The paper deals with the formal short-wavelength asymptotic solutions describing the
acoustic eigenoscillations in a vessel having a hard bottom, filled in by an acoustic medium, and
covered by a thin elastic membrane. The solutions are localized in the medium near the line of the
rigid contact of the membrane covering the vessel with the edge of the vessel. The coefficients in the
asymptotic expansion of the solutions satisfy a recurrent sequence of solvable problems, whereas the
frequencies, for which such nontrivial formal solutions exist, obey an asymptotic ‘quantization-type
condition.