Abstract
As we have often pointed out, when considering the propagation of sound in a shallow water waveguide, it is of primary importance to take into account interaction of the sound with the bottom. The hardest quantity to quantify in this interaction is absorption in the bottom, which leads to a fall in the signal intensity with distance. Formally, this decay is due to the complex eigenvalues in the Sturm–Liouville problem (3.7), or equivalently the constants of propagation \( {\xi_l} \) having an imaginary part \( {\gamma_l} = 2{\rm Im} {\xi_l} \) (i.e., the attenuation coefficients of the normal modes). Despite the fact that the values of \( {\gamma_l} \) are easily obtained numerically, though not experimentally, a more detailed study of their analytic behavior is advisable, particularly as a function of the parameters of the problem. We should first look at the attenuation as a function of mode number \( l \), since this dependence is crucial to explain the structure of the field formed at a large distance from the source, including its modal composition and total intensity. Moreover, in performing approximate calculations of the averaged intensity, simplified analytic formulae for \( {\gamma_l} \) are required. These simplified formulae can be obtained and interpreted in the framework of ray (or WKB) theory.
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Notes
- 1.
We used a liquid bottom here for simplicity. In the following, we will show that the expression for actual shallow water conditions can be reduced to the liquid bottom case by redefinition of the bottom parameters.
- 2.
- 3.
The ray approximation corresponds to comparatively high frequency.
- 4.
We note that the chirp parameters are not chosen arbitrarily in this example. The carrier frequency matches the “optimum frequency” or sound propagation in shallow water, and the frequency band corresponds to the frequency range of electromagnetic-type sound sources which are used for shallow water acoustic tomography (see Chap. 10, Table 10.1).
- 5.
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Katsnelson, B., Petnikov, V., Lynch, J. (2012). Examples Illustrating the Characteristics of Waveguide Propagation. In: Fundamentals of Shallow Water Acoustics. The Underwater Acoustics Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9777-7_4
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