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Valerian Tatarskii and Acoustic Wave Propagation in Random Media
Vladimir E. Ostashev, D. Keith Wilson, and John A. Colosi
Stars twinkling in the night sky due to atmospheric tur- bulence, dancing webs of bright sunlight on the bottom of a swimming pool with a wavy surface, and a handclap echoing through trees in a forest all exemplify the field of wave propagation in random media (WPRM). WPRM has applications throughout science and technology, including in seismology, optics, radio communication, medicine, global positioning system (GPS) navigation, and astronomy. Examples of WPRM furthermore abound in acoustics. Sound is randomly scattered by internal-wave fields in the ocean and upper atmosphere, turbulence in the lower atmosphere, buildings in an urban environment, fish schools, and biological organs and tissue.
Applications of WPRM generally fall into two categories. The first category is where a known signal is randomized or degraded by the medium, leading to a loss of otherwise recoverable signal information. An example application is underwater communication and navigation where random medium effects limit data rates and localization
precision of sources and receivers. In the second category, the signal randomization itself is of fundamental inter- est because it provides information about the random medium, for example, the size and density of a fish school or the strength of turbulence in the ocean or atmosphere.
In either case, the common objective of WPRM is to relate statistical characteristics of a sound signal (e.g., the variances of the amplitude and phase fluctuations) to random medium statistics and propagation parameters such as distance and signal frequency. This relationship is generally rather complex because sound signals are scattered and diffracted by many random inhomogeneities with different sizes.
Atmospheric turbulence consists of interacting eddies spanning six orders of magnitude, from 1 mm to about
1 km. In 1941, Kolmogorov formulated the famous −5/3 power law describing the middle part of this spectrum that is characterized by decay of large eddies into smaller ones.
The Kolmogorov spectrum set the stage for studies of acoustic and electromagnetic wave propagation through atmospheric turbulence, which was the origin of modern
WPRM. Using this spectrum and geometrical acous- tics, Krasilnikov (1945) calculated the variances of the amplitude and phase fluctuations of acoustic signals and compared the results with experimental data. Later, Obukhov (1953) realized that diffraction impacts these variances and adopted an approximation developed by Rytov (1937), originally for light diffraction by ultra- sound, to correct the calculations.
Despite these accomplishments and other studies, the field of WPRM did not reach maturity until the publica- tion of two monographs by Tatarskii (1959, 1967). The monographs clearly and rigorously presented the modern physical understanding of atmospheric turbulence spectra and showed how various statistical characteristics of electro- magnetic and acoustic waves can be expressed as functions of those spectra. These books were translated into English and ushered in several decades of highly productive, global, interdisciplinary WPRM research. As Wheelon (2001, 2003) stated in his own books on WPRM, “These volumes are dedicated to Valerian Tatarskii who taught us all.”
The main purposes of this article are to highlight Tatarskii’s role in the field of WPRM and to provide a summary of recent research on sound propagation through atmospheric turbulence and ocean internal waves that builds on his legacy.
Tatarskii as a Scientist and Mentor
Valerian Tatarskii’s (1929–2020) scientific career started at a fortuitous place and time. Vladimir Krasilnikov was
38 Acoustics Today • Winter 2021 | Volume 17, issue 4

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