In this work we construct and discuss special solutions of a homogeneous problem for
the Laplace equation in a domain with cone-shaped boundaries. The problem at hand is
interpreted as that describing oscillatory linear wave movement of a fluid under gravity in
such a domain. These solutions are found in terms of the Mellin transform and by means
of the reduction to some new functional-difference equations solved in an explicit form
(by quadrature). The behavior of the solutions at large distances is studied by use of the
saddle point technique. The corresponding eigenoscillations of a fluid are then interpreted
as generalized eigenfunctions of the continuous spectrum.