Abstract
Up to this point now we have considered sound propagation in shallow water as being through a deterministic, inhomogeneous medium. It is well known that these inhomogeneities can have a considerable effect on the propagation of sound. Moreover, we know from oceanography that these random inhomogeneities are primarily concentrated in the top kilometer of the water column (e.g., Babii 1983; Shavcho 1982), which leads to a particularly strong effect of random inhomogeneities in shallow water. Recently, great attention has been paid, both experimentally (Zhou et al. 1996; Headrick et al. 2000) and theoretically (Dozier and Tappert 1978; Gorskaya and Raevskii 1984; Ashley et al. 1987; Kuznetsova 1988; Derevyagina and Katsnel’son 1995; Creamer 1996; Beran and Frankenthal 1992, 1996; Frankenthal 1998), to the study of shallow water as a randomly inhomogeneous medium. Also, the propagation of waves of different kinds in a randomly inhomogeneous medium has been studied for a long time (beginning with Rayleigh) [see, for example, the monographs (Chernov 1958; Rytov et al. 1989; Ishimaru 1978; Flatte 1979)], etc. The major theme of this chapter will thus be the transition from the complex, detailed oceanography (see Chap. 2) to appropriate random medium representations, and then the description of propagation and scattering in these representations.
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Notes
- 1.
We remark that the coefficient Q depends on water properties (temperature, salinity etc.) and for another depth and region can be different (see Flatte et al. 1979).
- 2.
In some literature, there is another definition \( {\hbox{\it SI}} = {{{\left( {\left\langle {{I^2}} \right\rangle - {{\left\langle I \right\rangle }^2}} \right)}} \left/ {{{{\left\langle I \right\rangle }^2}}} \right.} \).
- 3.
Stochastic boundary conditions lead to analogous equations for mode interactions and so will not be considered separately in what follows.
- 4.
In measurements, i.e., in work with real CW signals, the amplitudes are determined somewhat differently: \( \left| {P(r,z,t)} \right| = \sqrt {{p_{{\cos }}^2(r,z,t) + p_{{\sin }}^2(r,z,t)}} \) where \( {p_{{\sin }}}(r,z,t) = \overline {p(r,z,t)*\sin \omega t} \) and \( {p_{{\cos }}}(r,z,t) = \overline {p(r,z,t)*\cos \omega t} \) are the sine and cosine quadrature components of the real (measured) field \( p(r,z,t) \), related to the field \( P(r,z,t) \) by the equation \( p(r,z,t) = {\hbox{Re}}\left( {P(r,z,t)} \right) \). The overline denotes averaging over the period. Simple transformations show that these two definitions are equivalent.
- 5.
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Katsnelson, B., Petnikov, V., Lynch, J. (2012). Sound Field in Shallow Water with Random Inhomogeneities. In: Fundamentals of Shallow Water Acoustics. The Underwater Acoustics Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9777-7_5
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