Propagation of acoustic waves in disordered flows composed of many vortices. II. Examples

摘要

The theory of acoustic wave propagation through systems of may vortices randomly distributed, developed in Part I, is applied to specific examples in two and three dimensions. Two classes of vortex blobs are considered; vortices with an axisymmetric distribution of vorticity, such as disks or tubes, and vortices with a nonvanishing dipolar moment such as dipoles or rings. The index of refraction and attenuation length are numerically computed as a function of wavelength for various values of vortex parameters. The asymptotic behavior of the dispersion relation for very short and very long wavelengths is also derived analytically. At short wavelengths #lambda# the attenuation length scales as #lambda#~(-2) in all examples studied. At long wavelengths the scaling depends on the lowest nonvanishing multipole moment of the vorticity distribution; say, for vortex rings, it is #lambda#~(-4) as in Thomson scattering. For an ideal gas, the phase velocity of the coherent acoustic wave is greater than in the undisturbed flow for long wavelengths and smaller than in the undisturbed flow for short wavelengths. This appears to be a robust feature. When properly normalized, the attenuation length does not depend very strongly on the ratio l/#epsilon#, where l is a vortex length scale and #epsilon# the thickness of the vorticity bearing region, both in two and three dimensions. The effective index of refraction, however, does depend on this ratio. The conditions of applicability of the results, which rely on a Born approximation scheme, are also determined. The expressions obtained in this paper for the scattering cross sections are used to discuss the properties of sound localization in two dimensional disordered flows.