It is indeed close to the Mathieu equation. The intent is to demonstrate how a wave equation when modified can produce a frequency modulation. The mechanism behind sloshing is that not only can the forcing function generate the necessary energy at the frequency to trigger a resonance, but that it can also perturb the parameters that define the resonance condition. This has the effect of modifying the resonance condition both subtly or strongly depending on the compliance of the fluid volume. That is the underlying physics basis for the application of the Mathieu equation.

One would think that it would take a massive forcing to get this effetc in motion. But in fact when one is dealing with sloshing with respect to the thermocline, then the small differences in density of water above and below the thermocline become very sensitive to momentum transfer, as the gravitational differences are small between layers of slightly differing density. See this paper:

Valentine, Daniel T., and Jannette B. Frandsen. ["Nonlinear Free-Surface and Viscous-Internal Sloshing."]( Journal of Offshore Mechanics and Arctic Engineering 127.2 (2005): 141-149.