Generalized Pekeris–Airy Waveguide and Its Properties

In this study, a generalization of the classical Pekeris-type shallow-water waveguide is investigated. This generalization is obtained by assuming that the sound speed in the bottom halfspace is increasing with the depth in such a way that the refractive index is a piecewise-linear function of the vertical coordinate. Arguably, this model is not only more realistic for many applications of underwater acoustics but also significantly more convenient, from the mathematical point of view, than that of Pekeris. It is proved that such waveguide admits only a countable set of modes. An elementary proof of the fact that the spectrum of the respective Sturm–Liouville problem for such waveguide is purely discrete is given. A generalization of this theorem to the case of a multi-layered waveguide in which each layer is an Airy medium with different parameters is also considered. The proof of the latter result is based on the methods of the spectral theory.