Geodesic
On the analytical properties of solutions of the dispersion equation of the Airy medium
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 101-113

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A problem of wave propagation near the interface between an isospeed water layer overlying a halfspace with a sound speed gradient, known as an Airy medium, characterized by a linear variation of the squared refractive index with depth is considered. A dispersion relation is derived, which is a transcendental equation containing Airy functions. For certain values of the problem parameters, it is proved that the dispersion equation has a countable set of solutions (horizontal wave numbers of normal modes). Asymptotic solutions of the dispersion equation for a geoacoustic shallow water waveguide, consisting of an unbounded homogeneous water layer and a bottom Airy halfspace, have been constructed and analyzed. 
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     title = {On the analytical properties of solutions of the dispersion equation of the {Airy} medium},
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G. L. Zavorokhin; A. A. Matskovskii. On the analytical properties of solutions of the dispersion equation of the Airy medium. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 101-113. http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a5/