and thermal diffusion processes occuring in the pores. The dissipative terms in the equations involve convolutions of the acoustic fields with the impulse response of the medium. It is shown that the equations reduce to previously known results in the limits of low and high frequencies. Alternative time-domain equations are also derived based on a Pade approximation.