On the analytical properties of solutions of the dispersion equation of the Airy medium

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.